{
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        "result": "Binomial expansion",
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        "result": "240",
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        "result": "2^n",
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        "status": "passed"
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      "grading": {
        "mode": "written",
        "sourceKeyPreserved": true,
        "automaticScore": false
      }
    },
    {
      "id": "3.1",
      "contentHash": "35a9fb28d00737f26e9d1b654af070420e199267c0708cc7025a8a23ef7a1e61",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          64,
          65,
          66
        ],
        "answerPdfPages": [],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "35a9fb28d00737f26e9d1b654af070420e199267c0708cc7025a8a23ef7a1e61",
        "method": "Construct labeled finite sets A=0..19 and B=9..33. Their union independently contains 34 labels, with overlap 11."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.2",
      "contentHash": "7ea44292231dd9d742fda4f5673e999313c2bae55d8b4194506bee82c88dd568",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          64,
          67
        ],
        "answerPdfPages": [],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "7ea44292231dd9d742fda4f5673e999313c2bae55d8b4194506bee82c88dd568",
        "method": "Independent arbitrary-element multiplicity audit: membership (0,0),(1,0),(0,1),(1,1) gives union weights 0,1,1,1 after a+b−ab. Finiteness explicit."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.3",
      "contentHash": "5563fed65ef9322e36c7c9718b1b3ddd0954e5f666f5d57f9424c145d3afe28b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          65,
          67
        ],
        "answerPdfPages": [],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "5563fed65ef9322e36c7c9718b1b3ddd0954e5f666f5d57f9424c145d3afe28b",
        "method": "Independent disjoint region construction: 17+11+12=40, with Spanish 17+11=28 and French 12+11=23; the linear constraints uniquely determine overlap 11."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.4",
      "contentHash": "631bf451bb9cea26bcdfae1ed6a98c12a9824b025e805aa483d827448e85c12e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          65,
          68
        ],
        "answerPdfPages": [],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "631bf451bb9cea26bcdfae1ed6a98c12a9824b025e805aa483d827448e85c12e",
        "method": "Independently enumerated every integer 100000..999999 and tested each of the four endpoint predicates. Counts 400000,450000,200000,650000."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.2.1",
      "contentHash": "3318d8af620ec82e3f138f5551ec5f0e985998adca1cfd6c3ce6d69f09999b5b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          68
        ],
        "answerPdfPages": [
          15
        ],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "3318d8af620ec82e3f138f5551ec5f0e985998adca1cfd6c3ce6d69f09999b5b",
        "method": "Disjoint cells: big-only 5, overlap 25, hairy-only 17; these satisfy total 47 and both given marginal counts uniquely."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.2.2",
      "contentHash": "ece93759c70b4b2ef9026652be2bd60ec7d6d19ef22eadf975fc8bd25c60bffe",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          69
        ],
        "answerPdfPages": [
          15
        ],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "ece93759c70b4b2ef9026652be2bd60ec7d6d19ef22eadf975fc8bd25c60bffe",
        "method": "Enumerated integers 512..1023, converted to base two, and directly tested startswith(11) or endswith(11). Exactly 320 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.2.3",
      "contentHash": "ede29be92c2a8559a468f8e06f7221e5a04ad12391861b1d0dd9e0e362cb4572",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          69
        ],
        "answerPdfPages": [
          15
        ],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "ede29be92c2a8559a468f8e06f7221e5a04ad12391861b1d0dd9e0e362cb4572",
        "method": "Independent four-atom feasibility: both=x, DVD-only=80−x, computer-only=70−x, neither=x−50. Nonnegative atoms summing to100 are possible exactly for 50≤x≤70; this constructs every value in the interval."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.2.4",
      "contentHash": "66472d204acc94a5cf744ac6c6f9b71d9555501817c9e69957ca8505de52cf36",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          69
        ],
        "answerPdfPages": [
          15,
          16
        ],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "66472d204acc94a5cf744ac6c6f9b71d9555501817c9e69957ca8505de52cf36",
        "method": "Built exact square and cube value sets by integer powering and directly counted all n=1..99999 outside their union; no floating roots used. Intersection values checked to equal six sixth powers."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.2.5",
      "contentHash": "a0a4ebcbe2d2ab4038732c542999f26d419fd01bcf070f0adbe432c7d9be8e3e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          69
        ],
        "answerPdfPages": [
          16
        ],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "a0a4ebcbe2d2ab4038732c542999f26d419fd01bcf070f0adbe432c7d9be8e3e",
        "method": "Independent atomic table: sons-only15,daughters-only10,both35,neither25; all nonnegative, total85, and marginals50/45."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.2.6",
      "contentHash": "bf351d4aa23b3c44ba210b83f0b00a8adb92d883c0258debe97bc86b81cdbe4b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          69
        ],
        "answerPdfPages": [
          16
        ],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "bf351d4aa23b3c44ba210b83f0b00a8adb92d883c0258debe97bc86b81cdbe4b",
        "method": "Independently enumerated all 90 legal ordered endpoint pairs and counted the 65 whose product is even. Each pair has 10^7 independent middle strings, giving650000000 without using PIE."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.2.7",
      "contentHash": "1be2b62db9831fdf05ee2106e240112111a7bc2e19bcf2bc455ab8504ed1e777",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          69
        ],
        "answerPdfPages": [
          16
        ],
        "method": "Original scans visually inspected; initial multi-part 3.4 retained, not replaced by its shorter repeated box."
      },
      "math": {
        "status": "passed",
        "contentHash": "1be2b62db9831fdf05ee2106e240112111a7bc2e19bcf2bc455ab8504ed1e777",
        "method": "Enumerated every feasible pair of French and Spanish counts0..100, every overlap, and all four nonnegative atomic populations. The unique satisfying table has French72,Spanish36,both12,neither4."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.5",
      "contentHash": "db987aa1e5c58ea549f09da09f8bb26ab9c74584d6792221305a5a3abbb37695",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          69,
          70
        ],
        "answerPdfPages": [],
        "method": "Scans inspected; lost language-cell8 restored; alternating coin strings repaired from actual answer scan."
      },
      "math": {
        "status": "passed",
        "contentHash": "db987aa1e5c58ea549f09da09f8bb26ab9c74584d6792221305a5a3abbb37695",
        "method": "Independent seven-atom linear constraints: parameter t gives atoms SF15−t,SC12−t,FC16−t,S-only2+t,F-only3+t,C-only5+t,ABCt; total53+t=57 forces t4, yielding all nonnegative cells6,7,9,11,8,12,4."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.6",
      "contentHash": "5752cc84fc7447bc0d6481fb001419162d66828709a9349cf986e16756bc260b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          69,
          70,
          71
        ],
        "answerPdfPages": [],
        "method": "Scans inspected; lost language-cell8 restored; alternating coin strings repaired from actual answer scan."
      },
      "math": {
        "status": "passed",
        "contentHash": "5752cc84fc7447bc0d6481fb001419162d66828709a9349cf986e16756bc260b",
        "method": "Independent exhaustive Boolean membership audit over all8 assignments: a+b+c−ab−ac−bc+abc equals indicator(a OR b OR c). This pointwise proof sums over any finite universe."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.7",
      "contentHash": "99c5626f69085f7eb11de042475840036690293cc1abdddb32c025ce5c45a731",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          69,
          71,
          72
        ],
        "answerPdfPages": [],
        "method": "Scans inspected; lost language-cell8 restored; alternating coin strings repaired from actual answer scan."
      },
      "math": {
        "status": "passed",
        "contentHash": "99c5626f69085f7eb11de042475840036690293cc1abdddb32c025ce5c45a731",
        "method": "Direct test of all999 integers against divisibility by2,3,5 yields266; independent residue-class count modulo30 corroborates. Strict endpoint and positive lower bound checked."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.3.1",
      "contentHash": "e3d1415685496ee185beca7f441a950910e1962a7f9c903fef4ff372cf47a68b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          72
        ],
        "answerPdfPages": [
          17
        ],
        "method": "Scans inspected; lost language-cell8 restored; alternating coin strings repaired from actual answer scan.",
        "fidelityAmendment": {
          "textbookPdfPage": 72,
          "answerPdfPage": 17,
          "change": "Restore64 white ribbons from textbook; explicitly retain answer-book62 discrepancy."
        },
        "chapter03FreshFidelity": "All textbook64–97 and manual15–37 visually compared; retained full supplied methods and restored figures/source discrepancy."
      },
      "math": {
        "status": "passed",
        "contentHash": "e3d1415685496ee185beca7f441a950910e1962a7f9c903fef4ff372cf47a68b",
        "method": "Independent disjoint regions: B-only25, G-only4, W-only9, BG-only7, BW-only27, GW-only14, all14 total100 and recover marginals73/39/64. With W-only7 instead, the supplied-key62 version totals98."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.3.2",
      "contentHash": "aa47242c01063fe5133453d7360cd91fe62191a58a556237080d61eea3999217",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          73
        ],
        "answerPdfPages": [
          17
        ],
        "method": "Scans inspected; lost language-cell8 restored; alternating coin strings repaired from actual answer scan."
      },
      "math": {
        "status": "passed",
        "contentHash": "aa47242c01063fe5133453d7360cd91fe62191a58a556237080d61eea3999217",
        "method": "Independently enumerated all26³ length-three strings over a26-letter alphabet and tested whether an A occurs. Exactly1951."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.3.3",
      "contentHash": "2939ae27a7b0838030b54d66219f3f513e8a5d84cc22c187bf6dfd49a04b5aeb",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          73
        ],
        "answerPdfPages": [
          17
        ],
        "method": "Scans inspected; lost language-cell8 restored; alternating coin strings repaired from actual answer scan."
      },
      "math": {
        "status": "passed",
        "contentHash": "2939ae27a7b0838030b54d66219f3f513e8a5d84cc22c187bf6dfd49a04b5aeb",
        "method": "Independent atom feasibility enumeration over triple count0..13: union and all class/pair constraints uniquely give x11; atoms50,6,3,7,6,2,11 sum85 and recover lacrosse22."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.3.4",
      "contentHash": "dcc956c760bd22a5d29b46b091c98e36caf664eeef07f7c5e5b341d839937c1c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          73
        ],
        "answerPdfPages": [
          17,
          18
        ],
        "method": "Scans inspected; lost language-cell8 restored; alternating coin strings repaired from actual answer scan."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "dcc956c760bd22a5d29b46b091c98e36caf664eeef07f7c5e5b341d839937c1c",
        "method": "Enumerated all16 independent fair-coin strings;8 contain TT,8 contain HH,2 contain both,2 contain neither. The supplied key is conditional on fairness and independence."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.3.5",
      "contentHash": "66a868322dee1ad55362610bd60c8f828b4dbca9dadfab23e588e3f529d088a8",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          73
        ],
        "answerPdfPages": [
          18
        ],
        "method": "Scans inspected; lost language-cell8 restored; alternating coin strings repaired from actual answer scan."
      },
      "math": {
        "status": "passed",
        "contentHash": "66a868322dee1ad55362610bd60c8f828b4dbca9dadfab23e588e3f529d088a8",
        "method": "Independent enumeration of symmetric nonnegative pair-only and singleton-only atoms:3a+3b+15=180 and3b=2(3a+15) uniquely yield a15,b40; each ribbon then contains85 dogs."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.3.6",
      "contentHash": "544cfa635c39a0b45300caf0dcf788172396f6f1a7d5c47862ccf7be644c520a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          73
        ],
        "answerPdfPages": [
          18
        ],
        "method": "Scans inspected; lost language-cell8 restored; alternating coin strings repaired from actual answer scan."
      },
      "math": {
        "status": "passed",
        "contentHash": "544cfa635c39a0b45300caf0dcf788172396f6f1a7d5c47862ccf7be644c520a",
        "method": "Direct enumeration of every integer100000..999999 and membership test for characters1,2,3 gives70110, independently of the missing-digit PIE derivation."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.8",
      "contentHash": "b03b09f90568e6a34305f8fac1393ba24ba587dc11890cc54c7dcbfc4e6b9e93",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          73,
          74,
          75
        ],
        "answerPdfPages": [],
        "method": "Original scans visually inspected; lost factorial denominators, malformed binary patterns, extra/missing gender letters and probability denominator825 repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "b03b09f90568e6a34305f8fac1393ba24ba587dc11890cc54c7dcbfc4e6b9e93",
        "method": "Independently tested gcd(n,180)=1 for each positive n<180. Exactly48; including180 changes nothing because its gcd is180."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.9",
      "contentHash": "197a6d79463dc78d4c07b9689087ba00e97848e428906bd3a28d79181eeb12f9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          73,
          75,
          76
        ],
        "answerPdfPages": [],
        "method": "Original scans visually inspected; lost factorial denominators, malformed binary patterns, extra/missing gender letters and probability denominator825 repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "197a6d79463dc78d4c07b9689087ba00e97848e428906bd3a28d79181eeb12f9",
        "method": "Enumerated all720 permutations of six distinct labels, with0–2 boys and3–5 girls. Directly rejected any contiguous triple consisting of all three boys or all three girls. Exactly504 remain."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.10",
      "contentHash": "ebdacd8485bf9bd1b5cf88d04cae04438b32a8aad12dca082adc7eb3b7dd6dae",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          73,
          76,
          77,
          78
        ],
        "answerPdfPages": [],
        "method": "Original scans visually inspected; lost factorial denominators, malformed binary patterns, extra/missing gender letters and probability denominator825 repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "ebdacd8485bf9bd1b5cf88d04cae04438b32a8aad12dca082adc7eb3b7dd6dae",
        "method": "Independently enumerated all34650 rank multiset patterns and all369600 suit multiset patterns, rejecting complete-rank or complete-suit runs directly. Counts32844 and308664 yield exact source fractions782/825 and12861/15400. Uniformity follows constant distinct-card multiplicity per pattern."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.11",
      "contentHash": "64aad3a5a987e1dd6989844108b8da3dc2a99693576dd96209d5563c409069f3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          74,
          78,
          79
        ],
        "answerPdfPages": [],
        "method": "Original scans visually inspected; lost factorial denominators, malformed binary patterns, extra/missing gender letters and probability denominator825 repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "64aad3a5a987e1dd6989844108b8da3dc2a99693576dd96209d5563c409069f3",
        "method": "Fixed biologist0 at seat0 and enumerated all120 permutations of the remaining five distinct scientists. Direct circular adjacency test leaves32. Exactly16 have the other biologist opposite and16 two seats away. Reflections remain distinct."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.12",
      "contentHash": "ace678c78ce4e39522edd845d6541f9c785deb5fd4019775aef0ea1d67260463",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          74,
          79
        ],
        "answerPdfPages": [],
        "method": "Original scans visually inspected; lost factorial denominators, malformed binary patterns, extra/missing gender letters and probability denominator825 repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "ace678c78ce4e39522edd845d6541f9c785deb5fd4019775aef0ea1d67260463",
        "method": "Enumerated integers32..63, each with a six-digit binary representation beginning1, and directly tested substring111. Exactly12 qualify; their explicit list matches the supplied one."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.4.1",
      "contentHash": "645affda4fc918877477017ec15d719cf16559ba494dfe2c4a2059ad70a960b4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          79
        ],
        "answerPdfPages": [
          18,
          19
        ],
        "method": "Original scans visually inspected; lost factorial denominators, malformed binary patterns, extra/missing gender letters and probability denominator825 repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "645affda4fc918877477017ec15d719cf16559ba494dfe2c4a2059ad70a960b4",
        "method": "Deduplicated all permutations of STRATA into180 distinct words. Direct substring rejection of AA or TT leaves84."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.4.2",
      "contentHash": "66e1ed662ae2fa1c86d736c3bab43a40266c61cba965aef8637259c64b898fbb",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          79
        ],
        "answerPdfPages": [
          19
        ],
        "method": "Original scans visually inspected; lost factorial denominators, malformed binary patterns, extra/missing gender letters and probability denominator825 repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "66e1ed662ae2fa1c86d736c3bab43a40266c61cba965aef8637259c64b898fbb",
        "method": "Direct gcd(n,126)=1 test on n=1..210 gives60, with strict upper bound211 included correctly."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.4.3",
      "contentHash": "d66a009feaa8c8c03e71514b40b50740a454abc06c8dc8e10b7852b6eeb542ef",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          79
        ],
        "answerPdfPages": [
          19
        ],
        "method": "Original scans visually inspected; lost factorial denominators, malformed binary patterns, extra/missing gender letters and probability denominator825 repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "d66a009feaa8c8c03e71514b40b50740a454abc06c8dc8e10b7852b6eeb542ef",
        "method": "Independently tested gcd(n,10)=gcd(n,12)=1 for every n=1..999;266 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.4.4",
      "contentHash": "06099e073ac38f64dd7ffe45fb14abaa7eb1319f9e9f39face9b044a9337d656",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          79
        ],
        "answerPdfPages": [
          19,
          20
        ],
        "method": "Original scans visually inspected; lost factorial denominators, malformed binary patterns, extra/missing gender letters and probability denominator825 repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "06099e073ac38f64dd7ffe45fb14abaa7eb1319f9e9f39face9b044a9337d656",
        "method": "Enumerated all9!=362880 permutations of nine labeled fans, mapped their schools by label//3, and directly rejected any same-school run of length3. Exactly283824 remain."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.4.5",
      "contentHash": "0c026af6dd8a3fe2e60fdd8cdb937fb2d52e61769254cc143cc511c81af20550",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          80
        ],
        "answerPdfPages": [
          20
        ],
        "method": "Original scans visually inspected; lost factorial denominators, malformed binary patterns, extra/missing gender letters and probability denominator825 repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "0c026af6dd8a3fe2e60fdd8cdb937fb2d52e61769254cc143cc511c81af20550",
        "method": "Independent occupancy-mask dynamic program starts with an empty mask and assigns students one at a time to three labeled courses. After15 transitions, full mask111 contains14250606 assignments; no PIE formula used."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.1",
      "contentHash": "4b9c943cd725368f998a8b064396889b16d8fb7b80b1a716f7bfae05a22a3c17",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          45,
          46
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Typed nested-set model distinguishes numbers from frozensets; enumerated all eight subsets of B. All a–e conclusions independently checked.",
        "subparts": [
          {
            "id": "2.1.a",
            "status": "checked-in-parent-proof",
            "evidence": "Typed nested-set model distinguishes numbers from frozensets; enumerated all eight subsets of B. All a–e conclusions independently checked."
          },
          {
            "id": "2.1.b",
            "status": "checked-in-parent-proof",
            "evidence": "Typed nested-set model distinguishes numbers from frozensets; enumerated all eight subsets of B. All a–e conclusions independently checked."
          },
          {
            "id": "2.1.c",
            "status": "checked-in-parent-proof",
            "evidence": "Typed nested-set model distinguishes numbers from frozensets; enumerated all eight subsets of B. All a–e conclusions independently checked."
          },
          {
            "id": "2.1.d",
            "status": "checked-in-parent-proof",
            "evidence": "Typed nested-set model distinguishes numbers from frozensets; enumerated all eight subsets of B. All a–e conclusions independently checked."
          },
          {
            "id": "2.1.e",
            "status": "checked-in-parent-proof",
            "evidence": "Typed nested-set model distinguishes numbers from frozensets; enumerated all eight subsets of B. All a–e conclusions independently checked."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2",
      "contentHash": "c04e248862f00e7e630c7771e50982fcacab7632a5a8ebfba5c3f762f2344223",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          46
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "General arbitrary-element audit: ∅ has no counterexample to the universal membership implication; vacuous inclusion, including A=∅.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.3",
      "contentHash": "a47f31e2c8cb7af78fbe3571bd9a6d9758d05c072608f648b591d895bdd09628",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          46
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Independent power-set enumeration and filter by absence of 1 yields exactly the supplied eight sets.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.4",
      "contentHash": "ade81a4ea45264b86a3697e91fe6837c714b6cf6a90f787f484f6de8fb70a381",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          49
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Exhaustive inclusions over all subsets of a four-element universe plus arbitrary-element proof: a subset contributes no extra union elements, and all its elements survive intersection.",
        "subparts": [
          {
            "id": "2.4.a",
            "status": "checked-in-parent-proof",
            "evidence": "Exhaustive inclusions over all subsets of a four-element universe plus arbitrary-element proof: a subset contributes no extra union elements, and all its elements survive intersection."
          },
          {
            "id": "2.4.b",
            "status": "checked-in-parent-proof",
            "evidence": "Exhaustive inclusions over all subsets of a four-element universe plus arbitrary-element proof: a subset contributes no extra union elements, and all its elements survive intersection."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.5",
      "contentHash": "3babe60c1986ac36c2161431143d9ab8cc97bce09939ed8a14468ca40638c6bb",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          50,
          51
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "All 4,096 membership-set triples checked. General proof audited via Boolean equivalence a∧(b∨c)=(a∧b)∨(a∧c); both membership directions are present.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.6",
      "contentHash": "56fd7a1cb292c254f7aed4214add64afdc9ccec1a7789e317d0b51b456ce8333",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          54,
          55
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "All truth assignments enumerated; a proposition and its negation have opposite values, so their conjunction is always false.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.7",
      "contentHash": "8d76086f06f2353911a6c7dcb19466a6b9ece049a9e99ad55ccb1cfb00246e04",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          55,
          56
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "All four p,q assignments independently computed; all intermediate truth-table columns checked by static QA.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2.1",
      "contentHash": "0f7d8b278c07c8947ead07fb89bac231f3d5ac7c3c41c3c9edd36d013fa1a802",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          47
        ],
        "answerPdfPages": [
          6
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "frozenset() and frozenset({frozenset()}) have cardinalities 0 and 1 and are unequal.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2.2",
      "contentHash": "89f426c3388a9d4a77b097a11d803d612baa416b4c45b0d60581f2c2d63e2053",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          47
        ],
        "answerPdfPages": [
          6
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Each a–g proposition checked against definitions, including typed nested singleton membership, strict empty inclusion, and odd integer 1 witnessing proper inclusion of even integers.",
        "subparts": [
          {
            "id": "2.2.2.a",
            "status": "checked-in-parent-proof",
            "evidence": "Each a–g proposition checked against definitions, including typed nested singleton membership, strict empty inclusion, and odd integer 1 witnessing proper inclusion of even integers."
          },
          {
            "id": "2.2.2.b",
            "status": "checked-in-parent-proof",
            "evidence": "Each a–g proposition checked against definitions, including typed nested singleton membership, strict empty inclusion, and odd integer 1 witnessing proper inclusion of even integers."
          },
          {
            "id": "2.2.2.c",
            "status": "checked-in-parent-proof",
            "evidence": "Each a–g proposition checked against definitions, including typed nested singleton membership, strict empty inclusion, and odd integer 1 witnessing proper inclusion of even integers."
          },
          {
            "id": "2.2.2.d",
            "status": "checked-in-parent-proof",
            "evidence": "Each a–g proposition checked against definitions, including typed nested singleton membership, strict empty inclusion, and odd integer 1 witnessing proper inclusion of even integers."
          },
          {
            "id": "2.2.2.e",
            "status": "checked-in-parent-proof",
            "evidence": "Each a–g proposition checked against definitions, including typed nested singleton membership, strict empty inclusion, and odd integer 1 witnessing proper inclusion of even integers."
          },
          {
            "id": "2.2.2.f",
            "status": "checked-in-parent-proof",
            "evidence": "Each a–g proposition checked against definitions, including typed nested singleton membership, strict empty inclusion, and odd integer 1 witnessing proper inclusion of even integers."
          },
          {
            "id": "2.2.2.g",
            "status": "checked-in-parent-proof",
            "evidence": "Each a–g proposition checked against definitions, including typed nested singleton membership, strict empty inclusion, and odd integer 1 witnessing proper inclusion of even integers."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2.3",
      "contentHash": "7ccaebd303cafa0a0238e89f5a511740199794641436e945fe890aec875b6c5f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          47
        ],
        "answerPdfPages": [
          6
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "General reflexivity: x∈A implies x∈A. Exhaustive finite-model regression includes empty A.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2.4",
      "contentHash": "72ea482ee7e41e4872cc358d4a2eb74385eff5a8a6efa499eaf03dc44ae484a6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          47
        ],
        "answerPdfPages": [
          6
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "General extensionality: each inclusion supplies one direction of membership; exhaustively checked finite pairs.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2.5",
      "contentHash": "46d3e80394486f3ca68ccb88847bc9402cc9afde1b4f7849d0439c866e498497",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          47
        ],
        "answerPdfPages": [
          6
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Arbitrary-element chain proves inclusion transitivity. For proper transitivity a C∖B witness is outside A; all finite triples checked.",
        "subparts": [
          {
            "id": "2.2.5.inclusion",
            "status": "checked-in-parent-proof",
            "evidence": "Arbitrary-element chain proves inclusion transitivity. For proper transitivity a C∖B witness is outside A; all finite triples checked."
          },
          {
            "id": "2.2.5.proper",
            "status": "checked-in-parent-proof",
            "evidence": "Arbitrary-element chain proves inclusion transitivity. For proper transitivity a C∖B witness is outside A; all finite triples checked."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2.6",
      "contentHash": "0e0146dc2d66aef98f45d09cc01e0a9d2cf25f09541017317fe06a128f7ffc66",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          47
        ],
        "answerPdfPages": [
          6
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "If B had an element it would belong to ∅, impossible; B=∅. Empty-boundary case checked.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2.7",
      "contentHash": "e0bacf343fcbcb803f01a3114d87aa48e7e639b1bd97670396d9a1e346673f37",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          47
        ],
        "answerPdfPages": [
          6,
          7
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Finite disjoint decomposition B=A ⊔ (B∖A) gives bound, equality criterion, and strict bound. Verified every finite A⊆B in the model. Finiteness retained.",
        "subparts": [
          {
            "id": "2.2.7.bound",
            "status": "checked-in-parent-proof",
            "evidence": "Finite disjoint decomposition B=A ⊔ (B∖A) gives bound, equality criterion, and strict bound. Verified every finite A⊆B in the model. Finiteness retained."
          },
          {
            "id": "2.2.7.equality",
            "status": "checked-in-parent-proof",
            "evidence": "Finite disjoint decomposition B=A ⊔ (B∖A) gives bound, equality criterion, and strict bound. Verified every finite A⊆B in the model. Finiteness retained."
          },
          {
            "id": "2.2.7.proper",
            "status": "checked-in-parent-proof",
            "evidence": "Finite disjoint decomposition B=A ⊔ (B∖A) gives bound, equality criterion, and strict bound. Verified every finite A⊆B in the model. Finiteness retained."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2.8",
      "contentHash": "075bb18ceb4875b6c3f7b2304c6f21f0da71080bd3f3ddc5a770a46ccf2b3c1a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          47
        ],
        "answerPdfPages": [
          7
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "A<B and B<A imply A<A by strict transitivity, contradicting equality with itself; all finite pairs checked.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2.9",
      "contentHash": "7fc80ff30b5785259468fe3d437451fdad5fe7ec14036394b5de0fc438d96db6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          47
        ],
        "answerPdfPages": [
          7
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Independent recursively typed power-set construction returns {∅,{∅}}, including both distinct elements.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.2.10",
      "contentHash": "f6d95a7284172292d2aabba0e498395774a6056b1f84798ef7b08a593e1eeef8",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          47
        ],
        "answerPdfPages": [
          7
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "General bijection to n-bit membership vectors checked: unique subset per vector and unique vector per subset; n=0 explicitly covered. Exhaustive finite subsets checked.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.3.1",
      "contentHash": "6d7ee2304d6e6cc28f79e34f754ca5d843fcf3a71e1cd5a0f9f435586d71e988",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          52
        ],
        "answerPdfPages": [
          7
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Every finite A checked with ∅; general membership: a∨false=a, a∧false=false.",
        "subparts": [
          {
            "id": "2.3.1.union",
            "status": "checked-in-parent-proof",
            "evidence": "Every finite A checked with ∅; general membership: a∨false=a, a∧false=false."
          },
          {
            "id": "2.3.1.intersection",
            "status": "checked-in-parent-proof",
            "evidence": "Every finite A checked with ∅; general membership: a∨false=a, a∧false=false."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.3.2",
      "contentHash": "ace0fe3207622d42a181ce002b9eab33c8bae91625f5a5985d815396cd36484a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          52
        ],
        "answerPdfPages": [
          7
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Every finite A checked; membership idempotence a∨a=a∧a=a proves arbitrary-set identity.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.3.3",
      "contentHash": "3557a1e8b80581424fddc9903f117aacc3f93b739cc3772df1ffd5be9c8c880f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          52
        ],
        "answerPdfPages": [
          7
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Given x∈S, singleton contributes no new union member and its sole element is in the intersection. Both results checked across finite sets and members.",
        "subparts": [
          {
            "id": "2.3.3.union",
            "status": "checked-in-parent-proof",
            "evidence": "Given x∈S, singleton contributes no new union member and its sole element is in the intersection. Both results checked across finite sets and members."
          },
          {
            "id": "2.3.3.intersection",
            "status": "checked-in-parent-proof",
            "evidence": "Given x∈S, singleton contributes no new union member and its sole element is in the intersection. Both results checked across finite sets and members."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.3.4",
      "contentHash": "5e4518cac96f709a9bf2ff0415f83bfdbdec0ed246ce5e1b745e82d49bef84ef",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          52
        ],
        "answerPdfPages": [
          7
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "All finite triples checked; arbitrary membership equivalence a∨(b∧c)=(a∨b)∧(a∨c) proves the general identity.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.3.5",
      "contentHash": "03da1c157372ce1c5bcfc0a30546c56152e62065334c3e8949cd8b8149748163",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          52
        ],
        "answerPdfPages": [
          7
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Both directions of union/intersection characterizations independently tested for every finite pair; arbitrary-element conclusions B⊆A and A⊆B audited.",
        "subparts": [
          {
            "id": "2.3.5.a",
            "status": "checked-in-parent-proof",
            "evidence": "Both directions of union/intersection characterizations independently tested for every finite pair; arbitrary-element conclusions B⊆A and A⊆B audited."
          },
          {
            "id": "2.3.5.b",
            "status": "checked-in-parent-proof",
            "evidence": "Both directions of union/intersection characterizations independently tested for every finite pair; arbitrary-element conclusions B⊆A and A⊆B audited."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.3.6",
      "contentHash": "4d874e93a46a9d55fe5f26567c613145ef808b162a7013c4d8ee4425f869b4e5",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          52
        ],
        "answerPdfPages": [
          7,
          8
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired.",
        "fidelityAmendment": {
          "textbookPdfPages": [
            42,
            43,
            44,
            45,
            46,
            47,
            48,
            49,
            50,
            51,
            52,
            53,
            54,
            55,
            56,
            57,
            58,
            59,
            60,
            61,
            62,
            63
          ],
          "answerPdfPages": [
            6,
            7,
            8,
            9,
            10,
            11,
            12,
            13,
            14
          ],
          "change": "Source follow-up restored;all lesson exposition,examples,and reference figures compared."
        }
      },
      "math": {
        "status": "passed",
        "method": "All finite pairs checked for power-set intersection and union inclusion. Explicit S={1},T={2} counterexample to union equality retained; membership proof covers arbitrary sets. The follow-up equality condition was independently checked on256 pairs of finite subsets and with a general witness proof.",
        "subparts": [
          {
            "id": "2.3.6.intersection",
            "status": "checked-in-parent-proof",
            "evidence": "All finite pairs checked for power-set intersection and union inclusion. Explicit S={1},T={2} counterexample to union equality retained; membership proof covers arbitrary sets."
          },
          {
            "id": "2.3.6.union",
            "status": "checked-in-parent-proof",
            "evidence": "All finite pairs checked for power-set intersection and union inclusion. Explicit S={1},T={2} counterexample to union equality retained; membership proof covers arbitrary sets."
          }
        ],
        "contentHash": "4d874e93a46a9d55fe5f26567c613145ef808b162a7013c4d8ee4425f869b4e5"
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.4.1",
      "contentHash": "f38010f8b70e8573331babcce55a5c5b5f031348452a9ee6416c24f810af00b2",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          57
        ],
        "answerPdfPages": [
          8
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Classified each a–f as assertion, opinion, question, or expression; truth is not required for statement status. The author-eye-color truth is attributed only to the supplied solution, not independently asserted.",
        "subparts": [
          {
            "id": "2.4.1.a",
            "status": "checked-in-parent-proof",
            "evidence": "Classified each a–f as assertion, opinion, question, or expression; truth is not required for statement status. The author-eye-color truth is attributed only to the supplied solution, not independently asserted."
          },
          {
            "id": "2.4.1.b",
            "status": "checked-in-parent-proof",
            "evidence": "Classified each a–f as assertion, opinion, question, or expression; truth is not required for statement status. The author-eye-color truth is attributed only to the supplied solution, not independently asserted."
          },
          {
            "id": "2.4.1.c",
            "status": "checked-in-parent-proof",
            "evidence": "Classified each a–f as assertion, opinion, question, or expression; truth is not required for statement status. The author-eye-color truth is attributed only to the supplied solution, not independently asserted."
          },
          {
            "id": "2.4.1.d",
            "status": "checked-in-parent-proof",
            "evidence": "Classified each a–f as assertion, opinion, question, or expression; truth is not required for statement status. The author-eye-color truth is attributed only to the supplied solution, not independently asserted."
          },
          {
            "id": "2.4.1.e",
            "status": "checked-in-parent-proof",
            "evidence": "Classified each a–f as assertion, opinion, question, or expression; truth is not required for statement status. The author-eye-color truth is attributed only to the supplied solution, not independently asserted."
          },
          {
            "id": "2.4.1.f",
            "status": "checked-in-parent-proof",
            "evidence": "Classified each a–f as assertion, opinion, question, or expression; truth is not required for statement status. The author-eye-color truth is attributed only to the supplied solution, not independently asserted."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.4.2",
      "contentHash": "43b5e64cf57e072d343fc7543efc71310d2e9946f0067341af2e6995bbfa1b2b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          57
        ],
        "answerPdfPages": [
          8
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Truth cases p=T,F both give p∨¬p=T; table reconstructed and checked.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.4.3",
      "contentHash": "2398df4a4c2c2b29079a435e5b863223b2b1e361ce6af9a8414a50d6aede8837",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          57
        ],
        "answerPdfPages": [
          8
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Independent exhaustive Boolean evaluation proves negation-of-conjunction identity; full intermediate columns reconstructed.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.4.4",
      "contentHash": "5bedd9e96f172fbe53a59111a1f19d598466cf4bb3b4ed3d6044b8c534204b0c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          58
        ],
        "answerPdfPages": [
          8
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Independent exhaustive Boolean evaluation: implication equals ¬p∨q, false only at p=T,q=F.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.4.5",
      "contentHash": "6a5c75bd93dd30705c1c52784651f9f8e4a36da810f3c137e05f4180dd2c9972",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          58
        ],
        "answerPdfPages": [
          8,
          9
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "All a–e original/converse/contrapositive forms audited. (a) subtract2; (b) factor(y−1)(y−2), counterexample y=2; (c) membership equivalence; (d) false conjunction as antecedent with proper OR negation; (e) definition within the nondegenerate-triangle domain. Coincident W=X=Y would violate the converse if arbitrary points were permitted; that domain caveat is displayed. All contrapositives match original truth values under the stated domains.",
        "subparts": [
          {
            "id": "2.4.5.a",
            "status": "checked-in-parent-proof",
            "evidence": "All a–e original/converse/contrapositive forms audited. (a) subtract2; (b) factor(y−1)(y−2), counterexample y=2; (c) membership equivalence; (d) false conjunction as antecedent with proper OR negation; (e) definition within the nondegenerate-triangle domain. Coincident W=X=Y would violate the converse if arbitrary points were permitted; that domain caveat is displayed. All contrapositives match original truth values under the stated domains."
          },
          {
            "id": "2.4.5.b",
            "status": "checked-in-parent-proof",
            "evidence": "All a–e original/converse/contrapositive forms audited. (a) subtract2; (b) factor(y−1)(y−2), counterexample y=2; (c) membership equivalence; (d) false conjunction as antecedent with proper OR negation; (e) definition within the nondegenerate-triangle domain. Coincident W=X=Y would violate the converse if arbitrary points were permitted; that domain caveat is displayed. All contrapositives match original truth values under the stated domains."
          },
          {
            "id": "2.4.5.c",
            "status": "checked-in-parent-proof",
            "evidence": "All a–e original/converse/contrapositive forms audited. (a) subtract2; (b) factor(y−1)(y−2), counterexample y=2; (c) membership equivalence; (d) false conjunction as antecedent with proper OR negation; (e) definition within the nondegenerate-triangle domain. Coincident W=X=Y would violate the converse if arbitrary points were permitted; that domain caveat is displayed. All contrapositives match original truth values under the stated domains."
          },
          {
            "id": "2.4.5.d",
            "status": "checked-in-parent-proof",
            "evidence": "All a–e original/converse/contrapositive forms audited. (a) subtract2; (b) factor(y−1)(y−2), counterexample y=2; (c) membership equivalence; (d) false conjunction as antecedent with proper OR negation; (e) definition within the nondegenerate-triangle domain. Coincident W=X=Y would violate the converse if arbitrary points were permitted; that domain caveat is displayed. All contrapositives match original truth values under the stated domains."
          },
          {
            "id": "2.4.5.e",
            "status": "checked-in-parent-proof",
            "evidence": "All a–e original/converse/contrapositive forms audited. (a) subtract2; (b) factor(y−1)(y−2), counterexample y=2; (c) membership equivalence; (d) false conjunction as antecedent with proper OR negation; (e) definition within the nondegenerate-triangle domain. Coincident W=X=Y would violate the converse if arbitrary points were permitted; that domain caveat is displayed. All contrapositives match original truth values under the stated domains."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.4.6",
      "contentHash": "23da1b4f5d70d6039945734ffb2b97ae3515aceeff9d90700c3a8bfa39ca6ad9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          58
        ],
        "answerPdfPages": [
          9
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Both identities checked for all eight p,q,r assignments. Every reconstructed intermediate truth-table column is generated directly from its named Boolean expression.",
        "subparts": [
          {
            "id": "2.4.6.a",
            "status": "checked-in-parent-proof",
            "evidence": "Both identities checked for all eight p,q,r assignments. Every reconstructed intermediate truth-table column is generated directly from its named Boolean expression."
          },
          {
            "id": "2.4.6.b",
            "status": "checked-in-parent-proof",
            "evidence": "Both identities checked for all eight p,q,r assignments. Every reconstructed intermediate truth-table column is generated directly from its named Boolean expression."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.4.7",
      "contentHash": "45c7565f705ac8ea32b06e3216744aee5db327764c62fa660d4f0781187d094d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          58
        ],
        "answerPdfPages": [
          9
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "NAND defined as ¬(p∧q); independent evaluation verifies (p↑p)↑(q↑q)=p∨q in all four rows.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.5.1",
      "contentHash": "ff07b4d05e16ed7b78df5eef8b5092800057c3e4d8289d876d8769a5d1190baa",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          60
        ],
        "answerPdfPages": [
          10
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Quantifier negation exhaustively checked on all 0..3 by 0..3 finite predicate matrices. General witness/counterexample dichotomy proves it, including empty S.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.5.2",
      "contentHash": "d653215cc52ea021ac221e9255939eed8896f5f389d20105a1237dd426d0ac96",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          60
        ],
        "answerPdfPages": [
          10
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Both supplied examples verified on integers: first choose x=0, counterexample x=−1; second x=1 is a universal witness, while x=0 forces noninteger y=1/2. Domain restriction retained.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.8",
      "contentHash": "8ca252a1d6ec7b162e87e5961532e486e80a170c72ec08e421d1af7cddf11132",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          61,
          62
        ],
        "answerPdfPages": [
          10
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Independent primality trial division and modular predicates reconstruct every requested set in (a), with exact list comparison; intersections in (b) have cardinalities 4 and 5.",
        "subparts": [
          {
            "id": "2.8.a",
            "status": "checked-in-parent-proof",
            "evidence": "Independent primality trial division and modular predicates reconstruct every requested set in (a), with exact list comparison; intersections in (b) have cardinalities 4 and 5."
          },
          {
            "id": "2.8.b",
            "status": "checked-in-parent-proof",
            "evidence": "Independent primality trial division and modular predicates reconstruct every requested set in (a), with exact list comparison; intersections in (b) have cardinalities 4 and 5."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.9",
      "contentHash": "01a96ac44482dca32f3e11b243ea9d5c1107c800fe29e2a7f0f6e81e7fa6ffd3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          62
        ],
        "answerPdfPages": [
          10
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Independent forward squaring and inverse integer-square filtering yield the supplied sets; |x|>2 cannot qualify in (b), so finite search is exhaustive.",
        "subparts": [
          {
            "id": "2.9.a",
            "status": "checked-in-parent-proof",
            "evidence": "Independent forward squaring and inverse integer-square filtering yield the supplied sets; |x|>2 cannot qualify in (b), so finite search is exhaustive."
          },
          {
            "id": "2.9.b",
            "status": "checked-in-parent-proof",
            "evidence": "Independent forward squaring and inverse integer-square filtering yield the supplied sets; |x|>2 cannot qualify in (b), so finite search is exhaustive."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.10",
      "contentHash": "e70eac26dd1debf77b349cc29998a202a35bdf89a0167c1e6049a909638a5992",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          62
        ],
        "answerPdfPages": [
          11
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Supplied counterexample checked directly: A=C={1,2}, B={3,4} satisfies both hypotheses while A∩C is nonempty.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.11",
      "contentHash": "e7d3857e66a5d5d5719d3858bf67af7d646ea37873d2614975dc6392980944af",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          62
        ],
        "answerPdfPages": [
          11
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Every finite pair tested for both converse characterizations; general arbitrary-element arguments independently confirm A⊆B in each part.",
        "subparts": [
          {
            "id": "2.11.a",
            "status": "checked-in-parent-proof",
            "evidence": "Every finite pair tested for both converse characterizations; general arbitrary-element arguments independently confirm A⊆B in each part."
          },
          {
            "id": "2.11.b",
            "status": "checked-in-parent-proof",
            "evidence": "Every finite pair tested for both converse characterizations; general arbitrary-element arguments independently confirm A⊆B in each part."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.12",
      "contentHash": "5152082eb25cced9f1bd058f4eb03cf2b03412874029dbab982185beb3a86f1a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          62
        ],
        "answerPdfPages": [
          11
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Finite equivalence checked, and arbitrary-element proof: retaining all of A after removal excludes every possible common element.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.13",
      "contentHash": "b73018e1596cdcccdaa66f2577cea7dfde1945ea4787b29a01ae4cb803ab6807",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          62
        ],
        "answerPdfPages": [
          11
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Finite disjoint cardinality decompositions with the same intersection prove equal sizes; all finite pairs checked. Infinite cancellation is not assumed.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.14",
      "contentHash": "fde1ef39973f2fa62ca9a392501d7486c555f8fc23697d9bff5148ee90a3a18e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          62
        ],
        "answerPdfPages": [
          11
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "All eight truth assignments checked, including every intermediate implication and final transitivity assertion.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.15",
      "contentHash": "6d89e2871392457684976880916cd31aa3ba96c42d504ce50f73d3394674f725",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          62
        ],
        "answerPdfPages": [
          11
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "All seven translations audited for correct subset direction, inclusive OR, self-photo predicate, and dependence versus a single common witness. Proper B⊂U preserves the existence of a non-snob.",
        "subparts": [
          {
            "id": "2.15.a",
            "status": "checked-in-parent-proof",
            "evidence": "All seven translations audited for correct subset direction, inclusive OR, self-photo predicate, and dependence versus a single common witness. Proper B⊂U preserves the existence of a non-snob."
          },
          {
            "id": "2.15.b",
            "status": "checked-in-parent-proof",
            "evidence": "All seven translations audited for correct subset direction, inclusive OR, self-photo predicate, and dependence versus a single common witness. Proper B⊂U preserves the existence of a non-snob."
          },
          {
            "id": "2.15.c",
            "status": "checked-in-parent-proof",
            "evidence": "All seven translations audited for correct subset direction, inclusive OR, self-photo predicate, and dependence versus a single common witness. Proper B⊂U preserves the existence of a non-snob."
          },
          {
            "id": "2.15.d",
            "status": "checked-in-parent-proof",
            "evidence": "All seven translations audited for correct subset direction, inclusive OR, self-photo predicate, and dependence versus a single common witness. Proper B⊂U preserves the existence of a non-snob."
          },
          {
            "id": "2.15.e",
            "status": "checked-in-parent-proof",
            "evidence": "All seven translations audited for correct subset direction, inclusive OR, self-photo predicate, and dependence versus a single common witness. Proper B⊂U preserves the existence of a non-snob."
          },
          {
            "id": "2.15.f",
            "status": "checked-in-parent-proof",
            "evidence": "All seven translations audited for correct subset direction, inclusive OR, self-photo predicate, and dependence versus a single common witness. Proper B⊂U preserves the existence of a non-snob."
          },
          {
            "id": "2.15.g",
            "status": "checked-in-parent-proof",
            "evidence": "All seven translations audited for correct subset direction, inclusive OR, self-photo predicate, and dependence versus a single common witness. Proper B⊂U preserves the existence of a non-snob."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.16",
      "contentHash": "242ab394428ad65e099f1abc5939b58edfe3dad8b7003f03cc854bd1362d8ad5",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          62,
          63
        ],
        "answerPdfPages": [
          11,
          12
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed-with-stated-interpretation",
        "method": "All seven symbolic translations audited for only-if direction, existential counterexample, implication scope, photo relation order, and nested quantifiers. Part f retains source intended reading and explicitly presents the weaker ordinary-language exemption reading for self-assessment.",
        "subparts": [
          {
            "id": "2.16.a",
            "status": "checked-in-parent-proof",
            "evidence": "All seven symbolic translations audited for only-if direction, existential counterexample, implication scope, photo relation order, and nested quantifiers. Part f retains source intended reading and explicitly presents the weaker ordinary-language exemption reading for self-assessment."
          },
          {
            "id": "2.16.b",
            "status": "checked-in-parent-proof",
            "evidence": "All seven symbolic translations audited for only-if direction, existential counterexample, implication scope, photo relation order, and nested quantifiers. Part f retains source intended reading and explicitly presents the weaker ordinary-language exemption reading for self-assessment."
          },
          {
            "id": "2.16.c",
            "status": "checked-in-parent-proof",
            "evidence": "All seven symbolic translations audited for only-if direction, existential counterexample, implication scope, photo relation order, and nested quantifiers. Part f retains source intended reading and explicitly presents the weaker ordinary-language exemption reading for self-assessment."
          },
          {
            "id": "2.16.d",
            "status": "checked-in-parent-proof",
            "evidence": "All seven symbolic translations audited for only-if direction, existential counterexample, implication scope, photo relation order, and nested quantifiers. Part f retains source intended reading and explicitly presents the weaker ordinary-language exemption reading for self-assessment."
          },
          {
            "id": "2.16.e",
            "status": "checked-in-parent-proof",
            "evidence": "All seven symbolic translations audited for only-if direction, existential counterexample, implication scope, photo relation order, and nested quantifiers. Part f retains source intended reading and explicitly presents the weaker ordinary-language exemption reading for self-assessment."
          },
          {
            "id": "2.16.f",
            "status": "checked-in-parent-proof",
            "evidence": "All seven symbolic translations audited for only-if direction, existential counterexample, implication scope, photo relation order, and nested quantifiers. Part f retains source intended reading and explicitly presents the weaker ordinary-language exemption reading for self-assessment."
          },
          {
            "id": "2.16.g",
            "status": "checked-in-parent-proof",
            "evidence": "All seven symbolic translations audited for only-if direction, existential counterexample, implication scope, photo relation order, and nested quantifiers. Part f retains source intended reading and explicitly presents the weaker ordinary-language exemption reading for self-assessment."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.17",
      "contentHash": "144ffc7b0c076fb6bd5ac6b758950b99a4a3b1ec598faba207765356f4ac495b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          63
        ],
        "answerPdfPages": [
          12
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Empty-domain quantifier semantics tested: empty all()=true, any()=false; general absence-of-counterexample/witness proof audited for both parts.",
        "subparts": [
          {
            "id": "2.17.a",
            "status": "checked-in-parent-proof",
            "evidence": "Empty-domain quantifier semantics tested: empty all()=true, any()=false; general absence-of-counterexample/witness proof audited for both parts."
          },
          {
            "id": "2.17.b",
            "status": "checked-in-parent-proof",
            "evidence": "Empty-domain quantifier semantics tested: empty all()=true, any()=false; general absence-of-counterexample/witness proof audited for both parts."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.18",
      "contentHash": "9df8b3c1c8e592d3ae6a83d98703b8826ae52ec1223e2560e4c5ec42bf58016c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          63
        ],
        "answerPdfPages": [
          12
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "All 4,096 finite-set triples checked. General forward proof uses C⊆A; reverse proof takes arbitrary x∈C and uses equality to force x∈A. Diagrams shade exact Boolean atoms.",
        "subparts": []
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.19",
      "contentHash": "24ec9886e856fecf561912113c79c2284f53cd3e4a4711ae0e8343a2390c90be",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          63
        ],
        "answerPdfPages": [
          12
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Union counterexample [0,1]∪[2,3] checked using missing 1.5. General intersection proof applies interval definition twice to arbitrary x≤z≤y. Empty intersection explicitly retained as an interval under the given definition; finite-order interval model also checked.",
        "subparts": [
          {
            "id": "2.19.a",
            "status": "checked-in-parent-proof",
            "evidence": "Union counterexample [0,1]∪[2,3] checked using missing 1.5. General intersection proof applies interval definition twice to arbitrary x≤z≤y. Empty intersection explicitly retained as an interval under the given definition; finite-order interval model also checked."
          },
          {
            "id": "2.19.b",
            "status": "checked-in-parent-proof",
            "evidence": "Union counterexample [0,1]∪[2,3] checked using missing 1.5. General intersection proof applies interval definition twice to arbitrary x≤z≤y. Empty intersection explicitly retained as an interval under the given definition; finite-order interval model also checked."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.20",
      "contentHash": "25af4369230d7f6c1fa8e985f6621e05d40c09244db40c4942bf7a9bb0e39781",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          63
        ],
        "answerPdfPages": [
          13
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Both XOR identities exhaustively verified over all eight truth assignments; associativity also follows parity of three truth bits.",
        "subparts": [
          {
            "id": "2.20.a",
            "status": "checked-in-parent-proof",
            "evidence": "Both XOR identities exhaustively verified over all eight truth assignments; associativity also follows parity of three truth bits."
          },
          {
            "id": "2.20.b",
            "status": "checked-in-parent-proof",
            "evidence": "Both XOR identities exhaustively verified over all eight truth assignments; associativity also follows parity of three truth bits."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.21",
      "contentHash": "5884366c12fb7b6092d3d7a9f6ad3d635a3bb6536abbe2d8708cd67d0a57dca7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          63
        ],
        "answerPdfPages": [
          13
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "All predicate matrices for independent domain sizes 0..3 checked: only ii and iv agree universally. Countermodels recorded for i,iii,v; general proof applies both quantifier-negation rules.",
        "subparts": [
          {
            "id": "2.21.i",
            "status": "checked-in-parent-proof",
            "evidence": "All predicate matrices for independent domain sizes 0..3 checked: only ii and iv agree universally. Countermodels recorded for i,iii,v; general proof applies both quantifier-negation rules."
          },
          {
            "id": "2.21.ii",
            "status": "checked-in-parent-proof",
            "evidence": "All predicate matrices for independent domain sizes 0..3 checked: only ii and iv agree universally. Countermodels recorded for i,iii,v; general proof applies both quantifier-negation rules."
          },
          {
            "id": "2.21.iii",
            "status": "checked-in-parent-proof",
            "evidence": "All predicate matrices for independent domain sizes 0..3 checked: only ii and iv agree universally. Countermodels recorded for i,iii,v; general proof applies both quantifier-negation rules."
          },
          {
            "id": "2.21.iv",
            "status": "checked-in-parent-proof",
            "evidence": "All predicate matrices for independent domain sizes 0..3 checked: only ii and iv agree universally. Countermodels recorded for i,iii,v; general proof applies both quantifier-negation rules."
          },
          {
            "id": "2.21.v",
            "status": "checked-in-parent-proof",
            "evidence": "All predicate matrices for independent domain sizes 0..3 checked: only ii and iv agree universally. Countermodels recorded for i,iii,v; general proof applies both quantifier-negation rules."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "2.22",
      "contentHash": "ada3e398423862318efd0b71f191425636d5682ab7efb32f4045c2bfddc6dda7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          63
        ],
        "answerPdfPages": [
          13,
          14
        ],
        "method": "Original textbook and supplied answer-book scans visually compared. Lost braces, negations, empty-set symbols, exponent, tables, and cross-page continuations repaired."
      },
      "math": {
        "status": "passed",
        "method": "Specific four symmetric-difference outputs independently calculated; associativity and intersection distributivity tested over all finite-set triples and proved via arbitrary-element XOR equivalences.",
        "subparts": [
          {
            "id": "2.22.a",
            "status": "checked-in-parent-proof",
            "evidence": "Specific four symmetric-difference outputs independently calculated; associativity and intersection distributivity tested over all finite-set triples and proved via arbitrary-element XOR equivalences."
          },
          {
            "id": "2.22.b",
            "status": "checked-in-parent-proof",
            "evidence": "Specific four symmetric-difference outputs independently calculated; associativity and intersection distributivity tested over all finite-set triples and proved via arbitrary-element XOR equivalences."
          },
          {
            "id": "2.22.c",
            "status": "checked-in-parent-proof",
            "evidence": "Specific four symmetric-difference outputs independently calculated; associativity and intersection distributivity tested over all finite-set triples and proved via arbitrary-element XOR equivalences."
          }
        ]
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.13",
      "contentHash": "a346c04d6a581bb22870227d8798bdbdea591bff9505b7244f5a6cb99452cfbe",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          80
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected. Alternating intersection signs, missing twin factor2, deal binomial coefficients and direct-sketch fractions repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "a346c04d6a581bb22870227d8798bdbdea591bff9505b7244f5a6cb99452cfbe",
        "method": "Directly tested n=1..999 for divisibility by2,3,5,7. Exactly228 survive; including1000 changes no survivor. This includes1 and composites with only larger prime factors."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.14",
      "contentHash": "a939491b54d2f2d22dbc7fbc3e624d1c5381d3e6663ea19263cbdb3eece639d3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          80,
          81,
          82
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected. Alternating intersection signs, missing twin factor2, deal binomial coefficients and direct-sketch fractions repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "a939491b54d2f2d22dbc7fbc3e624d1c5381d3e6663ea19263cbdb3eece639d3",
        "method": "Independent pointwise proof: sum over nonempty membership subsets of a k-member indicator support equals1−(1−1)^k. Exhaustively evaluated all support masks for n=1..8, plus the empty support whose union weight is0."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.15",
      "contentHash": "e4450ee5a44dfd06e3b5e75154a6ec8e18740e4e95c12dd43840fbd50ed9928c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          80,
          82,
          83
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected. Alternating intersection signs, missing twin factor2, deal binomial coefficients and direct-sketch fractions repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "e4450ee5a44dfd06e3b5e75154a6ec8e18740e4e95c12dd43840fbd50ed9928c",
        "method": "Independent dynamic program on the remaining two labeled members of each family and the last family seated. At every step choose a different family and multiply by its number of remaining distinct people. The full five-family count is1263360, without using PIE."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.5.1",
      "contentHash": "d4d7b587c5923f44f9f56b9f0465826453c8ea2983971c6ec2889155d29b05e2",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          83
        ],
        "answerPdfPages": [
          20,
          21
        ],
        "method": "Original scans inspected. Alternating intersection signs, missing twin factor2, deal binomial coefficients and direct-sketch fractions repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "d4d7b587c5923f44f9f56b9f0465826453c8ea2983971c6ec2889155d29b05e2",
        "method": "Direct gcd(n,462)=1 check for every integer1..528 yields137; no inclusion/exclusion derivation used."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.5.2",
      "contentHash": "c8adcfbdeb1b88939fc45a81f1c105120e3ca43b23d7ef9f67ac9cd158295641",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          83
        ],
        "answerPdfPages": [
          21,
          22
        ],
        "method": "Original scans inspected. Alternating intersection signs, missing twin factor2, deal binomial coefficients and direct-sketch fractions repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "c8adcfbdeb1b88939fc45a81f1c105120e3ca43b23d7ef9f67ac9cd158295641",
        "method": "Enumerated all2520 partitions of eight distinct cards into four labeled unordered two-card hands, and directly rejected equal-rank hands. Exactly1440 deals survive, giving4/7. The source direct sketch’s player-relabeling interpretation was checked separately."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.5.3",
      "contentHash": "f20bc82f6f88b228633299864b0899e5f4f96863204368acfd7d60b7081c8b68",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          83
        ],
        "answerPdfPages": [
          22
        ],
        "method": "Original scans inspected. Alternating intersection signs, missing twin factor2, deal binomial coefficients and direct-sketch fractions repaired against scans."
      },
      "math": {
        "status": "passed",
        "contentHash": "f20bc82f6f88b228633299864b0899e5f4f96863204368acfd7d60b7081c8b68",
        "method": "Independent row-by-row DP on the four remaining nationality counts. For each ordered pair of distinct countries, multiply by the remaining choices of individual occupants. Starting(3,3,3,3) yields154275840 labeled seatings."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.5.4",
      "contentHash": "8dedec48cd709c3b30774914d70336a0999d0c804e8a19e3c2b0e629bfd5759d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          83
        ],
        "answerPdfPages": [
          22,
          23
        ],
        "method": "Original scans inspected. Alternating intersection signs, missing twin factor2, deal binomial coefficients and direct-sketch fractions repaired against scans."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "8dedec48cd709c3b30774914d70336a0999d0c804e8a19e3c2b0e629bfd5759d",
        "method": "Enumerated all512 bit colorings of the nine labeled cells and directly tested each of the four2×2 blocks. Exactly417 avoid all-white blocks. Probability417/512 requires independence, explicitly retained as a condition."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.16",
      "contentHash": "8cb33d95f67210a1e10c5e17fa134b99fb4f1f49a298aa44c70fe6f4e657e2d0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          83,
          84,
          85
        ],
        "answerPdfPages": [],
        "method": "Original source and answer scans inspected; die icons transcribed as six (3.19) and five or greater (3.6.3); corrupt OCR formulas restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "8cb33d95f67210a1e10c5e17fa134b99fb4f1f49a298aa44c70fe6f4e657e2d0",
        "method": "Disjoint atoms: pair-only counts 11,8,6 and triple count 7 sum to 32. Pair incidence total 46 counts triple students three times. The unknown single-only counts sum to 18, but no individual single count is inferred."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.17",
      "contentHash": "f9ec1513c04091a750f1222294ca87ac7f37d860310ec4570f256a5ba5729ab3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          83,
          85
        ],
        "answerPdfPages": [],
        "method": "Original source and answer scans inspected; die icons transcribed as six (3.19) and five or greater (3.6.3); corrupt OCR formulas restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "f9ec1513c04091a750f1222294ca87ac7f37d860310ec4570f256a5ba5729ab3",
        "method": "Exhaustive truth table of all eight membership masks; pair-incidence count minus twice triple indicator equals indicator of membership count at least two."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.18",
      "contentHash": "140a569ba762765ac74d6f49af2df929af8252e2e6cba23c2aff9cc29fbb1b8f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          84,
          85
        ],
        "answerPdfPages": [],
        "method": "Original source and answer scans inspected; die icons transcribed as six (3.19) and five or greater (3.6.3); corrupt OCR formulas restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "140a569ba762765ac74d6f49af2df929af8252e2e6cba23c2aff9cc29fbb1b8f",
        "method": "Directly checked all integers 1..1999; exactly 533 have at least two of the three divisibilities, and exactly 467 have exactly two."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.19",
      "contentHash": "5bc96d128e463332a1fb5424842f098686f6e0f10ab6fce6550bde436299018b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          84,
          85,
          86
        ],
        "answerPdfPages": [],
        "method": "Original source and answer scans inspected; die icons transcribed as six (3.19) and five or greater (3.6.3); corrupt OCR formulas restored."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "5bc96d128e463332a1fb5424842f098686f6e0f10ab6fce6550bde436299018b",
        "method": "Enumerated all 7776 ordered five-die outcomes and directly counted at least three sixes: 276; exact rational reduction 23/648."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.20",
      "contentHash": "b6935135fdb3001055ccc008ae469a0233bca71a9817a66f96f5f54a5b5a5d6f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          84,
          86,
          87
        ],
        "answerPdfPages": [],
        "method": "Original source and answer scans inspected; die icons transcribed as six (3.19) and five or greater (3.6.3); corrupt OCR formulas restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "b6935135fdb3001055ccc008ae469a0233bca71a9817a66f96f5f54a5b5a5d6f",
        "method": "Checked every one of 128 seven-set membership masks: C(k,5)−5C(k,6)+15C(k,7) equals one exactly when k≥5, otherwise zero."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.6.1",
      "contentHash": "174f71235549afb47f640d35672779c1eaed896399979ec2c7dbcab06357a326",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          87
        ],
        "answerPdfPages": [
          23
        ],
        "method": "Original source and answer scans inspected; die icons transcribed as six (3.19) and five or greater (3.6.3); corrupt OCR formulas restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "174f71235549afb47f640d35672779c1eaed896399979ec2c7dbcab06357a326",
        "method": "Directly tested n=1..3150 for divisibility by at least three of 2,3,5,7; exactly 210 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.6.2",
      "contentHash": "2ecbe51d13de7a416a6c2e47d845cebc3fde7c28aba93b0cf870ee79358149f0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          87
        ],
        "answerPdfPages": [
          23,
          24
        ],
        "method": "Original source and answer scans inspected; die icons transcribed as six (3.19) and five or greater (3.6.3); corrupt OCR formulas restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "2ecbe51d13de7a416a6c2e47d845cebc3fde7c28aba93b0cf870ee79358149f0",
        "method": "Recovered all disjoint pair/triple/quadruple atoms by descending subtraction; pair-only total 265, triple-only 63, quadruple 4. Verified every supplied intersection from reconstructed atoms."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.6.3",
      "contentHash": "80b083b03e96c1f5a5480e916799f91f94bedcc3b18d4cacc536d3790ecb9bc1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          87
        ],
        "answerPdfPages": [
          24
        ],
        "method": "Original source and answer scans inspected; die icons transcribed as six (3.19) and five or greater (3.6.3); corrupt OCR formulas restored."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "80b083b03e96c1f5a5480e916799f91f94bedcc3b18d4cacc536d3790ecb9bc1",
        "method": "Enumerated all 1296 ordered four-die outcomes: 144 have at least three faces≥5, 128 have exactly three. Fractions reduce to 1/9 and 8/81."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.6.4",
      "contentHash": "aa39f4c220328d55b40fafd0b97c40cc03112d4ce81eadbab236fc172861727a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          88
        ],
        "answerPdfPages": [
          24,
          25
        ],
        "method": "Original source and answer scans inspected; die icons transcribed as six (3.19) and five or greater (3.6.3); corrupt OCR formulas restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "aa39f4c220328d55b40fafd0b97c40cc03112d4ce81eadbab236fc172861727a",
        "method": "For all 32 five-set membership masks checked C(k,2)−2C(k,3)+3C(k,4)−4C(k,5)=indicator(k≥2). Triangular linear equations give unique coefficients 1,−2,3,−4."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.21",
      "contentHash": "9aba2c5e923081430e0118b715d40f0ef111973ae1cb3c688d6cd27c60e78035",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          88,
          89,
          90
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; binomial OCR repaired, source figures reconstructed, source claim and supplied area-zero argument retained with explicit mathematical correction."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "9aba2c5e923081430e0118b715d40f0ef111973ae1cb3c688d6cd27c60e78035",
        "method": "Exhaustive enumeration of all 279936 target functions with no self-target: 141246 contain a directed 2-cycle; reduced probability 7847/15552."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.22",
      "contentHash": "690c497f2e52b1ba2eb7abfc236a8cbb37fa1892bb206435fc4b1e77f0e69e2c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          88,
          90
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; binomial OCR repaired, source figures reconstructed, source claim and supplied area-zero argument retained with explicit mathematical correction."
      },
      "math": {
        "status": "passed",
        "contentHash": "690c497f2e52b1ba2eb7abfc236a8cbb37fa1892bb206435fc4b1e77f0e69e2c",
        "method": "Independent dynamic program on remaining A/B/C counts, last letter and CA occurrence count (capped at3). Counts each distinct word once; result 91080."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.23",
      "contentHash": "fc12a029bc2b02ea240fa7768ae52ae6cc17477e4744eeda612586349ef533b3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          88,
          90,
          91
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; binomial OCR repaired, source figures reconstructed, source claim and supplied area-zero argument retained with explicit mathematical correction.",
        "chapter03FreshFidelity": "All textbook64–97 and manual15–37 visually compared; retained full supplied methods and restored figures/source discrepancy."
      },
      "math": {
        "status": "passed",
        "contentHash": "fc12a029bc2b02ea240fa7768ae52ae6cc17477e4744eeda612586349ef533b3",
        "method": "Enumerated every four-subset for n=5..22 and directly tested cyclic adjacency; all counts match the polynomial. Distinguishes incidence totals from counts of exactly-two-side quadrilaterals."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.24",
      "contentHash": "847e2f00477df63dc6618bdca0920e863b9bb02e97f110f5309dbcfdb10a6d61",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          88,
          91,
          92,
          93
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; binomial OCR repaired, source figures reconstructed, source claim and supplied area-zero argument retained with explicit mathematical correction.",
        "chapter03FreshFidelity": "All textbook64–97 and manual15–37 visually compared; retained full supplied methods and restored figures/source discrepancy."
      },
      "math": {
        "status": "passed",
        "contentHash": "847e2f00477df63dc6618bdca0920e863b9bb02e97f110f5309dbcfdb10a6d61",
        "method": "Independent pointwise inequality C(k,2)≥2k−3 for k=0..5, integrated over a coat of area5, proves pair-area sum≥10. Source elimination independently checked and all multiplicity identities verified."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.7.1",
      "contentHash": "f1a65bef3e5a1f486d3781a7c19e7ee4a1c6aeb716976e68cba307654af7c4b0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          93
        ],
        "answerPdfPages": [
          25
        ],
        "method": "Original scans inspected; binomial OCR repaired, source figures reconstructed, source claim and supplied area-zero argument retained with explicit mathematical correction."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "f1a65bef3e5a1f486d3781a7c19e7ee4a1c6aeb716976e68cba307654af7c4b0",
        "method": "All 15625 no-self-target functions on six children enumerated; 8265 have a directed 2-cycle; exact fraction 1653/3125."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.7.2",
      "contentHash": "ab9f75882ee180d5411abe2b81abc8ef0d1b8556d4d482c69280e2015f370633",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          94
        ],
        "answerPdfPages": [
          25,
          26,
          27
        ],
        "method": "Original scans inspected; binomial OCR repaired, source figures reconstructed, source claim and supplied area-zero argument retained with explicit mathematical correction.",
        "chapter03FreshFidelity": "All textbook64–97 and manual15–37 visually compared; retained full supplied methods and restored figures/source discrepancy."
      },
      "math": {
        "status": "passed",
        "contentHash": "ab9f75882ee180d5411abe2b81abc8ef0d1b8556d4d482c69280e2015f370633",
        "method": "Enumerated all C(10,4)=210 monotone grid paths, tested each directed step against the four shaded-cell edges; exactly145 qualify. Edge counts and both compatible edge-pair counts independently recovered."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.7.3",
      "contentHash": "c6330fb8163eb78b68075a6c1a2f78ede3d6316e5cd8b16d6e349676d925e393",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          94
        ],
        "answerPdfPages": [
          27
        ],
        "method": "Original scans inspected; binomial OCR repaired, source figures reconstructed, source claim and supplied area-zero argument retained with explicit mathematical correction."
      },
      "math": {
        "status": "passed",
        "contentHash": "c6330fb8163eb78b68075a6c1a2f78ede3d6316e5cd8b16d6e349676d925e393",
        "method": "All 64×64 ordered subset pairs enumerated with bit masks; 1394 meet either containment, 64 meet both. Exact probability697/2048."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.7.4",
      "contentHash": "ad84490409f179b82a29942f073d187d43214f3989f1fb52d2928a4745370e24",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          94
        ],
        "answerPdfPages": [
          27,
          28
        ],
        "method": "Original scans inspected; binomial OCR repaired, source figures reconstructed, source claim and supplied area-zero argument retained with explicit mathematical correction."
      },
      "math": {
        "status": "passed",
        "contentHash": "ad84490409f179b82a29942f073d187d43214f3989f1fb52d2928a4745370e24",
        "method": "All five-subsets of cyclic vertex sets n=6..22 enumerated; count subsets with at least two cyclic adjacent edges. Every result matches n³−11n²+31n."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.7.5",
      "contentHash": "64e582debac46ec2aec3fe5008546c61df2f49d6d15cdcb780059e22856587bb",
      "source": {
        "status": "passed-transcription-with-flagged-gap",
        "textbookPdfPages": [
          94
        ],
        "answerPdfPages": [
          28
        ],
        "method": "Original scans inspected; binomial OCR repaired, source figures reconstructed, source claim and supplied area-zero argument retained with explicit mathematical correction."
      },
      "math": {
        "status": "source-gap-corrected",
        "contentHash": "64e582debac46ec2aec3fe5008546c61df2f49d6d15cdcb780059e22856587bb",
        "method": "Verified equality example using all pair/triple membership atoms of area1/4. Patch areas2.5, pair areas1, total coat5; adding a common zero-area singleton refutes the literal pointwise assertion. Independently checked D=E=0 follows from source inequalities."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.25",
      "contentHash": "9d98bb41fce5acfa08c2cc147d8ea52f16be346da0a916209d89baf2f3d88bad",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          95
        ],
        "answerPdfPages": [
          28
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "9d98bb41fce5acfa08c2cc147d8ea52f16be346da0a916209d89baf2f3d88bad",
        "method": "Enumerated all26⁴ ordered four-letter tuples and tested the first/last vowel conditions;158860 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.26",
      "contentHash": "a8b8ed2e896f00deb26d55857b824ec3cb31e36df10fe51147dc84bdb5e21428",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          95
        ],
        "answerPdfPages": [
          28
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "a8b8ed2e896f00deb26d55857b824ec3cb31e36df10fe51147dc84bdb5e21428",
        "method": "Direct integer enumeration100..999 and adjacent-character comparison yields171."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.27",
      "contentHash": "e1e46575e3249f5e161401540820d63909b83104ebad828ca8ef35cddc581c41",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          95
        ],
        "answerPdfPages": [
          28
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "e1e46575e3249f5e161401540820d63909b83104ebad828ca8ef35cddc581c41",
        "method": "Disjoint membership-region reconstruction gives union21 and outside count−1, impossible for a20-person universe."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.28",
      "contentHash": "faaf475d99caa02a13628c9f14aa61f703052d5da8c596c41a6e6ecd5ffd519f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          95
        ],
        "answerPdfPages": [
          28
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "faaf475d99caa02a13628c9f14aa61f703052d5da8c596c41a6e6ecd5ffd519f",
        "method": "Probability atoms: keys-only15%,wallet-only25%,both-forgotten5%,neither-forgotten55%; all nonnegative and sum100%."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.29",
      "contentHash": "f4b802ac1734103901fcecb341ed5688f97cba5c84aee913aadea64c4da4e110",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          95
        ],
        "answerPdfPages": [
          28,
          29
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "f4b802ac1734103901fcecb341ed5688f97cba5c84aee913aadea64c4da4e110",
        "method": "Enumerated all26⁴ tuples, rejected equality at any of three adjacent positions;406250 survive. Symbolic factorization is the cube identity."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.30",
      "contentHash": "a6c619e79ef333b2fbce6711fb7ea9dae1907cc10307b41bd6a6377be83eb7bf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          96
        ],
        "answerPdfPages": [
          29
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "a6c619e79ef333b2fbce6711fb7ea9dae1907cc10307b41bd6a6377be83eb7bf",
        "method": "Explicitly constructed both900000-element string sets and took their intersection and union; intersection90, union1799910. Leading-zero constraint enforced."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.31",
      "contentHash": "1ba5d6effcc7e855a5878d8ef169336ef4c428bee993fd3575ce7c849dc0650a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          96
        ],
        "answerPdfPages": [
          29
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "1ba5d6effcc7e855a5878d8ef169336ef4c428bee993fd3575ce7c849dc0650a",
        "method": "Direct enumeration of all26³ letter triples and10³ digit triples gives676 and100 palindromes. Combining their independent uniform Cartesian blocks yields exact7/52."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.32",
      "contentHash": "9c8e3b1f34383791c53bce9e6bad13d1619c782a4a152720e9660106fdb74417",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          96
        ],
        "answerPdfPages": [
          29
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "9c8e3b1f34383791c53bce9e6bad13d1619c782a4a152720e9660106fdb74417",
        "method": "Enumerated all5040 permutations and directly tested fixed points at positions1,3,5,7;2250 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.33",
      "contentHash": "7925d1809ee00e3bb2ce428e781a2c883b4af3747c2d9b6bb74ab2396d1db624",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          96
        ],
        "answerPdfPages": [
          30
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "7925d1809ee00e3bb2ce428e781a2c883b4af3747c2d9b6bb74ab2396d1db624",
        "method": "Enumerated all2300 three-subsets of a25-cycle;550 contain a cyclic adjacent pair. Exact fraction11/46."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.34",
      "contentHash": "c9424b3045ea52688d5e0cbb266473d656ca5ad3bf93811812d6d798fc273af2",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          96
        ],
        "answerPdfPages": [
          30,
          31
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "c9424b3045ea52688d5e0cbb266473d656ca5ad3bf93811812d6d798fc273af2",
        "method": "Independent sum over all four suit-occupancy vectors totaling13, weighted by the product of C(13,occupancy); total C(52,13), favorable sum matches the PIE fraction exactly."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.35",
      "contentHash": "e04ff98f116da10512451ea0e76c2eb1bb8ef6935501f85072c4899846bcf338",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          96
        ],
        "answerPdfPages": [
          31
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "e04ff98f116da10512451ea0e76c2eb1bb8ef6935501f85072c4899846bcf338",
        "method": "Enumerated all100000 five-digit tuples and tested the maximum digit frequency;8560 meet the threshold."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.36",
      "contentHash": "a06a266020bd9b02a7158fb4c8abed634977dc22688fe467d9f0b0efe39f551d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          96
        ],
        "answerPdfPages": [
          31
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "a06a266020bd9b02a7158fb4c8abed634977dc22688fe467d9f0b0efe39f551d",
        "method": "Pointwise indicator identity1_(A∪B)=1_A+1_B−1_(A∩B) verified for all four Boolean membership cases; expectation proves the identity for any probability measure."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.37",
      "contentHash": "2a7bcc3de891cae6803d09f6e105d0bbb90d689a5da3f6ef971911ec8fa0a9b2",
      "source": {
        "status": "passed-with-recorded-domain-or-typo-note",
        "textbookPdfPages": [
          96
        ],
        "answerPdfPages": [
          32
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed-with-source-typo-repair",
        "contentHash": "2a7bcc3de891cae6803d09f6e105d0bbb90d689a5da3f6ef971911ec8fa0a9b2",
        "method": "All eight Boolean language-membership masks checked; pointwise weights0,2,3,3≤3 prove the inequality independently of any population sizes."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.38",
      "contentHash": "f770e1b24f0c1aaf2fbf415085c13bbfb97620ec8e0b762ddcd8fc41706d0999",
      "source": {
        "status": "passed-with-recorded-domain-or-typo-note",
        "textbookPdfPages": [
          96
        ],
        "answerPdfPages": [
          32
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "f770e1b24f0c1aaf2fbf415085c13bbfb97620ec8e0b762ddcd8fc41706d0999",
        "method": "Complement bound in a102-element universe: X,Y,Z miss2,2,1 elements, so at most5 elements miss the triple intersection; minimum97. Explicit supplied construction independently checked. Infinite-domain caveat recorded."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.39",
      "contentHash": "661bf94fcbb8a0a4ea594c069b2db744a79183d1abf6e7af7130456fe496cc3b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          96
        ],
        "answerPdfPages": [
          33
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "661bf94fcbb8a0a4ea594c069b2db744a79183d1abf6e7af7130456fe496cc3b",
        "method": "Independent occupancy DP: after each labeled object, place it in an occupied box or one of the9−occupied empty boxes. All-nonempty counts are0 for k1..8 and362880 for k9; checked against exact alternating expression."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.40",
      "contentHash": "cfab476a93df56deacb7c105c65cef6c1c03c2a51314b5d78df25fd4cb456cee",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          97
        ],
        "answerPdfPages": [
          33,
          34
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "cfab476a93df56deacb7c105c65cef6c1c03c2a51314b5d78df25fd4cb456cee",
        "method": "Generated actual integer divisor sets from all prime-exponent combinations and took their union;435 distinct integers."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.41",
      "contentHash": "9ede501396c036fc7ae0cccb45cb88636f86ed1e5dcc744ac01d3d6915e1e403",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          97
        ],
        "answerPdfPages": [
          34,
          35
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "9ede501396c036fc7ae0cccb45cb88636f86ed1e5dcc744ac01d3d6915e1e403",
        "method": "Enumerated all permutations n0..8, checked derangement counts against PIE, and checked the fixed-point partition identity for every n0..8."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.42",
      "contentHash": "6a670f8fa67b2b286aaa9bb01c60bc381ee73477b83e2801d2e30cd76ee64e75",
      "source": {
        "status": "passed-with-recorded-domain-or-typo-note",
        "textbookPdfPages": [
          97
        ],
        "answerPdfPages": [
          35
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "6a670f8fa67b2b286aaa9bb01c60bc381ee73477b83e2801d2e30cd76ee64e75",
        "method": "Enumerated all1024 orientations of the complete graph on five teams and counted those with every outdegree in1..3;544 qualify, yielding17/32."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.43",
      "contentHash": "5156a08240db32059707b407a92dc2ea1e5721a918ad7d3e8389ef56f327ae84",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          97
        ],
        "answerPdfPages": [
          35,
          36
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "5156a08240db32059707b407a92dc2ea1e5721a918ad7d3e8389ef56f327ae84",
        "method": "Direct gcd enumeration for every n1..500 compared against exact rational product formula; all match, including n1 empty-product boundary. Algebraic expansion proves all positive integers."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "3.44",
      "contentHash": "a906ecbc4b2247cff5771f804eb12732c864dfb5a4d5467caabc31a8da128c37",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          97
        ],
        "answerPdfPages": [
          36,
          37
        ],
        "method": "Original textbook and answer scans inspected. Binomial, exponent, factorial and sign OCR repaired; source sign typo3.37 and finite-set domain3.38 explicitly recorded; source star figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "a906ecbc4b2247cff5771f804eb12732c864dfb5a4d5467caabc31a8da128c37",
        "method": "Direct gcd check of every step2..499 for n1000 gives199. Independently checked orbit length n/gcd(a,n), primitive full tours and smaller n3..30 examples; constant-turn closure proves classification."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.1",
      "contentHash": "fcf8778624b08fff12fc58c64cafff7249a6e89e2f5d1b77f279c0b575a026ce",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          99
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; lost grid reconstructed, fractions/binomials and sequential rectangle probability restored. Both supplied alternative methods retained where present."
      },
      "math": {
        "status": "passed",
        "contentHash": "fcf8778624b08fff12fc58c64cafff7249a6e89e2f5d1b77f279c0b575a026ce",
        "method": "Independent positional radix count: four base10 positions and three base26 positions give exactly10⁴·26³ distinct strings, with a unique encoding and decoding."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.2",
      "contentHash": "295c74ea694882c1f47924647edc7130ff9d611f94afee72730a1b0cae838924",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          99,
          100
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; lost grid reconstructed, fractions/binomials and sequential rectangle probability restored. Both supplied alternative methods retained where present."
      },
      "math": {
        "status": "passed",
        "contentHash": "295c74ea694882c1f47924647edc7130ff9d611f94afee72730a1b0cae838924",
        "method": "Enumerated all integers10000..99999 and compared their strings with their reversals; exactly900 palindromes."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.3",
      "contentHash": "b3f8e5fd7d776eba539d8edaec071628bc0791e484ee38728cf8ab8aac4f9dfc",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          99,
          100
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; lost grid reconstructed, fractions/binomials and sequential rectangle probability restored. Both supplied alternative methods retained where present."
      },
      "math": {
        "status": "passed",
        "contentHash": "b3f8e5fd7d776eba539d8edaec071628bc0791e484ee38728cf8ab8aac4f9dfc",
        "method": "Independent digit DP on length and last digit, excluding leading0 and transitions to the same digit; seven-digit total4782969."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.4",
      "contentHash": "3d8fbb2e4419c6b4cb2bbb22c8912108205fa59a2ce2c71e65dd57e557242075",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          99,
          100,
          101
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; lost grid reconstructed, fractions/binomials and sequential rectangle probability restored. Both supplied alternative methods retained where present."
      },
      "math": {
        "status": "passed",
        "contentHash": "3d8fbb2e4419c6b4cb2bbb22c8912108205fa59a2ce2c71e65dd57e557242075",
        "method": "Enumerated all12650 four-subsets of25 labeled lattice points; checked whether their coordinate set equals the Cartesian product of two x and two y values. Exactly100 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.2.1",
      "contentHash": "6bae4143088b383eeb571c9e4942d972b8e9833c6d7efcfb84af7816ca1817e1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          101
        ],
        "answerPdfPages": [
          38
        ],
        "method": "Original scans inspected; lost grid reconstructed, fractions/binomials and sequential rectangle probability restored. Both supplied alternative methods retained where present."
      },
      "math": {
        "status": "passed",
        "contentHash": "6bae4143088b383eeb571c9e4942d972b8e9833c6d7efcfb84af7816ca1817e1",
        "method": "Placed15 exact rational boundary points on a nondegenerate right triangle and tested every455 triple by a zero determinant;357 are noncollinear. Any non-side line meets the convex triangle boundary in at most two points."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.2.2",
      "contentHash": "ec0d6d4a7fd2240d00fe62f73cb83e4a4031885446b4bd26f04471222bcbc910",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          101
        ],
        "answerPdfPages": [
          38
        ],
        "method": "Original scans inspected; lost grid reconstructed, fractions/binomials and sequential rectangle probability restored. Both supplied alternative methods retained where present."
      },
      "math": {
        "status": "passed",
        "contentHash": "ec0d6d4a7fd2240d00fe62f73cb83e4a4031885446b4bd26f04471222bcbc910",
        "method": "Directly enumerated all5984 three-subsets of1..34 and tested their sum modulo3;2002 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.2.3",
      "contentHash": "2dfb2f53035d5d069a44ce6cd78cb33c068538baf31d7badcdc7c8739926fb07",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          101
        ],
        "answerPdfPages": [
          38
        ],
        "method": "Original scans inspected; lost grid reconstructed, fractions/binomials and sequential rectangle probability restored. Both supplied alternative methods retained where present."
      },
      "math": {
        "status": "passed",
        "contentHash": "2dfb2f53035d5d069a44ce6cd78cb33c068538baf31d7badcdc7c8739926fb07",
        "method": "Enumerated distinct permutations of the half-multiset I,I,S,S,P, reflected around M, and checked30 unique palindromes with exactly the MISSISSIPPI letter counts."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.2.4",
      "contentHash": "c803fdc26c3b617ff85945e0d8a89ecc7ce2c105d5bc94aec7b3be27752c2001",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          101
        ],
        "answerPdfPages": [
          38,
          39
        ],
        "method": "Original scans inspected; lost grid reconstructed, fractions/binomials and sequential rectangle probability restored. Both supplied alternative methods retained where present."
      },
      "math": {
        "status": "passed",
        "contentHash": "c803fdc26c3b617ff85945e0d8a89ecc7ce2c105d5bc94aec7b3be27752c2001",
        "method": "Enumerated all1680 ordered partitions into three unlabeled-within-hand triples; directly summed tile labels and found360 deals with three odd hand sums."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.2.5",
      "contentHash": "65ef86b2e5d47a2e75f7180b054e6ad7e8106c23681ffb59d3603d578ff8dc72",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          101
        ],
        "answerPdfPages": [
          39
        ],
        "method": "Original scans inspected; lost grid reconstructed, fractions/binomials and sequential rectangle probability restored. Both supplied alternative methods retained where present."
      },
      "math": {
        "status": "passed",
        "contentHash": "65ef86b2e5d47a2e75f7180b054e6ad7e8106c23681ffb59d3603d578ff8dc72",
        "method": "Enumerated every70 metal-position pattern and all256 face-orientation patterns. Rejected a lower face-up followed by upper face-down; exactly9 orientations and630 total stacks survive."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.5",
      "contentHash": "4cb215e2c85b49af8b6e625d5b5a5692f71027b621aa40e074be940851ad1f82",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          101,
          102,
          103
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; corrupted digit-column sum, exponent3³, binomials and stack-list sums repaired. All supplied methods and scaffold subparts retained; trailing-block and classroom interpretations explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "4cb215e2c85b49af8b6e625d5b5a5692f71027b621aa40e074be940851ad1f82",
        "method": "Directly enumerated all900 five-digit palindromes, summed them to49500000 and separately recovered digit-column sums4500,4050,4050,4050,4500."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.6",
      "contentHash": "65a2a8be1055f821eb7564871a68dd2d3d0662529c90bef4ae0d014f6dce9548",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          102,
          103,
          104
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; corrupted digit-column sum, exponent3³, binomials and stack-list sums repaired. All supplied methods and scaffold subparts retained; trailing-block and classroom interpretations explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "65a2a8be1055f821eb7564871a68dd2d3d0662529c90bef4ae0d014f6dce9548",
        "method": "Exact hypergeometric fraction independently computed. Guarantee impossibility proven constructively by hitting sets: arbitrary≤10 representatives, or a three-ticket number, or two doubled numbers covering four disjoint tickets. Repeated selected representatives are explicitly deduplicated then padded."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.7",
      "contentHash": "d877937d06d8467bed1104c95de178b9141fd37a12ca3af93e94c3328e6e9a01",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          102,
          104,
          105
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; corrupted digit-column sum, exponent3³, binomials and stack-list sums repaired. All supplied methods and scaffold subparts retained; trailing-block and classroom interpretations explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "d877937d06d8467bed1104c95de178b9141fd37a12ca3af93e94c3328e6e9a01",
        "method": "Independently enumerated every permutation of digits0..9 with nonzero leading digit and tested divisibility by11111;3456 qualify. Also directly listed all distinct-digit two-digit multiples of9."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.8",
      "contentHash": "e2932a2e20e7bc35545a605c1d939b3d9f68a1ceebf5b0b49298212fbdcd04e9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          102,
          106,
          107,
          108
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; corrupted digit-column sum, exponent3³, binomials and stack-list sums repaired. All supplied methods and scaffold subparts retained; trailing-block and classroom interpretations explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "e2932a2e20e7bc35545a605c1d939b3d9f68a1ceebf5b0b49298212fbdcd04e9",
        "method": "Generated all16796 legal stack-pop orders for ten arrivals and every cut after racket9 had been popped. The set of distinct remaining suffixes has1536 elements, independently of the subset-sum formula."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.3.1",
      "contentHash": "5fce04d710e5906fa6dec02afe8a181e27890e41954a39969c586685aa8d76cc",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          108
        ],
        "answerPdfPages": [
          39
        ],
        "method": "Original scans inspected; corrupted digit-column sum, exponent3³, binomials and stack-list sums repaired. All supplied methods and scaffold subparts retained; trailing-block and classroom interpretations explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "5fce04d710e5906fa6dec02afe8a181e27890e41954a39969c586685aa8d76cc",
        "method": "Independent prefix enumeration count2^ceil(n/2)−2 for n1..29; first count exceeding2002 occurs at21 with2046, previous length20 has1022."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.3.2",
      "contentHash": "248bdab8d6a9121e6b8b98a7cf4deb41735033c22a8ebebfc271c0cea8683471",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          108
        ],
        "answerPdfPages": [
          39,
          40
        ],
        "method": "Original scans inspected; corrupted digit-column sum, exponent3³, binomials and stack-list sums repaired. All supplied methods and scaffold subparts retained; trailing-block and classroom interpretations explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "248bdab8d6a9121e6b8b98a7cf4deb41735033c22a8ebebfc271c0cea8683471",
        "method": "Enumerated all3¹⁰ colorings and tested all25 opposite-parity pairs;186 satisfy every restriction."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.3.3",
      "contentHash": "c4c70fcc36d03b7b66ec0be2385744bd88d3b8e47b34b72933580f3ad6a77a76",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          108
        ],
        "answerPdfPages": [
          40
        ],
        "method": "Original scans inspected; corrupted digit-column sum, exponent3³, binomials and stack-list sums repaired. All supplied methods and scaffold subparts retained; trailing-block and classroom interpretations explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "c4c70fcc36d03b7b66ec0be2385744bd88d3b8e47b34b72933580f3ad6a77a76",
        "method": "Independent maximum-independent-set recursion on25 vertices, with graph edges at differences1,4,9,16, returns10; the exhibited ten-element subset also tested pairwise."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.3.4",
      "contentHash": "712628ed1adcfb864e11f4f8e282073b2ebcb8fd7bbc72b013f5e37397e03abb",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          108
        ],
        "answerPdfPages": [
          40
        ],
        "method": "Original scans inspected; corrupted digit-column sum, exponent3³, binomials and stack-list sums repaired. All supplied methods and scaffold subparts retained; trailing-block and classroom interpretations explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "712628ed1adcfb864e11f4f8e282073b2ebcb8fd7bbc72b013f5e37397e03abb",
        "method": "Enumerated actual positive divisors of21600 and directly tested math.lcm(a,b) on every ordered divisor pair;385 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.3.5",
      "contentHash": "eadafb4753c67c7c7d2deff21434cf14c88c4ec632182f4ccc27a0b7c17805f7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          108
        ],
        "answerPdfPages": [
          40
        ],
        "method": "Original scans inspected; corrupted digit-column sum, exponent3³, binomials and stack-list sums repaired. All supplied methods and scaffold subparts retained; trailing-block and classroom interpretations explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "eadafb4753c67c7c7d2deff21434cf14c88c4ec632182f4ccc27a0b7c17805f7",
        "method": "Enumerated all720 permutations of digits1..6 and tested every prefix modulo its length; exactly123654 and321654 pass."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.3.6",
      "contentHash": "7a64be2a52ee7e89968ba6e1c59518e869d4349c090a7b25dfee20f716597ac7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          108
        ],
        "answerPdfPages": [
          40,
          41
        ],
        "method": "Original scans inspected; corrupted digit-column sum, exponent3³, binomials and stack-list sums repaired. All supplied methods and scaffold subparts retained; trailing-block and classroom interpretations explicit."
      },
      "math": {
        "status": "passed-with-stated-interpretation",
        "contentHash": "7a64be2a52ee7e89968ba6e1c59518e869d4349c090a7b25dfee20f716597ac7",
        "method": "Directly enumerated all occupancy boards for sizes1×1 through4×4 and tested the occupied-desk condition, recovering1+(2ⁿ−1)². General proof: nonempty empty-cell set equals its row projection × column projection; distinct nonempty projections give unique boards."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.9",
      "contentHash": "94fcaf91584312fba2cb052b26ecb8175522fd154cb2174e82bad98547f19473",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          110,
          111
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; power-set symbols, factorials, missing empty-set notation and inverse-function primes repaired. Source4-by-3 grids and RRURUUR path reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "94fcaf91584312fba2cb052b26ecb8175522fd154cb2174e82bad98547f19473",
        "method": "Explicit image sets and inverses checked for translation,squaring and ten listed distinct pairs; enumerated even integers and two-digit radix strings to confirm10 vs11 and90 vs56."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.10",
      "contentHash": "ca636a429419f7fcf1700e0d5372755d49fddba354ceca4450154e26da1f37cf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          110,
          111,
          112
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; power-set symbols, factorials, missing empty-set notation and inverse-function primes repaired. Source4-by-3 grids and RRURUUR path reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "ca636a429419f7fcf1700e0d5372755d49fddba354ceca4450154e26da1f37cf",
        "method": "Enumerated all35 choices of three U positions, decoded each to a distinct coordinate path, then re-encoded each path to recover exactly the original word."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.11",
      "contentHash": "4da416b0ece8ffe9120b7832ad7f8003cd277d6717d99a7db5c968da92264da6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          110,
          112
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; power-set symbols, factorials, missing empty-set notation and inverse-function primes repaired. Source4-by-3 grids and RRURUUR path reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "4da416b0ece8ffe9120b7832ad7f8003cd277d6717d99a7db5c968da92264da6",
        "method": "Enumerated every increasing four-tuple of distinct letters, giving14950 unique sorted words. Each decodes to a four-element set and sorts back to itself."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.4.1",
      "contentHash": "ba28368b43d29b8ddb289e3c0ce1f2819ea9e4a636c0abb2413bc003c9042f06",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          113
        ],
        "answerPdfPages": [
          41
        ],
        "method": "Original scans inspected; power-set symbols, factorials, missing empty-set notation and inverse-function primes repaired. Source4-by-3 grids and RRURUUR path reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "ba28368b43d29b8ddb289e3c0ce1f2819ea9e4a636c0abb2413bc003c9042f06",
        "method": "Enumerated all domains and explicit mapping images; binary/subset round trips for all16 values, pair matching on ten values, digit substitution on648 three-digit integers, and two-digit set sizes72 and81."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.4.2",
      "contentHash": "1511ca05aea195de4afaebcaa1b6e3c257e7a49538a6ef92f36bbeacd4fa1fd1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          113
        ],
        "answerPdfPages": [
          41
        ],
        "method": "Original scans inspected; power-set symbols, factorials, missing empty-set notation and inverse-function primes repaired. Source4-by-3 grids and RRURUUR path reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "1511ca05aea195de4afaebcaa1b6e3c257e7a49538a6ef92f36bbeacd4fa1fd1",
        "method": "Enumerated all26³ letter triples and directly tested that the middle index exceeds each outer index;5525 qualify. Independently summed m² for middle index m=0..25."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.4.3",
      "contentHash": "f9bca0e1815ee95cdd02d26a0e84d66460dca1f21267d984b76e7cc359183088",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          113
        ],
        "answerPdfPages": [
          41,
          42
        ],
        "method": "Original scans inspected; power-set symbols, factorials, missing empty-set notation and inverse-function primes repaired. Source4-by-3 grids and RRURUUR path reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "f9bca0e1815ee95cdd02d26a0e84d66460dca1f21267d984b76e7cc359183088",
        "method": "Independent three-dimensional lattice DP, adding counts from each predecessor and setting the forbidden midpoint count to0, gives26550; unrestricted DP gives34650."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.4.4",
      "contentHash": "10031097d2fe549f190276ad6f364c9db885b0a6522de2ea3d38c3be1dca2f76",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          113
        ],
        "answerPdfPages": [
          42
        ],
        "method": "Original scans inspected; power-set symbols, factorials, missing empty-set notation and inverse-function primes repaired. Source4-by-3 grids and RRURUUR path reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "10031097d2fe549f190276ad6f364c9db885b0a6522de2ea3d38c3be1dca2f76",
        "method": "General composition/inverse proof checked directly; compositions of every pair of permutations on sets of size0..5 exhaustively tested as finite instances."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.4.5",
      "contentHash": "d6db7513746165dd66a1f3246c10788a0d21d5fa8f38a3144338e02967d5a69d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          113
        ],
        "answerPdfPages": [
          42
        ],
        "method": "Original scans inspected; power-set symbols, factorials, missing empty-set notation and inverse-function primes repaired. Source4-by-3 grids and RRURUUR path reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "d6db7513746165dd66a1f3246c10788a0d21d5fa8f38a3144338e02967d5a69d",
        "method": "Pigeonhole/cardinality proof: a proper finite subset has at most n−1 elements, which cannot be covered injectively by n domain elements. Empty-set boundary and infinite counterexample checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.4.6",
      "contentHash": "f6d0332c3f67642c05a765ab911201497ab1f1a83a67d395f46c41fd94267526",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          113
        ],
        "answerPdfPages": [
          42
        ],
        "method": "Original scans inspected; power-set symbols, factorials, missing empty-set notation and inverse-function primes repaired. Source4-by-3 grids and RRURUUR path reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "f6d0332c3f67642c05a765ab911201497ab1f1a83a67d395f46c41fd94267526",
        "method": "For every n1..1000, directly enumerated odd/even divisors and checked equality exactly when n%4==2. General factor-of-two layering proof establishes all positive n."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.12",
      "contentHash": "0288c3630bff983c352d0bad4a62062246a6eb9810f33aae20c9e71d41adbad4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          114,
          115,
          116
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; lost product2¹²3⁸5⁶, binomial diagrams,61-variable sum and pairing fractions restored; supplied index endpoints explicitly corrected; Pascal table rebuilt.",
        "chapter04FreshFidelity": "All textbook98–132/manual38–57 visually compared; missing intermediate reference figures/tables restored as native HTML/SVG."
      },
      "math": {
        "status": "passed",
        "contentHash": "0288c3630bff983c352d0bad4a62062246a6eb9810f33aae20c9e71d41adbad4",
        "method": "Exact rational intersection computation on strictly convex parabolic vertex sets n3..9 gives C(n,4) distinct crossings. Each interleaved pair of diagonals has a unique four-endpoint subset. Finite-degeneracy perturbation argument establishes attainability for all n."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.13",
      "contentHash": "cf2ad0860414790a2bb71a10bdf587fb389905c136778b9554c2fba58eccd0fa",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          114,
          116,
          117
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; lost product2¹²3⁸5⁶, binomial diagrams,61-variable sum and pairing fractions restored; supplied index endpoints explicitly corrected; Pascal table rebuilt."
      },
      "math": {
        "status": "passed",
        "contentHash": "cf2ad0860414790a2bb71a10bdf587fb389905c136778b9554c2fba58eccd0fa",
        "method": "Enumerated all5525 exponent triples of divisors of the actual n² and compared their integer values with n;2762 below, one equal,2762 above. Checked the pairing d↦n²/d exactly."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.14",
      "contentHash": "fa7cf65ab81803fb40c6efc90a917d717e5f0658d9df121bd532db9fa166ab9c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          114,
          117,
          118
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; lost product2¹²3⁸5⁶, binomial diagrams,61-variable sum and pairing fractions restored; supplied index endpoints explicitly corrected; Pascal table rebuilt."
      },
      "math": {
        "status": "passed",
        "contentHash": "fa7cf65ab81803fb40c6efc90a917d717e5f0658d9df121bd532db9fa166ab9c",
        "method": "Direct divisor checks for odd k1..199 confirm the exact2-to-1 even/odd count; general unique factorization into2-adic layers proves the result for every positive odd k."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.15",
      "contentHash": "501348bcb54d8b08969bb7944b85ced522a5372db29049feb461b3d95ddee26a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          114,
          118,
          119
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; lost product2¹²3⁸5⁶, binomial diagrams,61-variable sum and pairing fractions restored; supplied index endpoints explicitly corrected; Pascal table rebuilt."
      },
      "math": {
        "status": "passed",
        "contentHash": "501348bcb54d8b08969bb7944b85ced522a5372db29049feb461b3d95ddee26a",
        "method": "General injection obtained by adjoining a distinguished member, with explicit inverse on its image and a nonempty omitted class. Pascal difference verified for all small n3..14 and applicable k; exact integer identity also holds for100,11."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.5.1",
      "contentHash": "38b9688049b28e5cce3987898206dc79dd7e45c3b1e1bdc5d3fff9e772e0d976",
      "source": {
        "status": "passed-with-recorded-assumption-or-repair",
        "textbookPdfPages": [
          119
        ],
        "answerPdfPages": [
          43
        ],
        "method": "Original scans inspected; lost product2¹²3⁸5⁶, binomial diagrams,61-variable sum and pairing fractions restored; supplied index endpoints explicitly corrected; Pascal table rebuilt."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "38b9688049b28e5cce3987898206dc79dd7e45c3b1e1bdc5d3fff9e772e0d976",
        "method": "Enumerated all64×64 pairs of six-step binary move sequences and directly compared endpoint coordinates;792 meet. Coordinate-sum invariant rules out all other meeting times."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.5.2",
      "contentHash": "863b6c887f5b4e25782b0ba2900934e629b3317682dfa27b55aa96fffd18bb46",
      "source": {
        "status": "passed-with-recorded-assumption-or-repair",
        "textbookPdfPages": [
          119
        ],
        "answerPdfPages": [
          43,
          44
        ],
        "method": "Original scans inspected; lost product2¹²3⁸5⁶, binomial diagrams,61-variable sum and pairing fractions restored; supplied index endpoints explicitly corrected; Pascal table rebuilt."
      },
      "math": {
        "status": "passed-with-index-repair",
        "contentHash": "863b6c887f5b4e25782b0ba2900934e629b3317682dfa27b55aa96fffd18bb46",
        "method": "Independent nested-sum DP over upper bounds0,1,2, repeated61 times, yields1953. Enumerating all nonnegative zero/one/two block-length triples summing61 gives the same count."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.5.3",
      "contentHash": "2c71cc111a81adc314356deac30b01cab0e11078cd0291e6fd15be1d116ef2a7",
      "source": {
        "status": "passed-with-recorded-assumption-or-repair",
        "textbookPdfPages": [
          119
        ],
        "answerPdfPages": [
          44
        ],
        "method": "Original scans inspected; lost product2¹²3⁸5⁶, binomial diagrams,61-variable sum and pairing fractions restored; supplied index endpoints explicitly corrected; Pascal table rebuilt."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "2c71cc111a81adc314356deac30b01cab0e11078cd0291e6fd15be1d116ef2a7",
        "method": "Exact convolution DP on101 dice verifies the total count6¹⁰¹, the strict half-tail at353, a positive count at354, and a smaller tail above354. Pointwise complement maps prove both symmetry claims."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.5.4",
      "contentHash": "4425aefd394baeb3b3a3d159b1cac6381fba50ac7ae1d593973c1887eac3ecd1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          119
        ],
        "answerPdfPages": [
          44
        ],
        "method": "Original scans inspected; lost product2¹²3⁸5⁶, binomial diagrams,61-variable sum and pairing fractions restored; supplied index endpoints explicitly corrected; Pascal table rebuilt.",
        "chapter04FreshFidelity": "All textbook98–132/manual38–57 visually compared; missing intermediate reference figures/tables restored as native HTML/SVG."
      },
      "math": {
        "status": "passed",
        "contentHash": "4425aefd394baeb3b3a3d159b1cac6381fba50ac7ae1d593973c1887eac3ecd1",
        "method": "For generic strictly convex point sets n3..9, enumerated all triples of chords and tested all three pairwise crossings. Counts equal C(n,6). Exact rational generic geometry confirms no triple concurrence; cyclic six-endpoint interleaving proves the classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.16",
      "contentHash": "5c666a67b8c40e3fe987a6f595b24c48343aeb65c9bf75a6d85baa0b9d40734a",
      "source": {
        "status": "passed-with-explicit-domain",
        "textbookPdfPages": [
          120,
          121,
          122,
          123
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired.",
        "chapter04FreshFidelity": "All textbook98–132/manual38–57 visually compared; missing intermediate reference figures/tables restored as native HTML/SVG."
      },
      "math": {
        "status": "passed",
        "contentHash": "5c666a67b8c40e3fe987a6f595b24c48343aeb65c9bf75a6d85baa0b9d40734a",
        "method": "Breadth-first enumeration from the full5-by-7 board using every legal cookie chomp reaches792 unique profiles, exactly all weakly increasing five-tuples in0..7. Full and empty profiles included."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.17",
      "contentHash": "b4aa2d914e549661c6c9c4bdd979a17faf7112f68faa8cd5372ea7fed332c338",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          120,
          123
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "b4aa2d914e549661c6c9c4bdd979a17faf7112f68faa8cd5372ea7fed332c338",
        "method": "Recursive generation in nonincreasing order independently gives partition counts5,7,11 for4,5,6 and verifies each displayed list is complete and duplicate-free."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.18",
      "contentHash": "b8a9c4fe3a8622be499689df0307f954463426fe275703eaf5e819e5fa157b60",
      "source": {
        "status": "passed-with-explicit-domain",
        "textbookPdfPages": [
          120,
          124,
          125
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired.",
        "chapter04FreshFidelity": "All textbook98–132/manual38–57 visually compared; missing intermediate reference figures/tables restored as native HTML/SVG."
      },
      "math": {
        "status": "passed",
        "contentHash": "b8a9c4fe3a8622be499689df0307f954463426fe275703eaf5e819e5fa157b60",
        "method": "Generated all three-part partitions for n1..24 and all restricted three-part partitions of2n; complementing/sorting gives exact equality of image and target for every n, and round trips restore all partitions."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.19",
      "contentHash": "d889c432c9fa4d48339eb688fa3068eb4143b1a9a8800a0ffbe88347355bd5dd",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          120,
          125,
          126,
          127
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired.",
        "chapter04FreshFidelity": "All textbook98–132/manual38–57 visually compared; missing intermediate reference figures/tables restored as native HTML/SVG."
      },
      "math": {
        "status": "passed",
        "contentHash": "d889c432c9fa4d48339eb688fa3068eb4143b1a9a8800a0ffbe88347355bd5dd",
        "method": "For all partitions of n1..24, conjugating row lengths preserves total and is an involution. For each r, image of exactly-r-part partitions is precisely the largest-part-r set."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.20",
      "contentHash": "a223a5028b0350000ad3fb4902fcc8db364c821d7f402a38228aa7bc7ce4fe82",
      "source": {
        "status": "passed-with-explicit-domain",
        "textbookPdfPages": [
          121,
          127,
          128
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired."
      },
      "math": {
        "status": "passed-under-stated-assumptions",
        "contentHash": "a223a5028b0350000ad3fb4902fcc8db364c821d7f402a38228aa7bc7ce4fe82",
        "method": "Enumerated every pair of binary flip strings for Steve0..6 coins and Marissa one more; exactly half are strict wins. Pointwise complement inequality proves all m, including499."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.6.1",
      "contentHash": "dc1824fca64aea8c6e98070e2ed9ec1f644634a202f39be76cc67229a2ff7551",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          128
        ],
        "answerPdfPages": [
          44
        ],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "dc1824fca64aea8c6e98070e2ed9ec1f644634a202f39be76cc67229a2ff7551",
        "method": "Breadth-first move enumeration for1×1,2×3,3×4,5×7 boards matches all monotone profiles and binomial counts; general boundary-path encoding and its inverse establish all positive m,n."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.6.2",
      "contentHash": "3e64fdedae8fd5887aba9248b2148789c98dd96e926ccd16843bb272548c9702",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          128
        ],
        "answerPdfPages": [
          44,
          45
        ],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "3e64fdedae8fd5887aba9248b2148789c98dd96e926ccd16843bb272548c9702",
        "method": "Enumerated all partitions n1..24; conjugation maps exactly the all-even-part set onto the even-multiplicity set and is its own inverse. The24-dot source example is checked with all four final2s restored."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.6.3",
      "contentHash": "4fd572152a8bac4e7ce4bf1f4656570a7f0cbbc0e8786228ab9572da381fb135",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          128
        ],
        "answerPdfPages": [
          45
        ],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "4fd572152a8bac4e7ce4bf1f4656570a7f0cbbc0e8786228ab9572da381fb135",
        "method": "Integer feasibility search gives the unique pair(N,x)=(6,6). Independently constructed a10×6 Boolean incidence matrix with first nine row sums4, last row sum6 and every column sum7."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.6.4",
      "contentHash": "2441f49919e8114e538d085a73ffed41d5153925ee46f51068e672deab3d46cf",
      "source": {
        "status": "passed-with-explicit-domain",
        "textbookPdfPages": [
          128
        ],
        "answerPdfPages": [
          45,
          46
        ],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired.",
        "chapter04FreshFidelity": "All textbook98–132/manual38–57 visually compared; missing intermediate reference figures/tables restored as native HTML/SVG."
      },
      "math": {
        "status": "passed",
        "contentHash": "2441f49919e8114e538d085a73ffed41d5153925ee46f51068e672deab3d46cf",
        "method": "Enumerated all integer starts i,j and positive side lengths a,b with i+j+a+b≤n for n1..10. Tripling the orientation count matches3C(n+2,4), including15 for n3."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.6.5",
      "contentHash": "28a97305d982bb81a1936b92940f4bc63e948be70fe5bd3da045bd91c620c958",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          128
        ],
        "answerPdfPages": [
          46,
          47
        ],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired.",
        "chapter04FreshFidelity": "All textbook98–132/manual38–57 visually compared; missing intermediate reference figures/tables restored as native HTML/SVG."
      },
      "math": {
        "status": "passed",
        "contentHash": "28a97305d982bb81a1936b92940f4bc63e948be70fe5bd3da045bd91c620c958",
        "method": "Exact integer computation for n1..1000 verifies the floor sum and its interpretation as counts of exact2-adic valuation classes; general disjoint-partition and telescoping proofs checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.6.6",
      "contentHash": "f5ecdb8c7407f1dbe8063c6f12b1ae7fdd3f5a0749abb53efd184c883cc41f61",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          129
        ],
        "answerPdfPages": [
          47
        ],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired.",
        "chapter04FreshFidelity": "All textbook98–132/manual38–57 visually compared; missing intermediate reference figures/tables restored as native HTML/SVG."
      },
      "math": {
        "status": "passed",
        "contentHash": "f5ecdb8c7407f1dbe8063c6f12b1ae7fdd3f5a0749abb53efd184c883cc41f61",
        "method": "For every n1..24, generated all even-part partitions and their images, verified image uniqueness and inverse pairing. General injection plus the explicit omitted partition2003+3 proves the2006 comparison without computing huge partition counts."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.6.7",
      "contentHash": "5edd4c4a131cdb344e22fa507dcb9be1f2bbca685301d1a7d73875171b3b0a52",
      "source": {
        "status": "passed-with-explicit-domain",
        "textbookPdfPages": [
          129
        ],
        "answerPdfPages": [
          47,
          48
        ],
        "method": "Original scans inspected; Chomp boundary and Ferrers dot figures rebuilt, complete source partition lists retained, triangular grid restored, floor powers and missing repeated parts repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "5edd4c4a131cdb344e22fa507dcb9be1f2bbca685301d1a7d73875171b3b0a52",
        "method": "Enumerated every nonempty subset for n1..16, checked integer-mean qualification and n+k parity, and verified toggling the mean pairs all nonsingletons while preserving the mean. General sum/size argument proves all n."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.21",
      "contentHash": "d684538bfb7280ce043745b411592ffd96532b5a5baf5f55fbbf112cbe473cc6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          130
        ],
        "answerPdfPages": [
          48
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "d684538bfb7280ce043745b411592ffd96532b5a5baf5f55fbbf112cbe473cc6",
        "method": "Directly enumerated all729 triples in1..9 and tested their integer product modulo20;102 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.22",
      "contentHash": "ffdc3c9cb1cb6b18740673a1bf5fe40f2883c270d31214ba0c72ca17f4c4ece5",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          130
        ],
        "answerPdfPages": [
          48
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "ffdc3c9cb1cb6b18740673a1bf5fe40f2883c270d31214ba0c72ca17f4c4ece5",
        "method": "Enumerated all4095 nonempty subsets and directly compared minimum+maximum with13;1365 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.23",
      "contentHash": "6e91fc5721838955d00133159ecf2a6c51689ec392463e705559c6b6a5ea7198",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          130
        ],
        "answerPdfPages": [
          49
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "6e91fc5721838955d00133159ecf2a6c51689ec392463e705559c6b6a5ea7198",
        "method": "Enumerated all64 face-color bit patterns and tested each of the three choices of an opposite pair as top/bottom;20 permit monochromatic remaining four faces."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.24",
      "contentHash": "6add407a3fd7dbdae8f1a0be7fc8d522f80f43c7ad74cc72d0de72dc92ffaf82",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          130
        ],
        "answerPdfPages": [
          49
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "6add407a3fd7dbdae8f1a0be7fc8d522f80f43c7ad74cc72d0de72dc92ffaf82",
        "method": "Enumerated all64000 exponent triples; exactly1331 have all exponents≥29."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.25",
      "contentHash": "d6e77bb6152ea537b00f5a7c8665b4470a49599404ba38dc2832f1e88b82bd65",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          130
        ],
        "answerPdfPages": [
          49
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "d6e77bb6152ea537b00f5a7c8665b4470a49599404ba38dc2832f1e88b82bd65",
        "method": "Enumerated all91390 physical four-slip subsets and tested their label-frequency multiset for[2,2];1620 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.26",
      "contentHash": "aac63e372179c978963f5b46036c52258d20c3c7f3ff0ee6f3be02a74f5b2972",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          130
        ],
        "answerPdfPages": [
          49
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "aac63e372179c978963f5b46036c52258d20c3c7f3ff0ee6f3be02a74f5b2972",
        "method": "Enumerated all1680 labeled three-hand deals of nine distinct rolls and tested one of each type in every hand;216 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.27",
      "contentHash": "d2a48de09011abc971f2e00722ef052bdd07dad5733d0800f0b8c70466fca817",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          130
        ],
        "answerPdfPages": [
          50
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "d2a48de09011abc971f2e00722ef052bdd07dad5733d0800f0b8c70466fca817",
        "method": "Directly generated every lower-left coordinate and positive side length fitting in the point grid;285 distinct squares."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.28",
      "contentHash": "3314e128ac21d596ca405b47d6ca4b6071ccd97afba1d009388e0526c8b5d8d0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          130
        ],
        "answerPdfPages": [
          50
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "3314e128ac21d596ca405b47d6ca4b6071ccd97afba1d009388e0526c8b5d8d0",
        "method": "Generated and sorted all nonempty subsets of the first seven powers of4; the50th value is1284. Binary/base4 encoding order checked independently."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.29",
      "contentHash": "f8cf27ed8d1e8923a5157e091d2442c7a5eaa79bf22dfb6dead44dce489e4457",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          130
        ],
        "answerPdfPages": [
          50
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "f8cf27ed8d1e8923a5157e091d2442c7a5eaa79bf22dfb6dead44dce489e4457",
        "method": "General fixed-point-free toggle involution; all subset masks of ground-set sizes1..12 can be paired by XOR with one fixed bit, reversing parity."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.30",
      "contentHash": "97be0685ad7bc3478dafc24cefe192c6018e5144e0499a319adcdeebc944f406",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          131
        ],
        "answerPdfPages": [
          50
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "97be0685ad7bc3478dafc24cefe192c6018e5144e0499a319adcdeebc944f406",
        "method": "Ferrers conjugation preserves total and swaps number of parts with largest part. Partition enumeration through n24 verifies the restricted image sets for every relevant r."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.31",
      "contentHash": "c4ff9c5078273731cf35b518524ff6dde9da59e57a999af9d3bc2709b041145f",
      "source": {
        "status": "passed-transcription-with-recorded-repair",
        "textbookPdfPages": [
          131
        ],
        "answerPdfPages": [
          50
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed-with-source-phrase-repair",
        "contentHash": "c4ff9c5078273731cf35b518524ff6dde9da59e57a999af9d3bc2709b041145f",
        "method": "Independent dynamic programming by allowed part value and exact part count verified the recurrence for every1≤k≤n≤30; both inverse maps checked analytically."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.32",
      "contentHash": "ce277a08c024018ea8a863870bcb03b0657cb6ef7c75a3cebee3bf96b1c77749",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          131
        ],
        "answerPdfPages": [
          50,
          51
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "ce277a08c024018ea8a863870bcb03b0657cb6ef7c75a3cebee3bf96b1c77749",
        "method": "Independent geometric DFS enumerated simple Hamiltonian paths of the twelve points, rejecting segment crossings, overlaps and skipped intermediate marked points;252 normalized paths result."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.33",
      "contentHash": "858a2e2ca5ac995e4e63300ed8acdb4e1a1569bcab127bbfef24458673cf1d56",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          131
        ],
        "answerPdfPages": [
          51
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "858a2e2ca5ac995e4e63300ed8acdb4e1a1569bcab127bbfef24458673cf1d56",
        "method": "Enumerated each square’s knight-attack set; maximum attack count8, while both attackers of a corner have only6. The capacity contradiction proves there are no successful placements."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.34",
      "contentHash": "882865a4dd82060e3f976a245cbe70721c4445f5a47e4fa2f78e09d8d4c6612b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          131
        ],
        "answerPdfPages": [
          51,
          52
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "882865a4dd82060e3f976a245cbe70721c4445f5a47e4fa2f78e09d8d4c6612b",
        "method": "For each of1024 keys, enumerated all252 balanced answer strings and tested its minimum score; exactly22 keys have minimum≥4."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.35",
      "contentHash": "f24e9bdbaf69795c7352b8126f061f0c663e3f0c730650ec786ffd98de8d56f7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          131
        ],
        "answerPdfPages": [
          52
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "f24e9bdbaf69795c7352b8126f061f0c663e3f0c730650ec786ffd98de8d56f7",
        "method": "Enumerated all24 purple row-to-column permutations and all24 green permutations, rejected overlapping cells;216 pairs remain."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.36",
      "contentHash": "44a4535f808d8df63834fb79eac359bcf70bb9afe22201aac6bfa915c363b482",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          131
        ],
        "answerPdfPages": [
          52
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "44a4535f808d8df63834fb79eac359bcf70bb9afe22201aac6bfa915c363b482",
        "method": "Enumerated all6⁴ matrices with exactly two positive positions per row and checked each column count;90 have two positives in every column."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.37",
      "contentHash": "8bf967804d8108215e38b8072faec2df66aab4691eea1a9740dba0be8709eef9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          131
        ],
        "answerPdfPages": [
          52,
          53
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "8bf967804d8108215e38b8072faec2df66aab4691eea1a9740dba0be8709eef9",
        "method": "Independent row-transition DP on all2048 width11 bit rows propagated the exactly-two-white constraint through11 rows;total4094."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.38",
      "contentHash": "8b2338e63d3747eb81dcdb33500d00d7e3f7501425dd5ba0a1b184461845939c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          131
        ],
        "answerPdfPages": [
          53
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "8b2338e63d3747eb81dcdb33500d00d7e3f7501425dd5ba0a1b184461845939c",
        "method": "Enumerated all148995 four-edge subsets of K10 and tested the four possible three-edge subcollections for three-vertex triangles;5040 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.39",
      "contentHash": "40cb7d4eafea64f543cf32235b39140b1c273844173dffaabffc59357e5e1c23",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          132
        ],
        "answerPdfPages": [
          53
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "40cb7d4eafea64f543cf32235b39140b1c273844173dffaabffc59357e5e1c23",
        "method": "Generated every maximal positive-step progression in1..100, normalized each as its full tuple, and deduplicated;2500 distinct sets."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.40",
      "contentHash": "f98bf5dbe12dd665f42c2d46549652c922e812af5aefb6a4577a563d46ec1e74",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          132
        ],
        "answerPdfPages": [
          54
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "f98bf5dbe12dd665f42c2d46549652c922e812af5aefb6a4577a563d46ec1e74",
        "method": "Directly enumerated all1024 subsets, sorted each descending and computed its signed sum;their total is5120."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.41",
      "contentHash": "76a6c057f8635438060da2aa830be0e860c60c1eabcf8e3b1f50e8a0bba1c8fe",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          132
        ],
        "answerPdfPages": [
          54
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "76a6c057f8635438060da2aa830be0e860c60c1eabcf8e3b1f50e8a0bba1c8fe",
        "method": "Explicitly counted grid color classes18 and17 and removed the central minority-color seat. The18 students requiring opposite-color destinations exceed the17 possible chairs; no matching exists."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.42",
      "contentHash": "9dd5414f3c748b1e52206e483df1a5e7fd72c71208771fc469d85c1e6764a136",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          132
        ],
        "answerPdfPages": [
          54
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "9dd5414f3c748b1e52206e483df1a5e7fd72c71208771fc469d85c1e6764a136",
        "method": "Exact row/column subset count, with unique decreasing-rank matching, gives12869; Vandermonde identity checked by exact integer arithmetic."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.43",
      "contentHash": "4d37908d64013929bcfe05945bd1f14811d8fd5b36e847c183d4174d80e27678",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          132
        ],
        "answerPdfPages": [
          54,
          55
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "4d37908d64013929bcfe05945bd1f14811d8fd5b36e847c183d4174d80e27678",
        "method": "Generated all determinant+1 signed coordinate permutations of three axes and their action on the dual cube’s eight vertices;24 distinct rotations. Distinct colors make stabilizers trivial, so the exact orbit count is8!/24."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.44",
      "contentHash": "0aec8102765cc51b413bc92938f5787c8c7a166d9b07d0bdb8b2cb06af948870",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          132
        ],
        "answerPdfPages": [
          55
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "0aec8102765cc51b413bc92938f5787c8c7a166d9b07d0bdb8b2cb06af948870",
        "method": "Explicitly partitioned all3000 integers into odd-root doubling chains; summed ceil(length/2) to1999 and constructed1999 alternate nodes with no doubled pair."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.45",
      "contentHash": "aa5fd84d753db2b2b2dd33948d893f6d02520c1c9be7f232f8a9181a2d5a8e6f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          132
        ],
        "answerPdfPages": [
          55,
          56
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "aa5fd84d753db2b2b2dd33948d893f6d02520c1c9be7f232f8a9181a2d5a8e6f",
        "method": "Exhaustively enumerated all binary boards for n1..4 and tested even row/column sums, obtaining2^((n−1)²). General forced-edge construction and shared corner-parity proof establish the2007 case."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.46",
      "contentHash": "b996a476050cf76f7b0cd206cc68bc0f4186dcfb10eeb6f8b99cee1f98e25cda",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          132
        ],
        "answerPdfPages": [
          56
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "b996a476050cf76f7b0cd206cc68bc0f4186dcfb10eeb6f8b99cee1f98e25cda",
        "method": "Enumerated all20 three-rank subsets of six ordered distinct values and tested coordinatewise comparison with their complementary sorted triple;exactly five pass."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "4.47",
      "contentHash": "3b51ce2476cd5dec271018873f9f3799883234c11cd1a345e1a8fc60418b6665",
      "source": {
        "status": "passed-transcription-with-recorded-repair",
        "textbookPdfPages": [
          132
        ],
        "answerPdfPages": [
          56,
          57
        ],
        "method": "Original source and answer scans inspected; fractions, factorials, binary/base4 exponents and parity powers restored; source diagrams reconstructed, false phrase4.31 and binary-expansion gap4.47 explicitly repaired."
      },
      "math": {
        "status": "source-proof-gap-corrected",
        "contentHash": "3b51ce2476cd5dec271018873f9f3799883234c11cd1a345e1a8fc60418b6665",
        "method": "General explicit integer enumeration, exhaustive reduced-fraction enumeration scheme, and analytic decimal tail bound prove the statements. Exact equality1/2=.1000…₂=.0111…₂ identifies the source binary-encoding gap; decimal0/1 injection repairs it."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.1",
      "contentHash": "9db84e354158c2e16490cde617b3b545b77aeab7495407ca013a3159d096f462",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          133,
          134
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "9db84e354158c2e16490cde617b3b545b77aeab7495407ca013a3159d096f462",
        "method": "Enumerated all4⁵ labeled-ball assignments to four boxes; every occupancy vector has a coordinate≥2. Independent capacity proof: four boxes of capacity1 hold at most4 objects."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.2.1",
      "contentHash": "82d8db43fdb4423d04fd61254830395881589e3b821225fee45e6b540b8166e4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          134
        ],
        "answerPdfPages": [
          58
        ],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "82d8db43fdb4423d04fd61254830395881589e3b821225fee45e6b540b8166e4",
        "method": "General counting contradiction: an injective assignment from n objects into k boxes would require n≤k; the capacity sum establishes the result for every n>k≥1."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.2.2",
      "contentHash": "1240440a9f15d6d6c7f6df75dece37ba8264f11f237d483985df30c5eef9dfb6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          134
        ],
        "answerPdfPages": [
          58
        ],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "1240440a9f15d6d6c7f6df75dece37ba8264f11f237d483985df30c5eef9dfb6",
        "method": "Enumerated all eight H/T strings of length3; the larger of the two face counts is always at least2."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.2.3",
      "contentHash": "9695024069855f008d497bd99c1fbf92045785fefb4e00949a0da708b608bba8",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          134
        ],
        "answerPdfPages": [
          58
        ],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "9695024069855f008d497bd99c1fbf92045785fefb4e00949a0da708b608bba8",
        "method": "Capacity bound: a pair-free draw has at most one card of each of13 ranks; an actual13-card one-per-rank hand realizes that bound, so14 is both sufficient and necessary."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.2",
      "contentHash": "5e6515d8bcb0426331e0c0648988a431cedf9d4e3cbff41d4e7ea961f1c7c383",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          134,
          135
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "5e6515d8bcb0426331e0c0648988a431cedf9d4e3cbff41d4e7ea961f1c7c383",
        "method": "Enumerated all3⁴ color strings and found a repeat in each; the three-color string W,Brown,Black is a length3 counterexample."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.3",
      "contentHash": "b44a451d80572801c2087ad4391bd321816e2f5552afca77a0f170a966aca4d2",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          135,
          136
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "b44a451d80572801c2087ad4391bd321816e2f5552afca77a0f170a966aca4d2",
        "method": "For every six-tuple of residue classes modulo5, checked a repeated class exists. General difference calculation x−y=5(a−b) proves the integer statement, including negative values."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.4",
      "contentHash": "adfd8082f32071664372a9844a337752ff64db69bca9815553bb35ea5d149dd7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          135,
          136
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "adfd8082f32071664372a9844a337752ff64db69bca9815553bb35ea5d149dd7",
        "method": "Analytic coordinate bound on the diameter of a half-unit square. A consistent half-open quadrant assignment handles boundaries; the corners-plus-center configuration checks sharpness at equality."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.5",
      "contentHash": "111be4596be774c44c65911c4cbff5aca1fce7f063ee2a24c78e92d25465df96",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          135,
          136
        ],
        "answerPdfPages": [],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "111be4596be774c44c65911c4cbff5aca1fce7f063ee2a24c78e92d25465df96",
        "method": "General simple-graph degree argument excludes simultaneous degree0 and n−1. Exhaustive graph instances through six vertices independently confirm duplicate degrees."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.3.1",
      "contentHash": "2a73352642c55926761f20f9e5c7f5f60beec4340c414041dcc9e93ba0a7e6f4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          137
        ],
        "answerPdfPages": [
          58
        ],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "2a73352642c55926761f20f9e5c7f5f60beec4340c414041dcc9e93ba0a7e6f4",
        "method": "Ten possible final decimal-digit labels give the capacity proof. For negative integers, the last digit of the magnitude still belongs to the same ten-label domain; no congruence claim is needed here."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.3.2",
      "contentHash": "3cd2fdac4aecea265a002f0fd453c3b9eb17edf5974669646b083e94f7f92392",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          137
        ],
        "answerPdfPages": [
          58,
          59
        ],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "3cd2fdac4aecea265a002f0fd453c3b9eb17edf5974669646b083e94f7f92392",
        "method": "The rook construction and row upper bound agree. For bishops, checked seven disjoint diagonal classes per color cover the board and tested all pairs of the exhibited14 positions for unequal diagonal slopes."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.3.3",
      "contentHash": "a9e3a0e84ff71f7e3f598aad8d9db368d235bd60cfe7a3fe9cedcfdb743de2a4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          137
        ],
        "answerPdfPages": [
          59
        ],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "a9e3a0e84ff71f7e3f598aad8d9db368d235bd60cfe7a3fe9cedcfdb743de2a4",
        "method": "Enumerated every positive eight-part composition of20, represented by7 cuts among19 positions; a subset-sum bitset for each composition always contains sum4."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.3.4",
      "contentHash": "705598efeb66a34fc8cbd930672188a402c5e25bbb0b23d3ad97d8c6bbc182c7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          137
        ],
        "answerPdfPages": [
          59
        ],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "705598efeb66a34fc8cbd930672188a402c5e25bbb0b23d3ad97d8c6bbc182c7",
        "method": "Built complementary residue classes and checked their populations; independent maximum-set construction of23 elements tested for all distinct pair sums, while complete cross-class conflicts prove the matching upper bound."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.3.5",
      "contentHash": "54585bb8104ba41811c79f42376af3332fc183336b6efe1fccd59e36daaa9b43",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          137
        ],
        "answerPdfPages": [
          59
        ],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "54585bb8104ba41811c79f42376af3332fc183336b6efe1fccd59e36daaa9b43",
        "method": "General remainder equivalence proof and pigeonhole capacity argument; boundary n1 and zero/negative differences explicitly checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.3.6",
      "contentHash": "f36318af30f2bf12b13bb8110db57aba0f5505274f4c79769e85e925e5b3ff61",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          137
        ],
        "answerPdfPages": [
          59
        ],
        "method": "Original scans inspected; radical√2/2, divisibility symbol, board constructions and residue counts restored. All scaffold subparts retained; boundary and graph-degree domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "f36318af30f2bf12b13bb8110db57aba0f5505274f4c79769e85e925e5b3ff61",
        "method": "Generated all38 disjoint forbidden pairs and24 unpaired values; verified the62-element construction pairwise and the one-per-pair upper bound."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.6",
      "contentHash": "8c55b5e4b71143cd9c106d01a113643e1e841d59ae731bb5354f6339e72c2bfb",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          137,
          138
        ],
        "answerPdfPages": [],
        "method": "Original textbook and answer scans inspected; all original subparts retained and OCR binomial/variable errors repaired. Domains and grandfather-town interpretation explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "8c55b5e4b71143cd9c106d01a113643e1e841d59ae731bb5354f6339e72c2bfb",
        "method": "Capacity contradiction and explicit balanced occupancy vector attain the claimed minimum possible largest box."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.7",
      "contentHash": "3326f0a84dc6e055e01710df06d2adc760f61ff018c138867f8491ddb14f68f2",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          137,
          138,
          139
        ],
        "answerPdfPages": [],
        "method": "Original textbook and answer scans inspected; all original subparts retained and OCR binomial/variable errors repaired. Domains and grandfather-town interpretation explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "3326f0a84dc6e055e01710df06d2adc760f61ff018c138867f8491ddb14f68f2",
        "method": "For n1..120 and k1..30, checked balanced integer occupancies sum to n and have maximum ceil(n/k), while the capacity at one less is strictly below n."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.8",
      "contentHash": "970dd4e22ba2a8d30a0d47dd728ba7ba54fd973f9c446d5cd3f21e54c8054c13",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          139,
          140
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        "method": "Original textbook and answer scans inspected; all original subparts retained and OCR binomial/variable errors repaired. Domains and grandfather-town interpretation explicit.",
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        "status": "passed",
        "contentHash": "970dd4e22ba2a8d30a0d47dd728ba7ba54fd973f9c446d5cd3f21e54c8054c13",
        "method": "Classified intersecting families of two-element subsets: a common vertex (star) or a triangle. Exhaustively checked all pairwise-intersecting edge families on up to six labels; tested the7/7/6 sharpness construction.",
        "attributionOnlyAmendment": {
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        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    {
      "id": "5.9",
      "contentHash": "fa75d432f195873758be9ea726e804fa0e93401cac8b9537a50b54a4ae738933",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          140,
          141
        ],
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        "method": "Original textbook and answer scans inspected; all original subparts retained and OCR binomial/variable errors repaired. Domains and grandfather-town interpretation explicit.",
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            140
          ]
        },
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        "contentHash": "fa75d432f195873758be9ea726e804fa0e93401cac8b9537a50b54a4ae738933",
        "method": "Independent all-subset946-box bound, plus explicit collision/cancellation on deterministic and seeded random ten-element samples. Positivity proves both remaining subsets are nonempty.",
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.10",
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          141,
          142
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        "method": "Original textbook and answer scans inspected; all original subparts retained and OCR binomial/variable errors repaired. Domains and grandfather-town interpretation explicit."
      },
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        "status": "passed",
        "contentHash": "a3284c6416a136d5fd95f87b2dae1eff989aae84d972c2ce482583471cb19e94",
        "method": "Independent translated-prefix proof: g_i≤88, so the114 distinct-within-list values g_i and g_i+23 lie in0..111 (112 boxes); a cross-list collision directly gives a23-game interval. Checked multiple valid sliding-week schedules and source remainder proof bounds."
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
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      "contentHash": "fa277eb86636f09ff415da07951904b3a8b6087c54758651240b6a931ef2cbf2",
      "source": {
        "status": "passed",
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        "method": "Original textbook and answer scans inspected; all original subparts retained and OCR binomial/variable errors repaired. Domains and grandfather-town interpretation explicit."
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        "status": "passed",
        "contentHash": "fa277eb86636f09ff415da07951904b3a8b6087c54758651240b6a931ef2cbf2",
        "method": "Four capacity3 suit classes hold only12; the exhibited13-card pattern is physically valid with at most13 per suit."
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.4.2",
      "contentHash": "228d8634b932bab1bc9e323a12d71130cba8caf1cd378776326239611cbe2498",
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        "status": "passed",
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          60
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        "method": "Original textbook and answer scans inspected; all original subparts retained and OCR binomial/variable errors repaired. Domains and grandfather-town interpretation explicit."
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        "status": "passed",
        "contentHash": "228d8634b932bab1bc9e323a12d71130cba8caf1cd378776326239611cbe2498",
        "method": "All5985 nonnegative count vectors of five remainder classes summing17 were checked: either all classes are nonempty or a class has at least5."
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
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        "status": "passed",
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        "method": "Original textbook and answer scans inspected; all original subparts retained and OCR binomial/variable errors repaired. Domains and grandfather-town interpretation explicit.",
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        "contentHash": "88b314a13502077184b55d81a023edd98765d96e4a9baa4b7374da6a369ae876",
        "method": "Enumerated all165 vertex triples; balanced four-color assignment has maximum42 and capacity4×41 is too small.",
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      "grading": {
        "mode": "numeric",
        "automaticScore": true,
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    {
      "id": "5.4.4",
      "contentHash": "511a81bd856c8d14a32c7bb246d40fd365f082ab3dc8708cf168d9049a0fa672",
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        "status": "passed",
        "textbookPdfPages": [
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        "method": "Original textbook and answer scans inspected; all original subparts retained and OCR binomial/variable errors repaired. Domains and grandfather-town interpretation explicit."
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        "status": "passed",
        "contentHash": "511a81bd856c8d14a32c7bb246d40fd365f082ab3dc8708cf168d9049a0fa672",
        "method": "Constructed all60 cyclic directed edges, checked every outdegree5 and no reciprocal edge. August proof counts at most one directed letter per unordered pair in a mutual-free configuration."
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.4.5",
      "contentHash": "a4f74c6ae2996e5baacba5a208b619f1b24f9dfd24339650ab7c099a1714c288",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          61
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        "method": "Original textbook and answer scans inspected; all original subparts retained and OCR binomial/variable errors repaired. Domains and grandfather-town interpretation explicit.",
        "restoredAttribution": {
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        "contentHash": "a4f74c6ae2996e5baacba5a208b619f1b24f9dfd24339650ab7c099a1714c288",
        "method": "Enumerated all1024 binary answer patterns and confirmed minimum20 same-status pair incidences. Enumerated the252 five-correct patterns; every one of90 box counts equals56.",
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      "grading": {
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        "automaticScore": true,
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    {
      "id": "5.11",
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        "status": "passed",
        "textbookPdfPages": [
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        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved.",
        "chapter05FreshFidelity": "All original133–144/manual58–66 scans compared; longer supplied proofs and original grid examples restored, independent alternatives retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "41b208ba650b10c62c6186b15255de261010143a62d3d39b5f6c31c38f51df88",
        "method": "Enumerated all four-color occupancy vectors with totals up to7 and counted disjoint pairs as sum floor(a_i/2); maximal count with at most one pair is6."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.12",
      "contentHash": "57a17c454ba94cf0811534379f7f29e029300116338898e550f59faf21658981",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
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        "status": "passed",
        "contentHash": "57a17c454ba94cf0811534379f7f29e029300116338898e550f59faf21658981",
        "method": "Simple-graph degree proof with incompatible extremes; independent graph enumeration for n2..6 verifies the general invariant."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.13",
      "contentHash": "d0de4cee94e8ba1156c8b889ef63618817ee55f7dc4a0822864c6dd593ddc965",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "d0de4cee94e8ba1156c8b889ef63618817ee55f7dc4a0822864c6dd593ddc965",
        "method": "Exhaustively checked all3⁵ residue sequences and all ten3-position subsets for a sum0 modulo3."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.14",
      "contentHash": "7c4cc6363927197fa1cd58c23f1716e53a8424bea34ff7a9b8cc8cc4ca301510",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "7c4cc6363927197fa1cd58c23f1716e53a8424bea34ff7a9b8cc8cc4ca301510",
        "method": "After removing the three specified scores, the residual capacity7×4=28 is strictly less than29."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.15",
      "contentHash": "6fa9c4fc95d4662127b6f2fa6165e01f38f581dccb94558d8eefdada14171807",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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          62
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        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved.",
        "chapter05FreshFidelity": "All original133–144/manual58–66 scans compared; longer supplied proofs and original grid examples restored, independent alternatives retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "6fa9c4fc95d4662127b6f2fa6165e01f38f581dccb94558d8eefdada14171807",
        "method": "Independent quotient/remainder capacity argument establishes thresholds29 and39 and validates their sharpness constructions."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.16",
      "contentHash": "ecf61fffd5d5dda9725fcf8c5f5dc6fab0894c23d013fd85bf985ff963b31967",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "ecf61fffd5d5dda9725fcf8c5f5dc6fab0894c23d013fd85bf985ff963b31967",
        "method": "General prefix-pair proof, including s_0 to cover the zero-remainder case. Tested seeded integer lists and found a nonempty divisible interval."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.17",
      "contentHash": "9f6528a2cf88378e1c61c1ac019963fc4a8b8562836202d84f6194edb6182393",
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        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "9f6528a2cf88378e1c61c1ac019963fc4a8b8562836202d84f6194edb6182393",
        "method": "Minimum sum of15 distinct nonnegative integers is105; signed counterexample sums100 and has15 distinct entries."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.18",
      "contentHash": "12d527674d5047282fa0b968d781a9394256c1f6272be894a6d1d89430428009",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "12d527674d5047282fa0b968d781a9394256c1f6272be894a6d1d89430428009",
        "method": "Enumerated all C(20,4)=4845 choices of empty seats; each leaves at least one of the five fixed four-seat blocks full."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.19",
      "contentHash": "741498dc2d2786bbcde81513962caa0a2b2eaadc9f20fd65b4935d27d032ca54",
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        "status": "passed",
        "textbookPdfPages": [
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        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "741498dc2d2786bbcde81513962caa0a2b2eaadc9f20fd65b4935d27d032ca54",
        "method": "Checked all pairs of the16 constructed squares for Chebyshev distance≥2; the partition proves the upper bound."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
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      "contentHash": "66434537fec4944f0d79131e862d963c498770d6d83a90cfa2cf25ce242cb684",
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        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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          63
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        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "66434537fec4944f0d79131e862d963c498770d6d83a90cfa2cf25ce242cb684",
        "method": "Enumerated all20 three-course subsets and all15 course pairs; every both-in and both-out count is4."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.21",
      "contentHash": "0fb7f8d739ce51e7878e4479862be83586645ddae5f3fb6f4a21735b4df0ae1e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          144
        ],
        "answerPdfPages": [
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        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved.",
        "restoredAttribution": {
          "name": "Mandelbrot",
          "pdfPages": [
            144
          ]
        },
        "chapter05FreshFidelity": "All original133–144/manual58–66 scans compared; longer supplied proofs and original grid examples restored, independent alternatives retained."
      },
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        "status": "passed",
        "contentHash": "0fb7f8d739ce51e7878e4479862be83586645ddae5f3fb6f4a21735b4df0ae1e",
        "method": "Enumerated all32768 subsets of the15 possible four-member concerts; checked that all pair intersections≤2. The largest legal family has size3, with15 extremal families.",
        "attributionOnlyAmendment": {
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        }
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      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.22",
      "contentHash": "3e7e775eb405c1a676f6d74c4981f27e5df1a16c0eda554b6000940e8510906d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          144
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        "answerPdfPages": [
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        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "3e7e775eb405c1a676f6d74c4981f27e5df1a16c0eda554b6000940e8510906d",
        "method": "Finite modular collision and invertibility proof; verified a^4000 mod10000 equals1 for every residue a coprime to10000."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.23",
      "contentHash": "5487bbef79229c9d7db81d2b64a98146c5d75bd51a8168a3833ea97e2b3cf52c",
      "source": {
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        "textbookPdfPages": [
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        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "5487bbef79229c9d7db81d2b64a98146c5d75bd51a8168a3833ea97e2b3cf52c",
        "method": "Boundary n1 is an empty existential domain. For n≥2, independent circular-gap proof on fractional parts of0,r,…,(n−1)r gives a gap≤1/n and a nonzero coefficient difference≤n−1. Exhaustively checked rational r with denominators1..80 and n2..40."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.24",
      "contentHash": "c46ab1d52a66998e116898fa7ee76c7ea7eb3912874d75ff42452f7b51d25323",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          64
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        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "c46ab1d52a66998e116898fa7ee76c7ea7eb3912874d75ff42452f7b51d25323",
        "method": "Finite pattern count81 plus duplicate row color; independent18-box incidence argument proves the stronger19-column sufficient bound."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.25",
      "contentHash": "1a4386467b9c05a7746b5027de8880872d661653fe5770dfde613fcf03f504c6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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          65
        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved.",
        "restoredAttribution": {
          "name": "Mandelbrot",
          "pdfPages": [
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          ]
        },
        "chapter05FreshFidelity": "All original133–144/manual58–66 scans compared; longer supplied proofs and original grid examples restored, independent alternatives retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "1a4386467b9c05a7746b5027de8880872d661653fe5770dfde613fcf03f504c6",
        "method": "Exhaustively checked all7⁶ bounded row-count vectors; pair-capacity≤15 permits total at most16. Independently verified the16-square construction has no repeated column pair and no rectangle.",
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        }
      },
      "grading": {
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        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "5.26",
      "contentHash": "600c499661f5fe1187dac31fecd85f01c4decdcdcc4ad8e496587f7249c9cd45",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          144
        ],
        "answerPdfPages": [
          65
        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved.",
        "restoredAttribution": {
          "name": "Putnam",
          "pdfPages": [
            144
          ]
        },
        "chapter05FreshFidelity": "All original133–144/manual58–66 scans compared; longer supplied proofs and original grid examples restored, independent alternatives retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "600c499661f5fe1187dac31fecd85f01c4decdcdcc4ad8e496587f7249c9cd45",
        "method": "Enumerated every list pair with1≤m,n≤4 satisfying the bounds (at most65536 pairs per shape); subset-sum bitsets always have a positive common sum. Verified the constructive prefix-gap witness.",
        "attributionOnlyAmendment": {
          "priorContentHash": "8fcead61522d771bdc02217b5abdfde9d0a966791c1f6ba779ec169cc462f2e2",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.27",
      "contentHash": "61fac3f67308e1787e5a6e1c46e59be9a207d8d456fbf67b2e6f84f2dfe20416",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          144
        ],
        "answerPdfPages": [
          65,
          66
        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved.",
        "restoredAttribution": {
          "name": "AIME",
          "pdfPages": [
            144
          ]
        }
      },
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        "status": "passed",
        "contentHash": "61fac3f67308e1787e5a6e1c46e59be9a207d8d456fbf67b2e6f84f2dfe20416",
        "method": "Exhaustively enumerated tournaments on2..6 teams; counts with distinct win totals equal n!. Independent induction classifies all such tournaments as unique total rankings. Random-ranking construction proves fair marginals alone insufficient.",
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          "priorContentHash": "9f0a32ccede35b2fe7aedfbe3231376356df437889aefdc924efe5264648d5ba",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "5.28",
      "contentHash": "416a45b656f7cff4c0b950c48436b8c15abb1861735a49426706c64b53377b66",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          144
        ],
        "answerPdfPages": [
          66
        ],
        "method": "Original source and answer scans inspected. Source5.23 n1 boundary error explicitly corrected;5.17 nonnegative domain and5.27 independence assumption explicit. Original subparts and source keys preserved.",
        "restoredAttribution": {
          "name": "IMO",
          "pdfPages": [
            144
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "416a45b656f7cff4c0b950c48436b8c15abb1861735a49426706c64b53377b66",
        "method": "Checked every ceiling step330→329→66→65→17→16→6→5→3→2. Proved preservation of pairwise-difference avoidance algebraically. Implemented recursive witness search and tested seeded six-colorings and extremal lower-color examples.",
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        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.1",
      "contentHash": "873f0646ea4632c33e10caf4507626632099e15c5126fa1e61147d29aca85615",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          146,
          147
        ],
        "answerPdfPages": [],
        "method": "Original scans checked; coin table, monetary amounts, grid5R/4U,all scaffold parts and exact source keys retained. Joint-choice assumptions and Henry conditional-mean gap explicit.",
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "873f0646ea4632c33e10caf4507626632099e15c5126fa1e61147d29aca85615",
        "method": "Enumerated all sixteen bitmasks and their weighted coin values, total328. Each denomination appears in eight outcomes. Also tested all-head/all-tail coupling, which has the same expectation but only two outcomes."
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        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
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      "contentHash": "5e91e367083f01890c352250f3ad093f5c90f84877489059cb450704284d4f35",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          147
        ],
        "answerPdfPages": [],
        "method": "Original scans checked; coin table, monetary amounts, grid5R/4U,all scaffold parts and exact source keys retained. Joint-choice assumptions and Henry conditional-mean gap explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "5e91e367083f01890c352250f3ad093f5c90f84877489059cb450704284d4f35",
        "method": "Enumerated all36 ordered two-die rolls, total252 and mean7. For ten dice, the finite linearity identity gives10×7/2."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.3",
      "contentHash": "1bb9cb04d67c9042b404169dbc7c2e2198b7a2bab8b33aecb5649d04bb39df44",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          146,
          147
        ],
        "answerPdfPages": [],
        "method": "Original scans checked; coin table, monetary amounts, grid5R/4U,all scaffold parts and exact source keys retained. Joint-choice assumptions and Henry conditional-mean gap explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "1bb9cb04d67c9042b404169dbc7c2e2198b7a2bab8b33aecb5649d04bb39df44",
        "method": "Exact rational outcome probabilities sum1; weighted prize sum200. Independent conditional-component calculation100+(1/10)×1000 agrees."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.4",
      "contentHash": "0eb7271c19008496444bc01dc3283199c733a5c525007a793183242b107d2938",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          146,
          148,
          149
        ],
        "answerPdfPages": [],
        "method": "Original scans checked; coin table, monetary amounts, grid5R/4U,all scaffold parts and exact source keys retained. Joint-choice assumptions and Henry conditional-mean gap explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "0eb7271c19008496444bc01dc3283199c733a5c525007a793183242b107d2938",
        "method": "Verified exact rational linearity on all small joint2×2 distributions with quarter-unit masses, including maximally dependent distributions. Algebra with arbitrary finite joint masses proves the identity."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.2.1",
      "contentHash": "1731b7b9f3e0722877cc5685250e8f27a7f031837f903643fc6d141b0096ba04",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          149
        ],
        "answerPdfPages": [
          67
        ],
        "method": "Original scans checked; coin table, monetary amounts, grid5R/4U,all scaffold parts and exact source keys retained. Joint-choice assumptions and Henry conditional-mean gap explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "1731b7b9f3e0722877cc5685250e8f27a7f031837f903643fc6d141b0096ba04",
        "method": "A single coin’s four equally likely two-flip histories give one TT. Linearity across twenty indicators gives5; no assumption of independence across the twenty indicators is needed."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.2.2",
      "contentHash": "bdede2d14f163800a9557b2bc8e4db808542068a7d61563926cdb20c8f9fc4ac",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          149,
          150
        ],
        "answerPdfPages": [
          67
        ],
        "method": "Original scans checked; coin table, monetary amounts, grid5R/4U,all scaffold parts and exact source keys retained. Joint-choice assumptions and Henry conditional-mean gap explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "bdede2d14f163800a9557b2bc8e4db808542068a7d61563926cdb20c8f9fc4ac",
        "method": "Enumerated all216 rolls for(a), mean7/2. General signed-linearity formula covers(b) and any coupling with fair die marginals."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.2.3",
      "contentHash": "7068bfbeee67468f27e8fc865d566128f303a088f2565743f50dec080cb08ae8",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          150
        ],
        "answerPdfPages": [
          67
        ],
        "method": "Original scans checked; coin table, monetary amounts, grid5R/4U,all scaffold parts and exact source keys retained. Joint-choice assumptions and Henry conditional-mean gap explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "7068bfbeee67468f27e8fc865d566128f303a088f2565743f50dec080cb08ae8",
        "method": "Expanded the arbitrary-coupling parameter t and canceled its coefficient. Exact rational joint-distribution tests verify both independent and dependent cases."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.2.4",
      "contentHash": "d6d0b24053159ccf71a50ab20363a4b1748294b806d5645b2eeda3d4959166f1",
      "source": {
        "status": "source-gap-corrected",
        "textbookPdfPages": [
          150
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        "answerPdfPages": [
          67,
          68
        ],
        "method": "Original scans checked; coin table, monetary amounts, grid5R/4U,all scaffold parts and exact source keys retained. Joint-choice assumptions and Henry conditional-mean gap explicit.",
        "restoredAttribution": {
          "name": "Mandelbrot",
          "pdfPages": [
            150
          ]
        }
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      "math": {
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        "contentHash": "d6d0b24053159ccf71a50ab20363a4b1748294b806d5645b2eeda3d4959166f1",
        "method": "Enumerated all1024 fair initial bit patterns and all10 independent selected indices; total final-head counts56320 over10240 equally likely cases,mean11/2. Verified conditional means differ from unconditional5.",
        "attributionOnlyAmendment": {
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        }
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.2.5",
      "contentHash": "531728e94e7aa88d351321465c23dd4906d6b5340f919866ab98827459582d8e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          150
        ],
        "answerPdfPages": [
          68
        ],
        "method": "Original scans checked; coin table, monetary amounts, grid5R/4U,all scaffold parts and exact source keys retained. Joint-choice assumptions and Henry conditional-mean gap explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "531728e94e7aa88d351321465c23dd4906d6b5340f919866ab98827459582d8e",
        "method": "Enumerated all126 words with five R and four U steps; counted560 turns in total and70 turns at each of the eight positions."
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      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "6.5",
      "contentHash": "d525c6356bcdc0f0d12289bba79ea62df9fa03f1f7cf57eef28de0e54510af69",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          150,
          151,
          152
        ],
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        "method": "Original textbook/answer scans checked;radius1/7 restored,binomial symbols reconstructed,all subparts retained. Local independence and uniform-area sampling assumptions explicit.",
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "d525c6356bcdc0f0d12289bba79ea62df9fa03f1f7cf57eef28de0e54510af69",
        "method": "Enumerated all1024 ten-flip strings; each of nine slots is HH in256 strings,total2304 pairs and mean9/4."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.6",
      "contentHash": "3be6d372ea3dd71a45a7704e99058d4818ca956fb56d73b7f714d4d2a85d509a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          150,
          152
        ],
        "answerPdfPages": [],
        "method": "Original textbook/answer scans checked;radius1/7 restored,binomial symbols reconstructed,all subparts retained. Local independence and uniform-area sampling assumptions explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "3be6d372ea3dd71a45a7704e99058d4818ca956fb56d73b7f714d4d2a85d509a",
        "method": "Fixed-point incidence formula n(n−1)!/n!=1 for n≥1; exhaustively verified permutations through n8."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "6.7",
      "contentHash": "98c25da052d5bc2c38f29e616039b755957407a26eb7d11d38aa3711ce6d4de9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          150,
          152,
          153
        ],
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        "method": "Original textbook/answer scans checked;radius1/7 restored,binomial symbols reconstructed,all subparts retained. Local independence and uniform-area sampling assumptions explicit.",
        "restoredAttribution": {
          "name": "USAMTS",
          "pdfPages": [
            150,
            152
          ]
        },
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "98c25da052d5bc2c38f29e616039b755957407a26eb7d11d38aa3711ce6d4de9",
        "method": "Enumerated barycentric lattice triples a,b,c≥1 with a+b+c=n to verify the interior count for n1..30; enumerated3⁶ local vertex choices,exactly one of729 makes a full circle.",
        "attributionOnlyAmendment": {
          "priorContentHash": "57010b7ec3edae73b09ec1abb35e3fbd75065ec301ec38ec75b929ba4cf869b0",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "6.8",
      "contentHash": "d61b11d37dfa89501f767165a87a1096a80be4a0c48664b763816d366c2ebc3d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          151,
          153,
          154
        ],
        "answerPdfPages": [],
        "method": "Original textbook/answer scans checked;radius1/7 restored,binomial symbols reconstructed,all subparts retained. Local independence and uniform-area sampling assumptions explicit.",
        "restoredAttribution": {
          "name": "PSS",
          "pdfPages": [
            151,
            153
          ]
        },
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "d61b11d37dfa89501f767165a87a1096a80be4a0c48664b763816d366c2ebc3d",
        "method": "Exact area-ratio calculation and containment by triangle inequality. Rational check3250/361−9=1/361>0; indicator-sum proof does not assume coverage events independent.",
        "attributionOnlyAmendment": {
          "priorContentHash": "833bda02f59057dea9854323d406e96c5d4e140af7aa7f144691c472d4b6944f",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.3.1",
      "contentHash": "d84f7113f9fb71bc908889712bd5b711da5d4d485891db271a810fe6a1fe01b2",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          154
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        "answerPdfPages": [
          68
        ],
        "method": "Original textbook/answer scans checked;radius1/7 restored,binomial symbols reconstructed,all subparts retained. Local independence and uniform-area sampling assumptions explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "d84f7113f9fb71bc908889712bd5b711da5d4d485891db271a810fe6a1fe01b2",
        "method": "For cycles n3..8,enumerated all4ⁿ colorings and counted n/4 average equal edges. General indicator proof gives20/4."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.3.2",
      "contentHash": "ab1a5bdffe5cb14b07877bb4f687fb5544b3ce7450b26198d00c83811c30cd46",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          155
        ],
        "answerPdfPages": [
          68
        ],
        "method": "Original textbook/answer scans checked;radius1/7 restored,binomial symbols reconstructed,all subparts retained. Local independence and uniform-area sampling assumptions explicit.",
        "restoredAttribution": {
          "name": "USAMTS",
          "pdfPages": [
            155
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        }
      },
      "math": {
        "status": "passed",
        "contentHash": "ab1a5bdffe5cb14b07877bb4f687fb5544b3ce7450b26198d00c83811c30cd46",
        "method": "Enumerated all10395 perfect matchings of twelve labeled ends. Union-find with the six initial rope pairs counts closed components; total loops19524,mean6508/3465.",
        "attributionOnlyAmendment": {
          "priorContentHash": "93008ba68184c0ad0142afa5f8c9e684bde7624a29933a2b654035921b8ad67f",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.3.3",
      "contentHash": "4eaf6c722712a9d7e42eac4456c17b5658f85744271d70c03354cc3db77672a4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          155
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        "answerPdfPages": [
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          69
        ],
        "method": "Original textbook/answer scans checked;radius1/7 restored,binomial symbols reconstructed,all subparts retained. Local independence and uniform-area sampling assumptions explicit.",
        "restoredAttribution": {
          "name": "HMMT",
          "pdfPages": [
            155
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "4eaf6c722712a9d7e42eac4456c17b5658f85744271d70c03354cc3db77672a4",
        "method": "Exact conditional means and tower identity. Enumerated first-stage a and all6^a second-stage rolls,weighting each by1/(6×6^a),to recover E(b)=49/4 and E(c)=343/8.",
        "attributionOnlyAmendment": {
          "priorContentHash": "c38b5bcdd0cdc1985414bd8dcc8abb2c1b6b16290941f930d70ea8ec046a7056",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.3.4",
      "contentHash": "bfe7204fda240ccbd62729354d4cbd55dcfbc5c9f2e84744772f30f9c83da038",
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        "status": "passed",
        "textbookPdfPages": [
          155
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        "answerPdfPages": [
          69
        ],
        "method": "Original textbook/answer scans checked;radius1/7 restored,binomial symbols reconstructed,all subparts retained. Local independence and uniform-area sampling assumptions explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "bfe7204fda240ccbd62729354d4cbd55dcfbc5c9f2e84744772f30f9c83da038",
        "method": "Capacity25×2=50<51; exact squared-radius comparison1/50<1/49 verifies the source radius1/7."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.3.5",
      "contentHash": "7d00c29214ebeee81330fd554a7b64b21010660b0c8d105795e6980202870a19",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          155
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        "answerPdfPages": [
          69,
          70
        ],
        "method": "Original textbook/answer scans checked;radius1/7 restored,binomial symbols reconstructed,all subparts retained. Local independence and uniform-area sampling assumptions explicit.",
        "restoredAttribution": {
          "name": "HMMT",
          "pdfPages": [
            155
          ]
        },
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "7d00c29214ebeee81330fd554a7b64b21010660b0c8d105795e6980202870a19",
        "method": "Enumerated all32768 subsets and counted anchors using the actual bit count k; total53248 and mean13/8.",
        "attributionOnlyAmendment": {
          "priorContentHash": "6a627c1593514546dde64ac3c96ddf80c016a8b30d13cc16c0ebffdd38665100",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.9",
      "contentHash": "fa68691953a3ae2da8ee54522bcf8e3d35b745ba0f0abbb3964015d9c2d5ff4f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          155,
          156,
          157
        ],
        "answerPdfPages": [],
        "method": "Original textbook/answer scans checked;radius1/7 restored,binomial symbols reconstructed,all subparts retained. Local independence and uniform-area sampling assumptions explicit.",
        "restoredAttribution": {
          "name": "PSS",
          "pdfPages": [
            155
          ]
        },
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "fa68691953a3ae2da8ee54522bcf8e3d35b745ba0f0abbb3964015d9c2d5ff4f",
        "method": "Verified integer inequality k0..5. Constructed20 equal-area tiles,one for each pair and each triple of five patches:each tile area1/4,each patch area2.5,and each pair overlap area1,proving sharpness.",
        "attributionOnlyAmendment": {
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          "reason": "Only source credit line restored;mathematical content unchanged."
        }
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.10",
      "contentHash": "75c6f67bd91af5b82890a3a1f0b39350dfdaf300abe97ce2209dce4fd6d27b79",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          157
        ],
        "answerPdfPages": [
          70
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit.",
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "75c6f67bd91af5b82890a3a1f0b39350dfdaf300abe97ce2209dce4fd6d27b79",
        "method": "Enumerated all4⁴ rotations around one vertex,exactly one success. Interior-vertex count and linearity give the formula."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.11",
      "contentHash": "b7600b24f0cb70a64bf423064a27f67f2287b4746fe90e7e96a9a8217801d759",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          157
        ],
        "answerPdfPages": [
          70,
          71
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "b7600b24f0cb70a64bf423064a27f67f2287b4746fe90e7e96a9a8217801d759",
        "method": "Enumerated all C(9,3)=84 valid integers; total57316 and average2047/3."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.12",
      "contentHash": "6e074fedee9c9fbdd6c3dcaa4d1f2b294b6268d4fa919a7d64ff41b64aa497b1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          157
        ],
        "answerPdfPages": [
          71
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "6e074fedee9c9fbdd6c3dcaa4d1f2b294b6268d4fa919a7d64ff41b64aa497b1",
        "method": "Enumerated all120 permutations,total120 fixed-point rewards and mean1."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.13",
      "contentHash": "4a0755d4265efd53403a096b9e606bcf6acc9d81a96fef997fa0f47f9b336330",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          157
        ],
        "answerPdfPages": [
          71
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit.",
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "4a0755d4265efd53403a096b9e606bcf6acc9d81a96fef997fa0f47f9b336330",
        "method": "Enumerated all C(18,6)=18564 dog-position subsets; total148512 species changes and mean8."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.14",
      "contentHash": "94a32d0572e330f66aa2ffcfe5591837232c39ff1ffc715c532fe7d8c9efd279",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          158
        ],
        "answerPdfPages": [
          71,
          72
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit.",
        "restoredAttribution": {
          "name": "AMC",
          "pdfPages": [
            158
          ]
        },
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "94a32d0572e330f66aa2ffcfe5591837232c39ff1ffc715c532fe7d8c9efd279",
        "method": "Exact polygon areas1/8 for each disagreement sign. Exhaustive rational midpoint lattice away from rounding ties checks sign-pair reflection; the100-input measure-preserving reflection proves zero mean.",
        "attributionOnlyAmendment": {
          "priorContentHash": "6bd19b4426f20f0f64f30e00895e6a543227466d525fa6071b65524e40d9aab0",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.15",
      "contentHash": "b499504d0231622b2aa8ca7984f691f53bfd5fe540cc423ef24e1a0f73f76f30",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          158
        ],
        "answerPdfPages": [
          73
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit.",
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "b499504d0231622b2aa8ca7984f691f53bfd5fe540cc423ef24e1a0f73f76f30",
        "method": "Expanded representative families into180 individual child records; total sibling count300 and mean5/3."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.16",
      "contentHash": "8e70a908c8f56dc5240e50abadbbb6c41886ee3c52eba02fba715c133e0bd443",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          158
        ],
        "answerPdfPages": [
          73,
          74
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit.",
        "restoredAttribution": {
          "name": "HMMT",
          "pdfPages": [
            158
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "8e70a908c8f56dc5240e50abadbbb6c41886ee3c52eba02fba715c133e0bd443",
        "method": "Enumerated all C(15,6)=5005 six-recipient subsets;125 dominate the15-cycle,and exactly50 contain a fixed student.",
        "attributionOnlyAmendment": {
          "priorContentHash": "26809c78b07d1ceebe81d00750083d64b5e5b810d629ce2f5f08da11f57f3667",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.17",
      "contentHash": "ff87ef1c23b51c9640d67d0156bfa400b8c0c42c3c75c6364a75634ee5013a6a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          158
        ],
        "answerPdfPages": [
          74
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit.",
        "restoredAttribution": {
          "name": "IMO",
          "pdfPages": [
            158
          ]
        },
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "ff87ef1c23b51c9640d67d0156bfa400b8c0c42c3c75c6364a75634ee5013a6a",
        "method": "Enumerated every positive(a,d)with a+9d≤2007,count222777. Exact probability denominator4⁹=262144 and numerator smaller imply a zero-count coloring exists.",
        "attributionOnlyAmendment": {
          "priorContentHash": "88fcff571f3098b98adf2d9ed3819d64a72999bcf4bc881edb3e0c9a9302b6f3",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.18",
      "contentHash": "fa26851ba12a73a0d61105ec6237ed319136bd9e4213c6cb064a67c3878304cb",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          158
        ],
        "answerPdfPages": [
          74
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit.",
        "restoredAttribution": {
          "name": "IMO",
          "pdfPages": [
            158
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "fa26851ba12a73a0d61105ec6237ed319136bd9e4213c6cb064a67c3878304cb",
        "method": "Computed fixed-point histograms for all permutations through n8 and verified the weighted sum equals n!; independent pair-count proof covers all positive n.",
        "attributionOnlyAmendment": {
          "priorContentHash": "c46f82142d40f17e51b761186858abcfef0319c08d0b75b50e5713a6a226f64f",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.19",
      "contentHash": "8e59e27410ad2a660f0518213b80e873598f39f49448351889d25c334669e841",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          159
        ],
        "answerPdfPages": [
          74,
          75
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit.",
        "chapter06FreshFidelity": "All original145–159/manual67–75 scans visually compared; source figures/tables/alternate derivations restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "8e59e27410ad2a660f0518213b80e873598f39f49448351889d25c334669e841",
        "method": "Projection-length probability d and finite linearity;sum diameters10/π>3 follows π<10/3. The integer-valued mean bound yields existence without circle-disjointness assumptions."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.20",
      "contentHash": "ac7c074b697bfbf7ea204bd76b212f59f47da2a8bb7defd282392a2c6117db43",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          159
        ],
        "answerPdfPages": [
          75
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "ac7c074b697bfbf7ea204bd76b212f59f47da2a8bb7defd282392a2c6117db43",
        "method": "Enumerated all C(52,4)=270725 sets of Ace positions;sum first positions2869685 gives mean53/5. Subtracting the Ace yields48/5."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "6.21",
      "contentHash": "e39462ba1baacc1cf6110482e133760c6057a3c63c7bfda7e1d158d350ab6440",
      "source": {
        "status": "source-gap-corrected",
        "textbookPdfPages": [
          159
        ],
        "answerPdfPages": [
          75
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Leading binary digit,uniform-child weights,first-Ace convention and corrected judge quadratic explicit.",
        "restoredAttribution": {
          "name": "IMO",
          "pdfPages": [
            159
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "e39462ba1baacc1cf6110482e133760c6057a3c63c7bfda7e1d158d350ab6440",
        "method": "For odd b3..101,enumerated p0..b and confirmed minimum(b−1)²/4. Exhaustive small binary judge matrices verify the pair-incidence identity and resulting bound.",
        "attributionOnlyAmendment": {
          "priorContentHash": "4997fc0028f0994c6bf70f94a03a1401a2e36d8526298b32d29974756a6ec69c",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.1",
      "contentHash": "25aebd886270296d68924c941f97b66e8243685fb00bd2cf4d7cb7802e14857e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          161,
          162
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;PDF162 damaged OCR replaced by original ten-bill divider diagram and source examples. All subparts retained;item identity and positivity domains explicit.",
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "25aebd886270296d68924c941f97b66e8243685fb00bd2cf4d7cb7802e14857e",
        "method": "Enumerated all positive quadruples summing10 and all three-gap subsets of nine positions;both counts84 and cumulative-sum encoding is bijective."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.2",
      "contentHash": "296b980bf3c9a7369ffb8e98e8a045c073227be777436728aaf2a2c089508775",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          161,
          162
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;PDF162 damaged OCR replaced by original ten-bill divider diagram and source examples. All subparts retained;item identity and positivity domains explicit.",
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "296b980bf3c9a7369ffb8e98e8a045c073227be777436728aaf2a2c089508775",
        "method": "Enumerated positive six-tuples summing11 by five cuts among ten gaps;252 distinct vectors result."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.3",
      "contentHash": "e5a1e06af4bf4a63db36622a2ab5b83a8525bcb41a467536a6f860ce47313678",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          161,
          162,
          163
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;PDF162 damaged OCR replaced by original ten-bill divider diagram and source examples. All subparts retained;item identity and positivity domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "e5a1e06af4bf4a63db36622a2ab5b83a8525bcb41a467536a6f860ce47313678",
        "method": "Enumerated all gap subsets for n1..15 and k1..min(n,7),checked positivity,sum n,and exact inverse encoding. Empty feasible families checked when n<k."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.4",
      "contentHash": "7a9bd009d1c9a8a9d2a31fec3db6b756988a19adf6591c1e5c7f2fb31f9b11b5",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          161,
          163
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;PDF162 damaged OCR replaced by original ten-bill divider diagram and source examples. All subparts retained;item identity and positivity domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "7a9bd009d1c9a8a9d2a31fec3db6b756988a19adf6591c1e5c7f2fb31f9b11b5",
        "method": "Independent nested-loop enumeration of positive a,b,c with d=17−a−b−c>0 gives560."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.2.1",
      "contentHash": "231ee951ec1e73d17f38ce54f621b17d90409f89c0affca0e1efee98fb21d0fb",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          163
        ],
        "answerPdfPages": [
          76
        ],
        "method": "Original scans checked;PDF162 damaged OCR replaced by original ten-bill divider diagram and source examples. All subparts retained;item identity and positivity domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "231ee951ec1e73d17f38ce54f621b17d90409f89c0affca0e1efee98fb21d0fb",
        "method": "Independent dynamic programming for positive compositions gives10015005,agreeing with C(29,9)."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.2.2",
      "contentHash": "7e2377610a5d41504b2197a1768c8b7dd7971335dfe026dbe8e57530a8a78b51",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          163
        ],
        "answerPdfPages": [
          76
        ],
        "method": "Original scans checked;PDF162 damaged OCR replaced by original ten-bill divider diagram and source examples. All subparts retained;item identity and positivity domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "7e2377610a5d41504b2197a1768c8b7dd7971335dfe026dbe8e57530a8a78b51",
        "method": "Enumerated positive quadruples summing10 with first entry≥3;35 vectors. Subtracting2 from first entry maps them bijectively to positive compositions of8 into4."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.2.3",
      "contentHash": "d16943db096df908acea582b451ef79811e32f64c362dc41d4d8f23687fc1659",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          164
        ],
        "answerPdfPages": [
          76
        ],
        "method": "Original scans checked;PDF162 damaged OCR replaced by original ten-bill divider diagram and source examples. All subparts retained;item identity and positivity domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "d16943db096df908acea582b451ef79811e32f64c362dc41d4d8f23687fc1659",
        "method": "Nested-loop enumeration over positive a,b and c=100−a−b>0 gives4851."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.2.4",
      "contentHash": "96c6f6b559ea5947d4c0d88f77e07c351a316fc01406910c90ece0742f3495f1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          164
        ],
        "answerPdfPages": [
          76
        ],
        "method": "Original scans checked;PDF162 damaged OCR replaced by original ten-bill divider diagram and source examples. All subparts retained;item identity and positivity domains explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "96c6f6b559ea5947d4c0d88f77e07c351a316fc01406910c90ece0742f3495f1",
        "method": "Enumerated all weak five-part allocations of8 and10;filtered the Cartesian pairs for both-positive and exactly-one-positive requirements,giving4410 and5005."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.5",
      "contentHash": "b2f938c422fcd30c0d54ed658ef3c0116acb2d2eb67fd9057c0165b56b706447",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          164,
          165
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "b2f938c422fcd30c0d54ed658ef3c0116acb2d2eb67fd9057c0165b56b706447",
        "method": "Independent dynamic programming counts12650 weak five-part compositions of21. Shift by the all-ones vector preserves bijection."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.6",
      "contentHash": "7fd815fa5e85aac6cf3b07309b5e55c432a7e0241089e38b81451b31f8e01c5e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          164,
          165,
          166
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "7fd815fa5e85aac6cf3b07309b5e55c432a7e0241089e38b81451b31f8e01c5e",
        "method": "Enumerated weak compositions for n0..12,k1..6 and verified countC(n+k−1,k−1),with exact positive-shift inverse."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.7",
      "contentHash": "380a6210e2a1f58936af2a12ec595612b37dae12894f948d19287cd1eb87278a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          164,
          166,
          167
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit.",
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "380a6210e2a1f58936af2a12ec595612b37dae12894f948d19287cd1eb87278a",
        "method": "Enumerated all positive seven-tuples summing20,filtered first two entries equal;4956 remain,with the seven listed case frequencies."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.8",
      "contentHash": "902b318efaa6bc13cf40de933970a4cd3fc36462c13cf06f9e46134d2884c179",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          164,
          167,
          168
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit.",
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "902b318efaa6bc13cf40de933970a4cd3fc36462c13cf06f9e46134d2884c179",
        "method": "Enumerated all364 positive quadruples summing15. Exactly81 have first entry strictly largest,324 have some unique maximum,and40 have a tied maximum;case frequencies match6,19,21,15,10,6,3,1."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.9",
      "contentHash": "e255f0ec4b1ab7a023770943d68ad79921f032805ad5a0f1b4cc5f6407b63f8d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          164,
          168
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit.",
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "e255f0ec4b1ab7a023770943d68ad79921f032805ad5a0f1b4cc5f6407b63f8d",
        "method": "Enumerated all even a0..26 and d1,3,9,27,then all nonnegative b,c with the required sum;466 solutions."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.3.1",
      "contentHash": "eb368bb691f94d4902f91572c5cbc06d1f6af69acfadb9a9b5bacd481b4f22c1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          169
        ],
        "answerPdfPages": [
          76
        ],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit.",
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored.",
        "attributionCorrection": "AMC credit was misplaced by canonical OCR carry-forward; source169 assigns it to7.3.2."
      },
      "math": {
        "status": "passed",
        "contentHash": "eb368bb691f94d4902f91572c5cbc06d1f6af69acfadb9a9b5bacd481b4f22c1",
        "method": "Weak-composition dynamic programming gives14950,agreeing with the positive-shift count.",
        "attributionOnlyAmendment": {
          "priorContentHash": "eb368bb691f94d4902f91572c5cbc06d1f6af69acfadb9a9b5bacd481b4f22c1",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.3.2",
      "contentHash": "a16dbcc787670ec64cf9cbd1809c08914650f3a7a2e6c0de710c7ac58cc49dd6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          169
        ],
        "answerPdfPages": [
          77
        ],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit.",
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored.",
        "restoredAttribution": {
          "name": "AMC",
          "pdfPages": [
            169
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "a16dbcc787670ec64cf9cbd1809c08914650f3a7a2e6c0de710c7ac58cc49dd6",
        "method": "Enumerated all nonnegative triples of total4 and6;counts15 and28."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.3.3",
      "contentHash": "d732b39895a7c4aa394b19a029ce5e15aa4c9d36d24e9b05e6668c5c7ee5e6d3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          169
        ],
        "answerPdfPages": [
          77
        ],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit.",
        "restoredAttribution": {
          "name": "ARML",
          "pdfPages": [
            169
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "d732b39895a7c4aa394b19a029ce5e15aa4c9d36d24e9b05e6668c5c7ee5e6d3",
        "method": "Dynamic programming for seven weak parts totaling7 gives1716;both lower-bound shifts are inverse bijections.",
        "attributionOnlyAmendment": {
          "priorContentHash": "5cd743c6279cec4afe41a3ed1348316c92892a851b296d06a52dc303e8175b2c",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.3.4",
      "contentHash": "ccd0fcc7a9d4c3270fa45469fab01a66c757c58897be27ed5447328852771985",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          169
        ],
        "answerPdfPages": [
          77
        ],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "ccd0fcc7a9d4c3270fa45469fab01a66c757c58897be27ed5447328852771985",
        "method": "Enumerated x,y,z in−28..19 satisfying sum10,obtaining1176,and checked the positive complement map."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.3.5",
      "contentHash": "c67aedcc5958b95c1678bc0d0855c2b6c799aeee1992a3c327cc64c79688b1ef",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          169
        ],
        "answerPdfPages": [
          77
        ],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "c67aedcc5958b95c1678bc0d0855c2b6c799aeee1992a3c327cc64c79688b1ef",
        "method": "Enumerated all positive quadruples with sum4..24,obtaining10626;each has exactly one positive slack value."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.3.6",
      "contentHash": "1638e47e76c6f55e7b395d34714b4dfae06dc03b05b574d12700b26c0e39e4bf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          169
        ],
        "answerPdfPages": [
          77
        ],
        "method": "Original scans checked;all eleven source records/scaffold parts retained. Strict bounds,negative domain,slack positivity and power3^0 convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "1638e47e76c6f55e7b395d34714b4dfae06dc03b05b574d12700b26c0e39e4bf",
        "method": "Enumerated all11⁴ bounded vectors and filtered sum20,obtaining891. Checked each excluded weak composition has exactly one count≥11."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.10",
      "contentHash": "81dcb06ca044d3c6730f4a4b61de09870cef4c97e14e92f96a839f80d3646d5e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          169,
          170
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;all nine source records retained. Multiset equivalence,leading digit,word-versus-parse distinction and model-limited lottery interpretation explicit.",
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "81dcb06ca044d3c6730f4a4b61de09870cef4c97e14e92f96a839f80d3646d5e",
        "method": "Enumerated the165 nondecreasing triples from1..9;classified9 equal,72 double and84 distinct."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.11",
      "contentHash": "a3c6ad83a81a2dd9713d90b339c1a874478d3aa1affd13061135d3f2db1a5468",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          169,
          171
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;all nine source records retained. Multiset equivalence,leading digit,word-versus-parse distinction and model-limited lottery interpretation explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "a3c6ad83a81a2dd9713d90b339c1a874478d3aa1affd13061135d3f2db1a5468",
        "method": "Enumerated positive four-part compositions of9,obtaining56 distinct coefficient vectors."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.12",
      "contentHash": "ec61052860109e143ccbb630b6a473ad78b8541969b824716e1257f858dc7135",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          169,
          171
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;all nine source records retained. Multiset equivalence,leading digit,word-versus-parse distinction and model-limited lottery interpretation explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "ec61052860109e143ccbb630b6a473ad78b8541969b824716e1257f858dc7135",
        "method": "Enumerated all length15 binary zero-position masks with eight zeros,first digit nonzero and exactly three zero runs;735 masks. Multiplying by3⁷ gives1607445."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.13",
      "contentHash": "37ea1696cbdad77e212d79aed865e4af721a9f7867db868ec2bc8147b62c7499",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          170,
          172
        ],
        "answerPdfPages": [],
        "method": "Original scans checked;all nine source records retained. Multiset equivalence,leading digit,word-versus-parse distinction and model-limited lottery interpretation explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "37ea1696cbdad77e212d79aed865e4af721a9f7867db868ec2bc8147b62c7499",
        "method": "Verified the source letter multiset;enumerated all C(13,3)=286 I-position masks,165 nonadjacent. Both factorial formulas agree exactly."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.4.1",
      "contentHash": "36ed71bf49dabcb310246d6cecdc942d190bba735786f655592f3c138172337a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          172
        ],
        "answerPdfPages": [
          78
        ],
        "method": "Original scans checked;all nine source records retained. Multiset equivalence,leading digit,word-versus-parse distinction and model-limited lottery interpretation explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "36ed71bf49dabcb310246d6cecdc942d190bba735786f655592f3c138172337a",
        "method": "Enumerated combinations with replacement of six entries from1..9,obtaining3003 distinct multisets."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.4.2",
      "contentHash": "81c35ec74b692f283461521815f9fddc9870122e3ff49792338739fa6bd1c115",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          172
        ],
        "answerPdfPages": [
          78
        ],
        "method": "Original scans checked;all nine source records retained. Multiset equivalence,leading digit,word-versus-parse distinction and model-limited lottery interpretation explicit.",
        "restoredAttribution": {
          "name": "ARML",
          "pdfPages": [
            172
          ]
        },
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "81c35ec74b692f283461521815f9fddc9870122e3ff49792338739fa6bd1c115",
        "method": "Enumerated all4⁷ words and validated them by an independent syllable-parsing dynamic program. Exactly1435 words;grouping by A count gives1,21,171,594,648. Multiple syllable parses are deduplicated as one word.",
        "attributionOnlyAmendment": {
          "priorContentHash": "7690eea25a4f366827a943fdf9fd7eca5297592d05b0ef4633d8e81110510786",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.4.3",
      "contentHash": "b740095fe0981a9929a6fc7446671425eecd0f278a5f79449377b629851fad49",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          172
        ],
        "answerPdfPages": [
          78,
          79
        ],
        "method": "Original scans checked;all nine source records retained. Multiset equivalence,leading digit,word-versus-parse distinction and model-limited lottery interpretation explicit.",
        "restoredAttribution": {
          "name": "AIME",
          "pdfPages": [
            172
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "b740095fe0981a9929a6fc7446671425eecd0f278a5f79449377b629851fad49",
        "method": "Enumerated all792 five-position birch sets;56 have no adjacent positions. Exact fraction reduces to7/99.",
        "attributionOnlyAmendment": {
          "priorContentHash": "4835d97752223670baa34c39c405f460085932febe48056ab379be5393573653",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.4.4",
      "contentHash": "367d07b1885e876664905e94fdca1e6c0350efcb6fa33a49fef5d8ac265d61e9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          172
        ],
        "answerPdfPages": [
          79
        ],
        "method": "Original scans checked;all nine source records retained. Multiset equivalence,leading digit,word-versus-parse distinction and model-limited lottery interpretation explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "367d07b1885e876664905e94fdca1e6c0350efcb6fa33a49fef5d8ac265d61e9",
        "method": "Enumerated positive four-part compositions of21 and three-part compositions of9;their31920 interleavings all satisfy the two evaluations."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.4.5",
      "contentHash": "21b09704fcb288d28ed16ba87eca314328011c26fb55294594406d5191e048c6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          172
        ],
        "answerPdfPages": [
          79
        ],
        "method": "Original scans checked;all nine source records retained. Multiset equivalence,leading digit,word-versus-parse distinction and model-limited lottery interpretation explicit.",
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "21b09704fcb288d28ed16ba87eca314328011c26fb55294594406d5191e048c6",
        "method": "Verified the shift map and inverse for small n,k by exhaustive subset enumeration;computed exact binomial counts for44,6. A gap-state dynamic program independently gives3262623 nonadjacent subsets."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.14",
      "contentHash": "5a4a0af83f1de81e9bf14263e886e946d9f449559cfbdd4805388b80d1b17206",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          174
        ],
        "answerPdfPages": [
          79,
          80
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "5a4a0af83f1de81e9bf14263e886e946d9f449559cfbdd4805388b80d1b17206",
        "method": "Enumerated all feasible triples independently in each domain;counts36,66,153,6."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.15",
      "contentHash": "5773ad73f387ef27c5969ac3edb2b6d75ea1ba890ac9fb8318f793c993fe7f8c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          174
        ],
        "answerPdfPages": [
          80
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "5773ad73f387ef27c5969ac3edb2b6d75ea1ba890ac9fb8318f793c993fe7f8c",
        "method": "Independent dynamic programming for five nonnegative parts totaling999 agrees with C(1003,4)."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.16",
      "contentHash": "c697814580a740402173793b216057f0ee7f1ef5b5562d891fb6ad3a42e77862",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          174
        ],
        "answerPdfPages": [
          80
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "c697814580a740402173793b216057f0ee7f1ef5b5562d891fb6ad3a42e77862",
        "method": "Enumerated all45 weak candy triples and15 weak cookie triples;filtered the three independent part conditions to get63,15,39."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.17",
      "contentHash": "1417573b8622e8d2bca53ced432552ebc4e175dbfcf698d3dfd4b4f1f6249390",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          174
        ],
        "answerPdfPages": [
          80
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "1417573b8622e8d2bca53ced432552ebc4e175dbfcf698d3dfd4b4f1f6249390",
        "method": "Enumerated all positive four-part compositions of30 and filtered maximum≤16;2510 remain."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.18",
      "contentHash": "f7865f59a9045103a022fb9b39411a6a13208d8391d347e837fd73685101b8ea",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          174
        ],
        "answerPdfPages": [
          80
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "f7865f59a9045103a022fb9b39411a6a13208d8391d347e837fd73685101b8ea",
        "method": "Bijection via ordered nonmultiples and occupied gaps;exhaustive small analogues with distinct special and ordinary labels verify the factorial factors."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.19",
      "contentHash": "9675cf0acfc831ecb007d290e1089420735c6bfd1e276d6548c6f5ff81a3fa78",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          174
        ],
        "answerPdfPages": [
          81
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "9675cf0acfc831ecb007d290e1089420735c6bfd1e276d6548c6f5ff81a3fa78",
        "method": "Enumerated all nonnegative exponent triples summing100,obtaining5151."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.20",
      "contentHash": "1efcb83ce65da6a3508e3d80ba40971093b11c8b2ca8fd43129c908bf333ef3c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          174
        ],
        "answerPdfPages": [
          81
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired.",
        "restoredAttribution": {
          "name": "AMC",
          "pdfPages": [
            174
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "1efcb83ce65da6a3508e3d80ba40971093b11c8b2ca8fd43129c908bf333ef3c",
        "method": "Enumerated all C(11,3)teacher-position sets;35 have consecutive teacher gaps≥3. Multiplying by the two identity permutations gives8467200.",
        "attributionOnlyAmendment": {
          "priorContentHash": "071760d17d84e3fbdce2797aebb7d6dc6dd709c2d1a9095adc6889a820f7a816",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.21",
      "contentHash": "4ed9ad7a465613706376784e35225f4a90d0f07a58afcd36b9785122844fb4d3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          174
        ],
        "answerPdfPages": [
          81
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired.",
        "restoredAttribution": {
          "name": "Mandelbrot",
          "pdfPages": [
            174
          ]
        },
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "4ed9ad7a465613706376784e35225f4a90d0f07a58afcd36b9785122844fb4d3",
        "method": "Enumerated all15504 five-seat subsets;4368 are nonadjacent,and the exact ratio is91/323.",
        "attributionOnlyAmendment": {
          "priorContentHash": "b40a7242a0165fd10c2657843a3b934c9817d1681f9ebff6d61e24ea6baef8fb",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.22",
      "contentHash": "b629d6ec3c5d2de9955d52b213ae32641cbced59d551044b6431c5954ad7c516",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          174
        ],
        "answerPdfPages": [
          81
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "b629d6ec3c5d2de9955d52b213ae32641cbced59d551044b6431c5954ad7c516",
        "method": "Enumerated all C(13,5)=1287 B-position masks;exactly41 have every maximal A-run and B-run of size≥2."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.23",
      "contentHash": "217285df1acff0709944cae98beff323411287f45449173c02ddf1863923df3f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          175
        ],
        "answerPdfPages": [
          82
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired."
      },
      "math": {
        "status": "passed",
        "contentHash": "217285df1acff0709944cae98beff323411287f45449173c02ddf1863923df3f",
        "method": "Enumerated all2²⁰ bit strings with an optimized transition count;exactly11550 have ordered counts(HH,TT,HT,TH)=(3,7,4,5)."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.24",
      "contentHash": "a86c3a38f776e93464a78f3794852656323deec54df9c81aece8f16f769eb9bf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          175
        ],
        "answerPdfPages": [
          82
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired.",
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "a86c3a38f776e93464a78f3794852656323deec54df9c81aece8f16f769eb9bf",
        "method": "Enumerated all64 six-color strings and their exact without-replacement probabilities via falling factorials;probabilities sum1 and adjacent-red probability is173/221."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "7.25",
      "contentHash": "ffdeb5397e22b0add6c816fa08b7a778e84f523f26684852c3522d3625921303",
      "source": {
        "status": "source-gap-corrected",
        "textbookPdfPages": [
          175
        ],
        "answerPdfPages": [
          82,
          83
        ],
        "method": "Original scans checked;all twelve review/challenge records and subparts retained. Transition order spelled out;source7.25 zero-probability omissions corrected;source shift-domain typos repaired.",
        "restoredAttribution": {
          "name": "Mandelbrot",
          "pdfPages": [
            175
          ]
        },
        "chapter07FreshFidelity": "All original160–175/manual76–83 scans compared; source tables, figures, examples and alternate methods restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "ffdeb5397e22b0add6c816fa08b7a778e84f523f26684852c3522d3625921303",
        "method": "Run-count formula for k selected labels and j adjacent pairs:C(k−1,j)C(n−k+1,k−j),with out-of-range binomials0. Exhaustive subset checks for n7..20 confirm the zero cases;integer formula sweep n7..500 finds exactly7,8,9,10,11,54.",
        "attributionOnlyAmendment": {
          "priorContentHash": "d4ec75c0679b5ab3365908d984d46c59800f94c253cab71e74bb21cc24b508e5",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "8.1",
      "contentHash": "22341c934b7156ac4184da2c6498cf975c723e18e9b3b7dae67b44c0f6bb07b9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          177
        ],
        "textbookSolutionPdfPages": [
          177,
          178
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "22341c934b7156ac4184da2c6498cf975c723e18e9b3b7dae67b44c0f6bb07b9",
        "method": "Symbolically checked base case and successor difference."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "8.2",
      "contentHash": "40dd2db14e4edbb9f06cfb9b112afba9068e754556a73004b0fd82ae7937f244",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          178
        ],
        "textbookSolutionPdfPages": [
          179
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "40dd2db14e4edbb9f06cfb9b112afba9068e754556a73004b0fd82ae7937f244",
        "method": "Symbolically checked the cube-sum successor identity; direct sums through100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.3",
      "contentHash": "b7148520ac0153364e988009893b8c6b2eb56515118a2ac23c5590fe167bad3d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          178
        ],
        "textbookSolutionPdfPages": [
          180
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed.",
        "chapter08FreshFidelity": "All original176–186/manual84–91 scans compared; native source diagrams/tables/matrices and source exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "b7148520ac0153364e988009893b8c6b2eb56515118a2ac23c5590fe167bad3d",
        "method": "Checked geometric dissection, convexity requirement and n=3 starting index."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "8.4",
      "contentHash": "310bf5946e464f59511096a0458c3eb815732aebf634aa8bdc549ec37bc1d085",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          178
        ],
        "textbookSolutionPdfPages": [
          181
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed.",
        "chapter08FreshFidelity": "All original176–186/manual84–91 scans compared; native source diagrams/tables/matrices and source exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "310bf5946e464f59511096a0458c3eb815732aebf634aa8bdc549ec37bc1d085",
        "method": "Recursive tiler checked for every hole on boards of side2,4,8,16; coverage and L shape independently checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.5",
      "contentHash": "854435697b78650dcc18e36a7de38c1f80890e5a68923f794cd97bc798816f1f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          178
        ],
        "textbookSolutionPdfPages": [
          182,
          183
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed.",
        "chapter08FreshFidelity": "All original176–186/manual84–91 scans compared; native source diagrams/tables/matrices and source exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "854435697b78650dcc18e36a7de38c1f80890e5a68923f794cd97bc798816f1f",
        "method": "Symbolic split invariant; all possible split totals computed recursively through25."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.6",
      "contentHash": "c480ad36b9e2e827e2378a08394c41b7382ddadb63609336bdd74b543ed92281",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          178
        ],
        "textbookSolutionPdfPages": [
          183,
          184
        ],
        "answerPdfPages": [],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "c480ad36b9e2e827e2378a08394c41b7382ddadb63609336bdd74b543ed92281",
        "method": "Checked both exhaustive cases and preservation of nearest-neighbor relation after deleting the closest pair."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.7",
      "contentHash": "88a4427834fa433fdcf46d8513552d8940075613efe2719449285880f0c1b11f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          184
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          84
        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "88a4427834fa433fdcf46d8513552d8940075613efe2719449285880f0c1b11f",
        "method": "Symbolic successor difference equals2k+1."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.8",
      "contentHash": "e0846d9636a6581a4fcda95b74bf7653a21dd445536de8d1e027eef91e95a609",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          184
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          84
        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "e0846d9636a6581a4fcda95b74bf7653a21dd445536de8d1e027eef91e95a609",
        "method": "Checked denominator restriction and separate r=1 case; exact rational sums for r=−2,0,1/2,1,2."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.9",
      "contentHash": "ee9166aba1e5bf3afdcd5186f9108a7d93ab77127361d70e873cbfb17a2130c0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          185
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "ee9166aba1e5bf3afdcd5186f9108a7d93ab77127361d70e873cbfb17a2130c0",
        "method": "Symbolic successor difference equals(k+1)²."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.10",
      "contentHash": "b534ab54ded5601a3a2ccdc25ad23bad80d5699d7bc7f48b31973aa49effb08b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          185
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          85
        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "b534ab54ded5601a3a2ccdc25ad23bad80d5699d7bc7f48b31973aa49effb08b",
        "method": "Independent modular check:7≡1(mod6), so all powers are1."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.11",
      "contentHash": "7f8f71d57d38a090d4e5eec264626a33256c293249eba98d9ec4dca359293050",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          185
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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          86
        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "7f8f71d57d38a090d4e5eec264626a33256c293249eba98d9ec4dca359293050",
        "method": "Checked factor count and positive cancellation; exact factorial inequalities for n1..30."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.12",
      "contentHash": "23a98395238b05033dd830eb849ae2e07477cd16988da57e7601a64a4c4e3917",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          185
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          86
        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "23a98395238b05033dd830eb849ae2e07477cd16988da57e7601a64a4c4e3917",
        "method": "Enumerated all graphs on4 and6 vertices at threshold n²+1; checked induction edge bound."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.13",
      "contentHash": "ac9d199ef61c0cbe23642d3402034eebdfe2be17ddf246abccd9ddf997e36406",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          185
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "ac9d199ef61c0cbe23642d3402034eebdfe2be17ddf246abccd9ddf997e36406",
        "method": "Checked local boundary cases, concurrent lines and duplicate-line reduction; equivalent sign-parity coloring."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.14",
      "contentHash": "fe73fc58772bed06657ba1b0237b5b8152559b972b5eceab394e87a54e49ae19",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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          87
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        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "fe73fc58772bed06657ba1b0237b5b8152559b972b5eceab394e87a54e49ae19",
        "method": "Checked squaring equivalence requires nonnegative sides and induction has no sign assumption on x_i."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.15",
      "contentHash": "aa6c6cd71f21ed90d0b55149bc585e086b009f82b7b7c78d3abb4c8a7ebeedaa",
      "source": {
        "status": "passed-original-scan-ocr-corrected",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          87,
          88
        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed.",
        "chapter08FreshFidelity": "All original176–186/manual84–91 scans compared; native source diagrams/tables/matrices and source exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "aa6c6cd71f21ed90d0b55149bc585e086b009f82b7b7c78d3abb4c8a7ebeedaa",
        "method": "Original scan resolves nested superscripts; verified finite squaring orbit of residue triples returns after6 steps, each sum0(mod19)."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.16",
      "contentHash": "4a137df6eed89326dad094cb0cd07b0adf9905cc0d6c17368bd8d1b3dbf9bb0c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed.",
        "chapter08FreshFidelity": "All original176–186/manual84–91 scans compared; native source diagrams/tables/matrices and source exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "4a137df6eed89326dad094cb0cd07b0adf9905cc0d6c17368bd8d1b3dbf9bb0c",
        "method": "Enumerated all eligible subsets for n≤14 and compared exact squared-product sums; checked recurrence identity."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.17",
      "contentHash": "b978bd70c9b75c00a4756d2759aa29865e415962759d52101cdabd49e96cea79",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          88,
          89
        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed.",
        "chapter08FreshFidelity": "All original176–186/manual84–91 scans compared; native source diagrams/tables/matrices and source exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "b978bd70c9b75c00a4756d2759aa29865e415962759d52101cdabd49e96cea79",
        "method": "Constructed circular words recursively through n20 and checked country multiplicities and all ordered neighbor pairs."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.18",
      "contentHash": "e699ad0781af4e4983f1038ee2b7f1a641dcf77dae0ccef05e59eb860cb148d5",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          185
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed."
      },
      "math": {
        "status": "passed",
        "contentHash": "e699ad0781af4e4983f1038ee2b7f1a641dcf77dae0ccef05e59eb860cb148d5",
        "method": "Checked contraction preserves total fuel and explicitly restores the removed pickup; independent exact prefix-minimum construction tested on rational distributions."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.19",
      "contentHash": "7de5ccb340e4d8b08ab81e188c7cba1934872aa685578e0310cc0992fee5e774",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          89,
          90
        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed.",
        "chapter08FreshFidelity": "All original176–186/manual84–91 scans compared; native source diagrams/tables/matrices and source exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "7de5ccb340e4d8b08ab81e188c7cba1934872aa685578e0310cc0992fee5e774",
        "method": "Checked gluing identity symbolically; simple-polygon and internal-diagonal hypotheses made explicit, including collinear redundant vertices."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "8.20",
      "contentHash": "5ba7efbbce9b96258e05310e81191fc62eaeeca124b7d70cda3a6baf613d3559",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          186
        ],
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        "answerPdfPages": [
          90
        ],
        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed.",
        "restoredAttribution": {
          "name": "USAMO",
          "pdfPages": [
            186
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        "status": "passed",
        "contentHash": "5ba7efbbce9b96258e05310e81191fc62eaeeca124b7d70cda3a6baf613d3559",
        "method": "Enumerated every desired white count for universes up to6 elements; checked both union-closure properties including empty/full counts.",
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          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "8.21",
      "contentHash": "c32cd806074e0b629dd2196522a03613b8cc6789d5a15eed9e79703492687f71",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          186
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          91
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        "method": "Original textbook and supplied solution scans visually compared; proof steps independently reviewed.",
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          "name": "IMO",
          "pdfPages": [
            186
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        },
        "chapter08FreshFidelity": "All original176–186/manual84–91 scans compared; native source diagrams/tables/matrices and source exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "c32cd806074e0b629dd2196522a03613b8cc6789d5a15eed9e79703492687f71",
        "method": "Constructed matrices through size128 and checked every row-column union and all entry bounds.",
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.1",
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "textbookSolutionPdfPages": [
          188,
          189,
          190
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        "answerPdfPages": [],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "44e0f8c579d7f53bc3195d9b04d4d8df90e613b19b8b731e8712032319f6fcd7",
        "method": "Enumerated all 1/2-step compositions through n16 and checked binomial sums."
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        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
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      "id": "9.2",
      "contentHash": "8cdc1c49b00aa264a307a263fdec9e9172f58d9babf05e2f2bdad761a5023563",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          192
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        "answerPdfPages": [],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
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      "math": {
        "status": "passed",
        "contentHash": "8cdc1c49b00aa264a307a263fdec9e9172f58d9babf05e2f2bdad761a5023563",
        "method": "All13 original table rows visually compared and independently computed from population transitions."
      },
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        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.3",
      "contentHash": "850f886ff374e537b244a4fb9c9eb6fa61e038c3461e6fcb7ff1e025be9312fa",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          191
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        "textbookSolutionPdfPages": [
          192,
          193,
          194
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        "answerPdfPages": [],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "850f886ff374e537b244a4fb9c9eb6fa61e038c3461e6fcb7ff1e025be9312fa",
        "method": "Enumerated nonconsecutive subsets through n16; checked split is disjoint and reversible."
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.4",
      "contentHash": "e78b0c674b5af4c899c94889276a5d22e41a35cdf7feb40a3c5ec60146493477",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          194,
          195
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        "answerPdfPages": [],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "e78b0c674b5af4c899c94889276a5d22e41a35cdf7feb40a3c5ec60146493477",
        "method": "Independent cell-by-cell tiling enumeration through n10 matches F(n+1)."
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    {
      "id": "9.5",
      "contentHash": "b314e85017698c6341077d9b167c82232d9d9c1af75f5ea1a8673a449171a8d0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          196
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        "answerPdfPages": [],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "b314e85017698c6341077d9b167c82232d9d9c1af75f5ea1a8673a449171a8d0",
        "method": "Exact identity checked through k100; symbolic induction uses defining recurrence."
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        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.6",
      "contentHash": "e0e1eca7772703bab9e43fe6fe8b749b96b13f0812db202eca9b3f52c18fcd4e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          197
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        "answerPdfPages": [],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
      },
      "math": {
        "status": "passed",
        "contentHash": "e0e1eca7772703bab9e43fe6fe8b749b96b13f0812db202eca9b3f52c18fcd4e",
        "method": "Exact identity checked through n100; induction factorization reviewed."
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
          92
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
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        "status": "passed",
        "contentHash": "b75d884028a15c1263fcbd0ef5974d10112d9eec6eec1adecd66c0403b70ec36",
        "method": "Enumerated ordered 1/2-step compositions; independent binomial sum for n15."
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        "automaticScore": true,
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    },
    {
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      "contentHash": "52d2faee975c8c7990a4be692e76591be9ca485d079af3335198546afae7820b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
          92
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "52d2faee975c8c7990a4be692e76591be9ca485d079af3335198546afae7820b",
        "method": "Enumerated reflection-sheet sequences with alternating vertical direction; counts through n12 agree with F(n+2)."
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      "grading": {
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        "automaticScore": true,
        "sourceKeyPreserved": true
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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          93
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "87fcbef343274b7c4a6c98dc42f6d11df283f2ef3ce10af9db6def94a23d2fb0",
        "method": "Enumerated all1024 coin strings; exactly144 have no HH."
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        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
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      "contentHash": "9a1c12edc985851e2c4ed924a1371660020880f30a994b025681b7c54078309c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "9a1c12edc985851e2c4ed924a1371660020880f30a994b025681b7c54078309c",
        "method": "Finite-state modular pair recurrence checked for its full period8."
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        "automaticScore": false,
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    },
    {
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
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      "math": {
        "status": "passed",
        "contentHash": "f9ee2ff0a16c81a3a152cf2946c9de2eaf9b8aab9276a1f19ac7b719ab13fbec",
        "method": "Exact identity checked through k100; base and successor reviewed."
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      "grading": {
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        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.3.6",
      "contentHash": "ccdf1d502a6900c1676f9bac387b3e49da86eadf18b5c7966bc44f09389f6182",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "restoredAttribution": {
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        "status": "passed",
        "contentHash": "ccdf1d502a6900c1676f9bac387b3e49da86eadf18b5c7966bc44f09389f6182",
        "method": "Enumerated all subsets through n14 using minimum=size characterization; independently checked no selfish proper subset for n≤8.",
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.3.7",
      "contentHash": "d92f185c89884020be9b4af8ff43708450c773ff7f3f2816838c1e994d6cd18f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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          94
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
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        "status": "passed",
        "contentHash": "d92f185c89884020be9b4af8ff43708450c773ff7f3f2816838c1e994d6cd18f",
        "method": "Enumerated round trips and highest-common-level classes through n10; verified each class size F(j+1)^2."
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
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      "contentHash": "3dafda6aaa42185ea36db51acb63d938f87fc613b37716ff88cddb08b6b97d61",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          200,
          201
        ],
        "answerPdfPages": [],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "3dafda6aaa42185ea36db51acb63d938f87fc613b37716ff88cddb08b6b97d61",
        "method": "Exact algebra in Q(sqrt5) checks formula and recurrence; ratios independently computed."
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
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      "contentHash": "055776f0c6626f8a5bce5bf4e59b0501317e58974a6f5a5eab4634894e04c2eb",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
      },
      "math": {
        "status": "passed",
        "contentHash": "055776f0c6626f8a5bce5bf4e59b0501317e58974a6f5a5eab4634894e04c2eb",
        "method": "High-precision nearest-integer formula checked for n0–200; uniform error bound reviewed."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
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          204
        ],
        "answerPdfPages": [],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
      },
      "math": {
        "status": "passed",
        "contentHash": "0157bb21f45d7b7fb3f09ccd34588c235de4e3844898a788cef86512c54f9862",
        "method": "Exact identity through n200 and recurrence sign-flip checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
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        "answerPdfPages": [
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
      },
      "math": {
        "status": "passed",
        "contentHash": "456fefdde2a7fa8d79e0403729a7b199f5ff3f6f4fcb2f4d4d119963bd063d27",
        "method": "Exact identities checked through n100; cancellation and mixed-term signs reviewed."
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      "grading": {
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        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
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      "source": {
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        "textbookPdfPages": [
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        "answerPdfPages": [
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          95
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
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      "math": {
        "status": "passed",
        "contentHash": "0a7733491463cb329204be616762d8a59f56a16edffe226742d246ef62941042",
        "method": "Compared approximation and exact integer digit lengths through n500; recorded n1 exception and coefficient-rounding limitation."
      },
      "grading": {
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        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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          96
        ],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "b043d5ce8d8ecaf04d4cc4d3c1000be76e0ae000d7d3ab9f755f577c06c9d88b",
        "method": "Verified two-base induction and strict inequalities; exact Lucas identities through n200; all four source subparts retained."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.10",
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
      },
      "math": {
        "status": "passed",
        "contentHash": "c0c4d523fc5efd049d74dc090625e8bfde82712b48e17f162d06bc0a280e9b73",
        "method": "Independent compositions and binomial sum for n12."
      },
      "grading": {
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        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.11",
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
      },
      "math": {
        "status": "passed",
        "contentHash": "09fec07a18058d97d4594c94fa0578630fb8f76a37610a43821b65336e614a94",
        "method": "Enumerated ordered compositions with parts≥2 through n18; boundary n3 explicitly checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.12",
      "contentHash": "3769470778caef215436c883abbfbcf345607a0a5c2d1480643f216349ddc0e5",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
      },
      "math": {
        "status": "passed",
        "contentHash": "3769470778caef215436c883abbfbcf345607a0a5c2d1480643f216349ddc0e5",
        "method": "Enumerated displacement-restricted permutations by subset dynamic programming; row14=610 and cycle14=845."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.13",
      "contentHash": "2e384430588451a23131ff56ec6dc3143e4d8178ba5f26c7d42a242a0d231741",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
      },
      "math": {
        "status": "passed",
        "contentHash": "2e384430588451a23131ff56ec6dc3143e4d8178ba5f26c7d42a242a0d231741",
        "method": "Independently expanded sex-count recurrence through generation30; pedigree-position assumption disclosed."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "9.14",
      "contentHash": "454842d3ab8a5d00cace213091d038c1c9e85fb5509fb2066790c2e0c7d95b90",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          205
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          97
        ],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed."
      },
      "math": {
        "status": "passed",
        "contentHash": "454842d3ab8a5d00cace213091d038c1c9e85fb5509fb2066790c2e0c7d95b90",
        "method": "Exact sums through n200; telescoping endpoints checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "9.15",
      "contentHash": "33f9bbfb0915cb2037c23cee143122f7c1ebc9568c5fbb4f0f1daa63bf91acdc",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          205
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          97,
          98
        ],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "33f9bbfb0915cb2037c23cee143122f7c1ebc9568c5fbb4f0f1daa63bf91acdc",
        "method": "Independent adjacency-matrix path counts on digits0,1,2,3 through length30."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "9.16",
      "contentHash": "bfa70577f70ab6e6aa592901a9da3d7ebcf32741ccb9513e22bf2a440cd00f86",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          205
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          98
        ],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "bfa70577f70ab6e6aa592901a9da3d7ebcf32741ccb9513e22bf2a440cd00f86",
        "method": "Native14-hex graph adjacency checked geometrically; directed-path dynamic programming gives377."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.17",
      "contentHash": "2cb24292f4d4f5265df3786c185998e749e1744379d24c391d77e16f74b460a6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          206
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          98
        ],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "2cb24292f4d4f5265df3786c185998e749e1744379d24c391d77e16f74b460a6",
        "method": "Enumerated all2048 binary strings; exactly110 satisfy neighbor condition."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.18",
      "contentHash": "4b259acd0bbe0beb1ed4395d86f4e6d69529703974aaf34c0bfe295db332adab",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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          99
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "4b259acd0bbe0beb1ed4395d86f4e6d69529703974aaf34c0bfe295db332adab",
        "method": "Exact binomial sums through n200; corrected lower summation bound in supplied solution."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.19",
      "contentHash": "a9c17b5fc3494fce91d2c9fce52e5669bc30a451b4cc7991b2955372e8b8ff5f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          99
        ],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "restoredAttribution": {
          "name": "Mandelbrot",
          "pdfPages": [
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          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "a9c17b5fc3494fce91d2c9fce52e5669bc30a451b4cc7991b2955372e8b8ff5f",
        "method": "Exact rational telescoping products checked for endpoints2–100.",
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          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.20",
      "contentHash": "0dcc74a72c51a85d0029afa4aab0ec73306d5ebcd1a00d87a1df4b08cee64cdf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          99
        ],
        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "restoredAttribution": {
          "name": "Mandelbrot",
          "pdfPages": [
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        }
      },
      "math": {
        "status": "passed",
        "contentHash": "0dcc74a72c51a85d0029afa4aab0ec73306d5ebcd1a00d87a1df4b08cee64cdf",
        "method": "Exact rational partial sums and geometric tail bound checked.",
        "attributionOnlyAmendment": {
          "priorContentHash": "fc6dc907f114fc42a6be6ef84f77a2997940f97b8d01f2f9463aa09fc63b377f",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "9.21",
      "contentHash": "dbc698efe3c6237ad677b444de54d38f31b91bcfa55659f638dce2a57d15e019",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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          100
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        "method": "Original scans compared, all source subparts preserved, supplied proof reviewed with corrections disclosed.",
        "restoredAttribution": {
          "name": "AIME",
          "pdfPages": [
            206
          ]
        },
        "chapter09FreshFidelity": "All original187–206 and manual92–100 scans compared; source tables/figures/methods restored; rabbit table and ancestry bijection preserved."
      },
      "math": {
        "status": "passed",
        "contentHash": "dbc698efe3c6237ad677b444de54d38f31b91bcfa55659f638dce2a57d15e019",
        "method": "Exact polynomial reduction modulo a²−a−1; verified positive root and strict bounds.",
        "attributionOnlyAmendment": {
          "priorContentHash": "a14c2d03c8d735ff00043036ec3efedfa3c0535382503dff9d766cc499d81146",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.1",
      "contentHash": "42876a791266664a87f88f2aaf8a55d11157c310a83263cdb315bdbed573bc90",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [
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          210,
          211
        ],
        "answerPdfPages": [],
        "method": "Original scans visually compared, complete subparts retained.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "42876a791266664a87f88f2aaf8a55d11157c310a83263cdb315bdbed573bc90",
        "method": "All legal states searched by BFS through n6; recursive solution validated through n8; lower-bound proof reviewed."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.2",
      "contentHash": "ffbc9d02bd51d8bea25b223f631d7f241160810fd84468193cbd9cfa870be4f9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [
          211,
          212,
          213
        ],
        "answerPdfPages": [],
        "method": "Original scans visually compared, complete subparts retained.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "ffbc9d02bd51d8bea25b223f631d7f241160810fd84468193cbd9cfa870be4f9",
        "method": "Enumerated typed compositions through n12 and verified closed formula against recurrence through n500."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.3",
      "contentHash": "b467b2a955722aa7b202f058feed5c55c1282b447f58bcacff6b1e824dd68369",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [
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        ],
        "answerPdfPages": [],
        "method": "Original scans visually compared, complete subparts retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "b467b2a955722aa7b202f058feed5c55c1282b447f58bcacff6b1e824dd68369",
        "method": "Enumerated all59049 ternary strings of length10; matched recurrence."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.4",
      "contentHash": "dab20b1418ce5d5ffb9e080f0a449b9dfa214942c2a94116e52c487e4b64e35b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          208
        ],
        "textbookSolutionPdfPages": [
          213,
          214,
          215
        ],
        "answerPdfPages": [],
        "method": "Original scans visually compared, complete subparts retained.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "dab20b1418ce5d5ffb9e080f0a449b9dfa214942c2a94116e52c487e4b64e35b",
        "method": "Independently enumerated ordered set partitions via surjective rank assignments for six labeled runners."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.2.1",
      "contentHash": "cba560d6f7724b334716447cbf50392b00686c2f9b70eba0cfc77578535e69c8",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original scans visually compared, complete subparts retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "cba560d6f7724b334716447cbf50392b00686c2f9b70eba0cfc77578535e69c8",
        "method": "Independent sum over number of double flags: Σ C(10−k,k)3^k2^(10−2k)=44287."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.2.2",
      "contentHash": "ebcf09dd34927a1c47cb646647e7373122ee2f919fde838f781c5bedf1ce867e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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          102
        ],
        "method": "Original scans visually compared, complete subparts retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "ebcf09dd34927a1c47cb646647e7373122ee2f919fde838f781c5bedf1ce867e",
        "method": "Exact repeated multiplication for k1–100 agrees with2^(k(k−1)/2); nested source subscript preserved."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.2.3",
      "contentHash": "a9bc6715f1eaf2653120a51d23b4f3a0b435c3f686e5473e926b87ef5127cff9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original scans visually compared, complete subparts retained."
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      "math": {
        "status": "passed",
        "contentHash": "a9bc6715f1eaf2653120a51d23b4f3a0b435c3f686e5473e926b87ef5127cff9",
        "method": "Enumerated six-digit base-four numerals with nonzero leading digit and explicit first-occurrence condition."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.2.4",
      "contentHash": "82af0b7ef4cf9ebe749f708b100f52d7bddffbdedd7d8d5aa5a5c516c8e4061a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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          103
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        "method": "Original scans visually compared, complete subparts retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "82af0b7ef4cf9ebe749f708b100f52d7bddffbdedd7d8d5aa5a5c516c8e4061a",
        "method": "Compared exact products to exponent formula for p,q in1…5 through n10; base-case range corrected explicitly."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.2.5",
      "contentHash": "da191b9435309bf231c2142b909e7ed0a639ab2bc51855be0f60a10443a4e5c3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original scans visually compared, complete subparts retained.",
        "restoredAttribution": {
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          "pdfPages": [
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        }
      },
      "math": {
        "status": "passed",
        "contentHash": "da191b9435309bf231c2142b909e7ed0a639ab2bc51855be0f60a10443a4e5c3",
        "method": "Enumerated all4096 subsets and checked pairwise selected positions differ by at least3.",
        "attributionOnlyAmendment": {
          "priorContentHash": "a58e64099220abc9541627ada6743c8555b8053213a40858b3a25e0a9e9ff707",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.2.6",
      "contentHash": "11fd2bf85edd8db09dcebd20c3b806f5e46e32dbbbf003268191b61d5dbe7915",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original scans visually compared, complete subparts retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "11fd2bf85edd8db09dcebd20c3b806f5e46e32dbbbf003268191b61d5dbe7915",
        "method": "Enumerated all1024 strings using overlapping adjacent-pair count; exactly235 qualify."
      },
      "grading": {
        "mode": "numeric",
        "automaticScore": true,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.5",
      "contentHash": "0149ae68c4db99b0a331583ff594d55a73b22b81b1d70d4e1cbc02f840c2afe7",
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        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [
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        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "0149ae68c4db99b0a331583ff594d55a73b22b81b1d70d4e1cbc02f840c2afe7",
        "method": "Exact rational recurrence versus formula through n100; both initial values checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.6",
      "contentHash": "57772956db1420726042dbb5e2ee2336382dba119c9276c45644d4d56244e6ab",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [
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          218
        ],
        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "57772956db1420726042dbb5e2ee2336382dba119c9276c45644d4d56244e6ab",
        "method": "Exact rational formula checked against recurrence through n100; normalized first differences checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.3.1",
      "contentHash": "4401ba9f324148ab00bf5f740703d7d13f9be4f5f93c6902c13e448c3b5b89d4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "4401ba9f324148ab00bf5f740703d7d13f9be4f5f93c6902c13e448c3b5b89d4",
        "method": "Exact formula checked against initial values and recurrence through n100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.3.2",
      "contentHash": "b45fdad732991578e849c2a9ae502b78b65bfac0ab299ad95eec0be9841aa279",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
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        "answerPdfPages": [
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        "method": "Original textbook and manual scans visually compared."
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      "math": {
        "status": "passed",
        "contentHash": "b45fdad732991578e849c2a9ae502b78b65bfac0ab299ad95eec0be9841aa279",
        "method": "Exact constant sequence and characteristic coefficients checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.3.3",
      "contentHash": "872f417d055e7abacdbb7d93d1d87a1879dd05d4cd7ebeed532ded41e88d0727",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "872f417d055e7abacdbb7d93d1d87a1879dd05d4cd7ebeed532ded41e88d0727",
        "method": "Exact rational formula checked against recurrence through n100, including n0."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.3.4",
      "contentHash": "fd20b462ada78ac9bf599dd0e1379237b20ed0bffe0beb9d30d2fa79ce5df23f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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          105
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        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "fd20b462ada78ac9bf599dd0e1379237b20ed0bffe0beb9d30d2fa79ce5df23f",
        "method": "Symbolic substitution checked without dividing by c; zero-root boundary handled separately."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.3.5",
      "contentHash": "d306aab06b357900d65fc1fa8857cdef12db1f0dd92d9b639fba5b51bd56fc44",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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        ],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "d306aab06b357900d65fc1fa8857cdef12db1f0dd92d9b639fba5b51bd56fc44",
        "method": "Exact Gaussian-integer powers and rational coefficients checked through n50 against integer recurrence."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.3.6",
      "contentHash": "57a6ee1a116bd7ec0a54a1063c6fb93c79a25f132759cbc5eff611076e39ae5d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          218
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          105,
          106
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        "method": "Original textbook and manual scans visually compared."
      },
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        "status": "passed",
        "contentHash": "57a6ee1a116bd7ec0a54a1063c6fb93c79a25f132759cbc5eff611076e39ae5d",
        "method": "Exact formula checked against order-three recurrence through n100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.7",
      "contentHash": "22b542fb2178cfbc026c641644914202e1e8125270dbd89d54ef69f305bf69e1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          219
        ],
        "textbookSolutionPdfPages": [
          219,
          220,
          221
        ],
        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared.",
        "restoredAttribution": {
          "name": "AIME",
          "pdfPages": [
            219
          ]
        },
        "chapter10FreshFidelity": "Original prompt figures reconciled and mobile table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "22b542fb2178cfbc026c641644914202e1e8125270dbd89d54ef69f305bf69e1",
        "method": "Direct stack simulation compared with recurrence and closed form for powers of two through 2¹⁴.",
        "attributionOnlyAmendment": {
          "priorContentHash": "52efb4812df1f7ffce6f33bd31766058e62bc117a9fa51771adf5db1e3c725e1",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.4.1",
      "contentHash": "48e1f59ce97acc7222541c069cdeb2eb95a71a1ee5d13d8291439b00579813df",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          221
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          106
        ],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "48e1f59ce97acc7222541c069cdeb2eb95a71a1ee5d13d8291439b00579813df",
        "method": "Paired odd/even terms verified through n200; two-step recurrence derived algebraically."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.4.2",
      "contentHash": "1e4a2b7df3f22860827bd8a0bc0b6a1ae4076b5db2bde442e6941d33f6c6577c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          221
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          106,
          107
        ],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "1e4a2b7df3f22860827bd8a0bc0b6a1ae4076b5db2bde442e6941d33f6c6577c",
        "method": "Exact integer formula checked against recurrence through n200 and direct stack simulation through n14."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.8",
      "contentHash": "ecefff9a379d75091f600437485af461aa99dc01f81b5410518a9ebe5e784f67",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          221
        ],
        "textbookSolutionPdfPages": [
          223,
          224,
          225
        ],
        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "Original prompt figures reconciled and mobile table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "ecefff9a379d75091f600437485af461aa99dc01f81b5410518a9ebe5e784f67",
        "method": "All perfect matchings independently enumerated and filtered for crossings through12 people."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.9",
      "contentHash": "62eff4f0dc0dd920a230325bd6abd10db7e68a8605538070c6ac37a8b7e473ae",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          222
        ],
        "textbookSolutionPdfPages": [
          225,
          226,
          227
        ],
        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "Original prompt figures reconciled and mobile table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "62eff4f0dc0dd920a230325bd6abd10db7e68a8605538070c6ac37a8b7e473ae",
        "method": "Balanced words independently enumerated through n8; stack pairing and inverse encoding checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.10",
      "contentHash": "788768c1602a093f3f81902588d69968495a4422bbca82108a96bb3f415c659c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          222
        ],
        "textbookSolutionPdfPages": [
          227,
          228,
          229
        ],
        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "Original prompt figures reconciled and mobile table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "788768c1602a093f3f81902588d69968495a4422bbca82108a96bb3f415c659c",
        "method": "All binary step strings independently checked for endpoint and every-prefix diagonal constraint through n8."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.5.2",
      "contentHash": "c260d8d6f9391c24825c4834c90b7121b8ae7642117b097ec315af887fa5be3f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          230
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          107
        ],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "c260d8d6f9391c24825c4834c90b7121b8ae7642117b097ec315af887fa5be3f",
        "method": "All weak compositions of5 into5 boxes independently filtered by every prefix bound."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.5.3",
      "contentHash": "5751fb432dc9dcef0e8c3a052c3d519480f4e192430e9214657198685e19b92d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          230
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          107
        ],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "5751fb432dc9dcef0e8c3a052c3d519480f4e192430e9214657198685e19b92d",
        "method": "All five-digit candidates independently enumerated with nondecreasing and position-bound conditions."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.5.4",
      "contentHash": "f2f77e56f66f5a1f8e4d3cea0e6c879570a505efa7eb6048e6f64639b45309f0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          231
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          108
        ],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "f2f77e56f66f5a1f8e4d3cea0e6c879570a505efa7eb6048e6f64639b45309f0",
        "method": "Exact cover enumeration of all contained integer rectangles, with exactly four tiles, gives14."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.5.5",
      "contentHash": "48dfe70e18d5783bd384d9d3193465cff451f70645c2320823da655678eb9b78",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          231
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
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          109
        ],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "48dfe70e18d5783bd384d9d3193465cff451f70645c2320823da655678eb9b78",
        "method": "Independent blocked-vertex dynamic programming and enumeration give132."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "10.12",
      "contentHash": "15492a37a9cdb31a21cf52af8551c46036f9a13c59de4275e4a961d61f199523",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          231
        ],
        "textbookSolutionPdfPages": [
          231,
          232,
          233
        ],
        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "15492a37a9cdb31a21cf52af8551c46036f9a13c59de4275e4a961d61f199523",
        "method": "Convolution checked against exact binomial formula through n100; empty case and first-return decomposition checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.13",
      "contentHash": "4e4253502ee9fbeccd0df6a4dc7a24849277ff0bc5f9e1b060035f812ecca4bf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          231
        ],
        "textbookSolutionPdfPages": [
          233,
          234
        ],
        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "4e4253502ee9fbeccd0df6a4dc7a24849277ff0bc5f9e1b060035f812ecca4bf",
        "method": "Both table rows and ratios computed with exact integers."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.14",
      "contentHash": "a774c1f50d6cf1992f7468ed086b485b9fabd78050b875fc9a2e59b772ea5d6f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          231
        ],
        "textbookSolutionPdfPages": [
          234,
          235,
          236
        ],
        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "Original prompt figures reconciled and mobile table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "a774c1f50d6cf1992f7468ed086b485b9fabd78050b875fc9a2e59b772ea5d6f",
        "method": "Every bad path and every target path enumerated through n8; suffix reflection and its inverse checked exactly."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.15",
      "contentHash": "4149490c659fbe3406545adb0f3a158232c84b30b4a3751acbe4d968107090b4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          231
        ],
        "textbookSolutionPdfPages": [
          236
        ],
        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "4149490c659fbe3406545adb0f3a158232c84b30b4a3751acbe4d968107090b4",
        "method": "Exact formula, binomial difference and convolution compared through n100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.6.1",
      "contentHash": "2d7e9cea9782086dffe6a1bb23f299f6989ec30bf416a950276c769cf92624dd",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          236
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          109
        ],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "2d7e9cea9782086dffe6a1bb23f299f6989ec30bf416a950276c769cf92624dd",
        "method": "Eight convolution terms independently computed and compared to integer formula."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.6.2",
      "contentHash": "fae22c3971882a8bf3d88e2ca974e79978640e628acf3bef8fd0d4e298306cbe",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          236
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          109,
          110
        ],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "fae22c3971882a8bf3d88e2ca974e79978640e628acf3bef8fd0d4e298306cbe",
        "method": "All supported row subsets independently enumerated through bottom size8 and compared with Cn."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.6.3",
      "contentHash": "30a16cf7655e38f65cb0cdb980954bebb7b5030eb3b007611e0b680a715d3795",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          110
        ],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "30a16cf7655e38f65cb0cdb980954bebb7b5030eb3b007611e0b680a715d3795",
        "method": "All3003 goal strings enumerated;429 satisfy strict prefix lead."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.6.4",
      "contentHash": "ee54c0bbda6d7cc94afdb2bf59e8b6700ba7b01ef5e61e804653c977c7e95fa3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          237
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          110
        ],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "ee54c0bbda6d7cc94afdb2bf59e8b6700ba7b01ef5e61e804653c977c7e95fa3",
        "method": "Ordered tree shapes independently constructed through n8 and counted."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.6.5",
      "contentHash": "0c72ad68c761574d97a4f2b417342e6b1e1bd6bc0fccdeae0caeac1062d3e6b7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          237
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          110,
          111
        ],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "0c72ad68c761574d97a4f2b417342e6b1e1bd6bc0fccdeae0caeac1062d3e6b7",
        "method": "Catalan parity checked through n1000 against the power-of-two characterization."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.11",
      "contentHash": "d3f390ff4f0ae0f67acbcfa92aae195cc46219affb13a4906049bc1a5f0b1a02",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          222
        ],
        "textbookSolutionPdfPages": [
          229,
          230
        ],
        "answerPdfPages": [],
        "method": "Original textbook and manual scans visually compared.",
        "chapter10FreshFidelity": "All original207–241 and manual101–124 scans compared; source instructional figures, tables and exposition restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "d3f390ff4f0ae0f67acbcfa92aae195cc46219affb13a4906049bc1a5f0b1a02",
        "method": "Noncrossing diagonal sets of size v−3 independently enumerated for polygons through8 vertices."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.5.1",
      "contentHash": "483808add3330ca2512a13370f83258487d68707180c3e7bf5bd2ac6c97e5375",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          230
        ],
        "textbookSolutionPdfPages": [],
        "answerPdfPages": [
          107
        ],
        "method": "Original textbook and manual scans visually compared."
      },
      "math": {
        "status": "passed",
        "contentHash": "483808add3330ca2512a13370f83258487d68707180c3e7bf5bd2ac6c97e5375",
        "method": "Direct enumeration of noncrossing5-diagonal sets in an octagon gives132."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.16",
      "contentHash": "3259bac73b5e87c95f4adaea8f1a0dbe0196a803f2d445a3919959b8db61f742",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
          111
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "3259bac73b5e87c95f4adaea8f1a0dbe0196a803f2d445a3919959b8db61f742",
        "method": "Recursive pairing enumeration through n6 and factorial/double-factorial agreement through n100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.17",
      "contentHash": "78a0a0b909ea8ab299f8712289090b72fad12c2b3e5ebb3dab9e387d6e73097d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          238
        ],
        "answerPdfPages": [
          111,
          112
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "78a0a0b909ea8ab299f8712289090b72fad12c2b3e5ebb3dab9e387d6e73097d",
        "method": "General-position incremental intersection counts and independent Euler-formula counts for line arrangements;recurrence sums checked n0..100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.18",
      "contentHash": "6f9ef57c71d5ec423977968f3c12d7394286a6fcaf9dd9df846f73a20a4c4589",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          239
        ],
        "answerPdfPages": [
          112,
          113
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "6f9ef57c71d5ec423977968f3c12d7394286a6fcaf9dd9df846f73a20a4c4589",
        "method": "Exact symbolic recurrence/root and initial-value checks;integer recurrence values compared through n100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.19",
      "contentHash": "24c6bcda960b413e8660f0d29a13be76c89db3bdda2c1f4e7b41c0b8b8abc7ab",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          239
        ],
        "answerPdfPages": [
          113
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed.",
        "restoredAttribution": {
          "name": "AIME",
          "pdfPages": [
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          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "24c6bcda960b413e8660f0d29a13be76c89db3bdda2c1f4e7b41c0b8b8abc7ab",
        "method": "Exact independent Markov recursion on remaining multiplicities and unmatched hand,through six pairs,returns9/385.",
        "attributionOnlyAmendment": {
          "priorContentHash": "de7fcb853ab82de729dedba0ee19cafb5196c22c6c646eadf39fa53f8bceeb2b",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.20",
      "contentHash": "675b1d40717a0f0ace6e35570a89d61b8e202603d3f366dcc9c79d57e8751a32",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          239
        ],
        "answerPdfPages": [
          113,
          114
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "675b1d40717a0f0ace6e35570a89d61b8e202603d3f366dcc9c79d57e8751a32",
        "method": "Independent cell-by-cell exact-cover/profile enumeration returns1,3,11,41,153,571 for widths0,2,4,6,8,10."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.21",
      "contentHash": "2f6ceb94dd2f7a0533a6b4c5889b9cfbb4b961b6a9baf42ab91713e4279bd227",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          239
        ],
        "answerPdfPages": [
          114
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed.",
        "restoredAttribution": {
          "name": "AIME",
          "pdfPages": [
            239
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "2f6ceb94dd2f7a0533a6b4c5889b9cfbb4b961b6a9baf42ab91713e4279bd227",
        "method": "Enumerated all8!=40320 bottom-to-top permutations;1458 satisfy every adjacent inequality.",
        "attributionOnlyAmendment": {
          "priorContentHash": "5ff370f73921b1ca54f2b76e477772a05123460237fafa2bf3fc4834c2fc3c86",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.22",
      "contentHash": "5b6f4353495135849a1af5232e8694d0bdc455560cfe774571d7f368eb2b3764",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          239
        ],
        "answerPdfPages": [
          114
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed.",
        "chapter10FreshFidelity": "Original prompt figures reconciled and mobile table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "5b6f4353495135849a1af5232e8694d0bdc455560cfe774571d7f368eb2b3764",
        "method": "Triangular-array dynamic program checked against C_n for n0..100;grid path recurrence establishes the interpretation."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.23",
      "contentHash": "59d195050238cf6810824d41866477452ab90a887e2f32cd1eb682647f652d96",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          239
        ],
        "answerPdfPages": [
          114,
          115
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "59d195050238cf6810824d41866477452ab90a887e2f32cd1eb682647f652d96",
        "method": "Enumerated all3^8 words,filtering AC and CA;1393 remain."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.24",
      "contentHash": "d5e6f71c73d480b179e2975b915042807abdf043e8fd0759c136619a4e1107df",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          239
        ],
        "answerPdfPages": [
          115,
          116
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed.",
        "restoredAttribution": {
          "name": "AIME",
          "pdfPages": [
            239
          ]
        },
        "chapter10FreshFidelity": "Original prompt figures reconciled and mobile table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "d5e6f71c73d480b179e2975b915042807abdf043e8fd0759c136619a4e1107df",
        "method": "Exhaustive enumeration of all2^19 strings avoiding11 and000 returns351;each ending-state table entry independently checked.",
        "attributionOnlyAmendment": {
          "priorContentHash": "436175c3c35579bce04ca6fde3da755791970e7d09575be06b39dc5ef555ae46",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.25",
      "contentHash": "89b5eddfcb957860a36aadf5941d08ae21da567eebdbbbc11dc1f68bd5da9ca2",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          239
        ],
        "answerPdfPages": [
          116
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed.",
        "restoredAttribution": {
          "name": "Putnam",
          "pdfPages": [
            239
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "89b5eddfcb957860a36aadf5941d08ae21da567eebdbbbc11dc1f68bd5da9ca2",
        "method": "Enumerated exact weighted independent outcomes through n12 and compared to n/(2n+1).",
        "attributionOnlyAmendment": {
          "priorContentHash": "ba09d2b4724ad9309c369fb97c19cd03682c6e34664a792fead656a893e49c97",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.26",
      "contentHash": "33ba8fdd195a9f7c529e4d15a33840643c2c6827f848c87615402042abd4f9cb",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          240
        ],
        "answerPdfPages": [
          116,
          117
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "33ba8fdd195a9f7c529e4d15a33840643c2c6827f848c87615402042abd4f9cb",
        "method": "Independent composition/word enumeration for sums1..20 and recurrence/Fibonacci identities through n100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.27",
      "contentHash": "fab786c0b4539ec8b9f80b385a8ca8da1343b19e2e4f3726d4bd8d85da92b591",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          240
        ],
        "answerPdfPages": [
          117,
          118,
          119
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed.",
        "restoredAttribution": {
          "name": "USAMTS",
          "pdfPages": [
            240
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "fab786c0b4539ec8b9f80b385a8ca8da1343b19e2e4f3726d4bd8d85da92b591",
        "method": "Independent Matrix-Tree determinant of the ladder Laplacian through n10;exhaustive edge-subset tree enumeration through n5.",
        "attributionOnlyAmendment": {
          "priorContentHash": "6189ba2993d878035abeebd5d2e7619a678ef668e0a8d9819224c4e718ad2c32",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.28",
      "contentHash": "788b4bb2222b468c891b0ff49ddd84821b994b0a934d27070ef9d3bc53a2e49f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          240
        ],
        "answerPdfPages": [
          119,
          120
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed.",
        "chapter10FreshFidelity": "Original prompt figures reconciled and mobile table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "788b4bb2222b468c891b0ff49ddd84821b994b0a934d27070ef9d3bc53a2e49f",
        "method": "Enumerated all3^10 fixed-position colorings and checked every cycle edge,including the closing edge;1026 valid."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.29",
      "contentHash": "62e22ac190d91f6e36e2af3c2fa9959a6f824d73014c43b4f3a7de89b9d08821",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          240
        ],
        "answerPdfPages": [
          120,
          121
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed.",
        "restoredAttribution": {
          "name": "ARML",
          "pdfPages": [
            240
          ]
        },
        "chapter10FreshFidelity": "Original prompt figures reconciled and mobile table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "62e22ac190d91f6e36e2af3c2fa9959a6f824d73014c43b4f3a7de89b9d08821",
        "method": "Generated sequences through n15;maxima match F_(n+1),the stronger adjacent pair is present,and even-index parity strings split exactly into101 blocks.",
        "attributionOnlyAmendment": {
          "priorContentHash": "1542a497cc14e7ca055088222cdc70e78205e9fadee34706abec4b04d38196ae",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.30",
      "contentHash": "587c5d55bee3fde31ea9038d89cc355881fd68097ca630991792653684d4fff9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          240
        ],
        "answerPdfPages": [
          121,
          122
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "587c5d55bee3fde31ea9038d89cc355881fd68097ca630991792653684d4fff9",
        "method": "Exhaustively enumerated prefix-valid words through n7;counts equal Fuss-Catalan formula and binomial difference;generated the complete n2 example list."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.31",
      "contentHash": "80782c5f09d6cf6fa1387ae1826f4cc64b8aa6b8d545fbfabadc202f30cb175d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          241
        ],
        "answerPdfPages": [
          123,
          124
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "80782c5f09d6cf6fa1387ae1826f4cc64b8aa6b8d545fbfabadc202f30cb175d",
        "method": "Enumerated every goal word for n0..8,each m0..n;prefix-bound counts equal both formulas and reflected suffix maps are reversible."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "10.32",
      "contentHash": "8aa8f77ce64c60d73086c770e985c5e669a8e04e9270773cfc32aa008b73fd9e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          241
        ],
        "answerPdfPages": [
          124
        ],
        "method": "Original scans PDF238–241/manual111–124 visually compared;all subparts,source tables and essential figures reconstructed.",
        "restoredAttribution": {
          "name": "HMMT",
          "pdfPages": [
            241
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "8aa8f77ce64c60d73086c770e985c5e669a8e04e9270773cfc32aa008b73fd9e",
        "method": "Independent prefix-height recursion enumerates all132 length-five sequences;append/delete and first-return reconstruction checked.",
        "attributionOnlyAmendment": {
          "priorContentHash": "a9039671278fb85b2fbcd4a823e549cef870b23770f2aa066f72a19a04575275",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.1",
      "contentHash": "72086177e876d37cee93b1a08dbb76c620a3a7ce8d32564e797c55ba46f053ff",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          242,
          243
        ],
        "answerPdfPages": [],
        "method": "Original textbook242–245 and answer125–126 scans visually compared;die faces6/3/1/2 and heart suit recovered,all subparts and units-pair lists retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "72086177e876d37cee93b1a08dbb76c620a3a7ce8d32564e797c55ba46f053ff",
        "method": "Enumerated the six equiprobable faces;conditional set has3 elements and success set1."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.2",
      "contentHash": "b3e0056997ee220444c068974ce798611b4cba16acc4bbf45780f83e0c0446c0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          243,
          244
        ],
        "answerPdfPages": [],
        "method": "Original textbook242–245 and answer125–126 scans visually compared;die faces6/3/1/2 and heart suit recovered,all subparts and units-pair lists retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "b3e0056997ee220444c068974ce798611b4cba16acc4bbf45780f83e0c0446c0",
        "method": "Enumerated all36 ordered rolls;11 satisfy the condition and2 satisfy it together with sum7."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.3",
      "contentHash": "231216f3819e2e51a720caa752aa8a5244ec5efc280a916683ab27a186af571d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          243,
          244,
          245
        ],
        "answerPdfPages": [],
        "method": "Original textbook242–245 and answer125–126 scans visually compared;die faces6/3/1/2 and heart suit recovered,all subparts and units-pair lists retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "231216f3819e2e51a720caa752aa8a5244ec5efc280a916683ab27a186af571d",
        "method": "Enumerated all eight independent fair outcomes and both conditioning subsets;ratios1/4 and1/7."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.2.1",
      "contentHash": "ed2578277777de5917b963b861fb99664c99d309ef882974083349571c1a2e94",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          245
        ],
        "answerPdfPages": [
          125
        ],
        "method": "Original textbook242–245 and answer125–126 scans visually compared;die faces6/3/1/2 and heart suit recovered,all subparts and units-pair lists retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "ed2578277777de5917b963b861fb99664c99d309ef882974083349571c1a2e94",
        "method": "Enumerated36 rolls;25 condition outcomes and4 successes."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.2.2",
      "contentHash": "cc57a33e4a29d2390a2347847dab0f712915f86485326e870db7ab0b7652125b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          245
        ],
        "answerPdfPages": [
          125
        ],
        "method": "Original textbook242–245 and answer125–126 scans visually compared;die faces6/3/1/2 and heart suit recovered,all subparts and units-pair lists retained.",
        "restoredAttribution": {
          "name": "AMC",
          "pdfPages": [
            245
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "cc57a33e4a29d2390a2347847dab0f712915f86485326e870db7ab0b7652125b",
        "method": "Enumerated216 ordered triples;15 satisfy the additive condition and8 contain2.",
        "attributionOnlyAmendment": {
          "priorContentHash": "a96620baf9381557f81ad5403ffa223da35751dc40bafc545868eb6029039494",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.2.3",
      "contentHash": "e392524364811df6d2ee388ba67ed7094d027962627cf670c0603a308e4e178e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          245
        ],
        "answerPdfPages": [
          125
        ],
        "method": "Original textbook242–245 and answer125–126 scans visually compared;die faces6/3/1/2 and heart suit recovered,all subparts and units-pair lists retained.",
        "restoredAttribution": {
          "name": "HMMT",
          "pdfPages": [
            245
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "e392524364811df6d2ee388ba67ed7094d027962627cf670c0603a308e4e178e",
        "method": "Exact geometric-series normalization and enumeration of terminal words through finite rounds,with common tail probability.",
        "attributionOnlyAmendment": {
          "priorContentHash": "1b364699614ccc6dd77a4019977a48b77c30c0011eb9140f14c96c52c274fe05",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.2.4",
      "contentHash": "93208d90d795d74e6ef42bab155a3aa1f97c72dd17de968337d422058d6ff9d8",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          245
        ],
        "answerPdfPages": [
          125
        ],
        "method": "Original textbook242–245 and answer125–126 scans visually compared;die faces6/3/1/2 and heart suit recovered,all subparts and units-pair lists retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "93208d90d795d74e6ef42bab155a3aa1f97c72dd17de968337d422058d6ff9d8",
        "method": "Hypergeometric hand-class counts independently checked by rank/suit occupancy enumeration;exact ratio5/83."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.2.5",
      "contentHash": "4963c70cdc72b0730d087365963c9492eba46d812cd916dbab1a4b9e8e246020",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          245
        ],
        "answerPdfPages": [
          125,
          126
        ],
        "method": "Original textbook242–245 and answer125–126 scans visually compared;die faces6/3/1/2 and heart suit recovered,all subparts and units-pair lists retained.",
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "4963c70cdc72b0730d087365963c9492eba46d812cd916dbab1a4b9e8e246020",
        "method": "Enumerated all9801 ordered integer pairs;conditioning count4901 and favorable count2401. Weighted units-pair counts1401+1000 agree."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.4",
      "contentHash": "73555aa3abb5ed279f25f23e5c670fcd371c6f1d8c136838a78333da0e63e5e3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          245,
          246,
          247
        ],
        "answerPdfPages": [],
        "method": "Original textbook245–247/manual126–128 scans visually compared;all subparts and four-branch urn table retained;source zero-intersection omission explicitly corrected."
      },
      "math": {
        "status": "passed",
        "contentHash": "73555aa3abb5ed279f25f23e5c670fcd371c6f1d8c136838a78333da0e63e5e3",
        "method": "Enumerated all15 labeled-ball leaves with exact two-stage weights;posterior3/4."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.3.1",
      "contentHash": "3fb0f7eb5d0343c438cebe93acaf64ea315853e01d86a2783937a17be5866ef1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          247
        ],
        "answerPdfPages": [
          126
        ],
        "method": "Original textbook245–247/manual126–128 scans visually compared;all subparts and four-branch urn table retained;source zero-intersection omission explicitly corrected."
      },
      "math": {
        "status": "passed",
        "contentHash": "3fb0f7eb5d0343c438cebe93acaf64ea315853e01d86a2783937a17be5866ef1",
        "method": "Exact labeled-ball enumeration:weights1/20 for each X-ball,1/4 for each Y-ball;posterior1/11."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.3.2",
      "contentHash": "46506a3f68793627aa9a49174c2ffc088469f79339a2c20e00fad7712e603bad",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          247
        ],
        "answerPdfPages": [
          126,
          127
        ],
        "method": "Original textbook245–247/manual126–128 scans visually compared;all subparts and four-branch urn table retained;source zero-intersection omission explicitly corrected.",
        "restoredAttribution": {
          "name": "AMC",
          "pdfPages": [
            247
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "46506a3f68793627aa9a49174c2ffc088469f79339a2c20e00fad7712e603bad",
        "method": "Exact four-atom probability model verifies nonnegativity at both endpoints and the full interval;conditional endpoint ratios checked.",
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          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.3.3",
      "contentHash": "22d659097566ff161157f89327e96eeff63e10b35a3ba02e098bc5ebb73cac13",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          247
        ],
        "answerPdfPages": [
          127
        ],
        "method": "Original textbook245–247/manual126–128 scans visually compared;all subparts and four-branch urn table retained;source zero-intersection omission explicitly corrected."
      },
      "math": {
        "status": "passed",
        "contentHash": "22d659097566ff161157f89327e96eeff63e10b35a3ba02e098bc5ebb73cac13",
        "method": "Enumerated four oriented labeled-card outcomes;three red-up,two red-hidden successes."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.3.4",
      "contentHash": "a9448c6845bbac35095a0f966ef2560126ac939f821233fc0b050f25b905f14d",
      "source": {
        "status": "source-gap-corrected",
        "textbookPdfPages": [
          247
        ],
        "answerPdfPages": [
          127
        ],
        "method": "Original textbook245–247/manual126–128 scans visually compared;all subparts and four-branch urn table retained;source zero-intersection omission explicitly corrected."
      },
      "math": {
        "status": "passed",
        "contentHash": "a9448c6845bbac35095a0f966ef2560126ac939f821233fc0b050f25b905f14d",
        "method": "Algebraic cross multiplication gives P(A∩B)(P(A)−P(B))=0. Exact disjoint counterexample verifies the source’s omitted case."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "11.3.5",
      "contentHash": "d123c70e654ef6d7bdfc032eb080e79128d628017b626905eda7f35f4e67620e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          247
        ],
        "answerPdfPages": [
          127,
          128
        ],
        "method": "Original textbook245–247/manual126–128 scans visually compared;all subparts and four-branch urn table retained;source zero-intersection omission explicitly corrected.",
        "restoredAttribution": {
          "name": "HMMT",
          "pdfPages": [
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        },
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
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      "math": {
        "status": "passed",
        "contentHash": "d123c70e654ef6d7bdfc032eb080e79128d628017b626905eda7f35f4e67620e",
        "method": "Exhaustively enumerated two-stage urn choices and labeled balls with exact without-replacement weights;joint7/60,condition1/4,posterior7/15.",
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          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "11.5",
      "contentHash": "1ea095b297c26eb8d986bd556025233a51ac86a72d5d10bb8bc172753664618d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          247,
          248,
          249
        ],
        "answerPdfPages": [],
        "method": "Original textbook248–253/manual128–130 scans visually compared;source geometric figures reconstructed;all subparts retained;conditional independence and meeting information made explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "1ea095b297c26eb8d986bd556025233a51ac86a72d5d10bb8bc172753664618d",
        "method": "Enumerated every unordered labeled pair for0≤m,n≤8 with possible condition;all ratios match the formula."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.6",
      "contentHash": "f895bf84c4d2451a1cbca88ccb22c046653d4ea3f0a21825c85e36c1baa24bf3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          248,
          249,
          250
        ],
        "answerPdfPages": [],
        "method": "Original textbook248–253/manual128–130 scans visually compared;source geometric figures reconstructed;all subparts retained;conditional independence and meeting information made explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "f895bf84c4d2451a1cbca88ccb22c046653d4ea3f0a21825c85e36c1baa24bf3",
        "method": "Exact two-status outcome table:flu-positive9/200,healthy-positive19/200;posterior9/28."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.7",
      "contentHash": "a35d396235c67d30e99d32ec329210327590fe3868699c50f0e640764e044dbc",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          248,
          250,
          251
        ],
        "answerPdfPages": [],
        "method": "Original textbook248–253/manual128–130 scans visually compared;source geometric figures reconstructed;all subparts retained;conditional independence and meeting information made explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "a35d396235c67d30e99d32ec329210327590fe3868699c50f0e640764e044dbc",
        "method": "Exact rational posterior at n14 and15 brackets19/20;conditional likelihood from independent Bernoulli sequence."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.8",
      "contentHash": "a85d711af93d7f614e96fd79d3e7af6c4cf707e8a83ce544c11f127211acabf1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          248,
          251,
          252
        ],
        "answerPdfPages": [],
        "method": "Original textbook248–253/manual128–130 scans visually compared;source geometric figures reconstructed;all subparts retained;conditional independence and meeting information made explicit.",
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "a85d711af93d7f614e96fd79d3e7af6c4cf707e8a83ce544c11f127211acabf1",
        "method": "Exact polygon areas:condition11025,success2925/2;ratio13/98. Verified additional-information alternative13/49."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.4.1",
      "contentHash": "28faf73986eb04c979090902fa2b2a25e87aacd9f5c3ebf5dbb97743ad7c5492",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          252
        ],
        "answerPdfPages": [
          128
        ],
        "method": "Original textbook248–253/manual128–130 scans visually compared;source geometric figures reconstructed;all subparts retained;conditional independence and meeting information made explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "28faf73986eb04c979090902fa2b2a25e87aacd9f5c3ebf5dbb97743ad7c5492",
        "method": "Exact kernel-color and popping branch enumeration;2/9 divided by5/9."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.4.2",
      "contentHash": "d2d469a1e79f0116774a6ad87530adf187014ef75dfcd1051aacb31b17788f0b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          252
        ],
        "answerPdfPages": [
          128
        ],
        "method": "Original textbook248–253/manual128–130 scans visually compared;source geometric figures reconstructed;all subparts retained;conditional independence and meeting information made explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "d2d469a1e79f0116774a6ad87530adf187014ef75dfcd1051aacb31b17788f0b",
        "method": "Enumerated all720 permutations;600 satisfy condition and96 also have third term3."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.4.3",
      "contentHash": "6885733257c46d2d5076942984a6864605cb9d63cf872295721db5855925802e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          252
        ],
        "answerPdfPages": [
          128
        ],
        "method": "Original textbook248–253/manual128–130 scans visually compared;source geometric figures reconstructed;all subparts retained;conditional independence and meeting information made explicit.",
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "6885733257c46d2d5076942984a6864605cb9d63cf872295721db5855925802e",
        "method": "Exact rational polygon area and triangle vertices;ratio1/14."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.4.4",
      "contentHash": "5431cc229d93cf207cb5ae062a4ea01a6e7bcf601c73c19ebfc7d524f8a967d0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          252
        ],
        "answerPdfPages": [
          128,
          129
        ],
        "method": "Original textbook248–253/manual128–130 scans visually compared;source geometric figures reconstructed;all subparts retained;conditional independence and meeting information made explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "5431cc229d93cf207cb5ae062a4ea01a6e7bcf601c73c19ebfc7d524f8a967d0",
        "method": "Enumerated all16 truth/lie patterns under each king status using exact Bernoulli weights;posterior8/89."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.4.5",
      "contentHash": "af5f9bcf691a52cf782b80e772ad75c8ce71d9c903acb93f6eee48320f435d09",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          252,
          253
        ],
        "answerPdfPages": [
          129
        ],
        "method": "Original textbook248–253/manual128–130 scans visually compared;source geometric figures reconstructed;all subparts retained;conditional independence and meeting information made explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "af5f9bcf691a52cf782b80e772ad75c8ce71d9c903acb93f6eee48320f435d09",
        "method": "Exact rational posteriors at11 and12 straddle19/20."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.4.6",
      "contentHash": "5aae5efa8314d86a203aa8843544701b52de78eed6dc85d349910c767d6d178e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          253
        ],
        "answerPdfPages": [
          129,
          130
        ],
        "method": "Original textbook248–253/manual128–130 scans visually compared;source geometric figures reconstructed;all subparts retained;conditional independence and meeting information made explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "5aae5efa8314d86a203aa8843544701b52de78eed6dc85d349910c767d6d178e",
        "method": "Enumerated all eight response strings under each status with exact probabilities;posteriors3/3587,2187/2243,243/467."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.9",
      "contentHash": "0940dc5560b7b80bec79b1508d67041b4642886d91979354fc791f3c15f9265d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          253,
          254,
          255
        ],
        "answerPdfPages": [],
        "method": "Original253–256/manual130–131 scans visually compared;four-row Monty table,all prisoner parts,and all four strategies retained;host/random-draw protocols explicit.",
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "0940dc5560b7b80bec79b1508d67041b4642886d91979354fc791f3c15f9265d",
        "method": "Exact enumeration of car location and host goat choices;summed switching mass2/3."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.10",
      "contentHash": "f683e0eb347d512bceaa2978336cfb72cf2c977c067592e3be396bc0c73568e4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          253,
          255
        ],
        "answerPdfPages": [],
        "method": "Original253–256/manual130–131 scans visually compared;four-row Monty table,all prisoner parts,and all four strategies retained;host/random-draw protocols explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "f683e0eb347d512bceaa2978336cfb72cf2c977c067592e3be396bc0c73568e4",
        "method": "Enumerated each freed prisoner and each of two scraps with weight1/6;four surviving leaves,two freed-self;deliberate non-self disclosure always possible."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.5.1",
      "contentHash": "4f43726c557aebd196d61d49b335e14c7c40a95916e2db5bfe54469e34318e2f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          256
        ],
        "answerPdfPages": [
          130
        ],
        "method": "Original253–256/manual130–131 scans visually compared;four-row Monty table,all prisoner parts,and all four strategies retained;host/random-draw protocols explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "4f43726c557aebd196d61d49b335e14c7c40a95916e2db5bfe54469e34318e2f",
        "method": "Enumerated10 equally likely car locations;9 wins from switch."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.5.2",
      "contentHash": "a740b9505bc64265fb49c14032124a53f6ffc92e4a5ce1eda4131e72a60d65ee",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
          130,
          131
        ],
        "method": "Original253–256/manual130–131 scans visually compared;four-row Monty table,all prisoner parts,and all four strategies retained;host/random-draw protocols explicit.",
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "a740b9505bc64265fb49c14032124a53f6ffc92e4a5ce1eda4131e72a60d65ee",
        "method": "Exact enumeration of car position,eligible host reveals and uniform first-switch targets reproduces1/4,3/8,3/4,5/8."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.11",
      "contentHash": "351a39eb71acd4b6d3f0d55aca6d8093313cf126f6c872f72406634f3c9f960f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          256
        ],
        "answerPdfPages": [
          131
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "351a39eb71acd4b6d3f0d55aca6d8093313cf126f6c872f72406634f3c9f960f",
        "method": "Enumerated primes and parity;one successful outcome among eight."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.12",
      "contentHash": "09d2462d7992ba0936312661580731c26455043fa2d68ec0b1757e001290964c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          257
        ],
        "answerPdfPages": [
          131
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed.",
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "09d2462d7992ba0936312661580731c26455043fa2d68ec0b1757e001290964c",
        "method": "Enumerated5 labeled transfers and8 labeled second draws;23 eligible leaves,15 white-transfer successes."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.13",
      "contentHash": "0693592ee3dd48e7f6ddbf455ea8761fc5a979eda981709d2440b468853e976f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          257
        ],
        "answerPdfPages": [
          131
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "0693592ee3dd48e7f6ddbf455ea8761fc5a979eda981709d2440b468853e976f",
        "method": "Exact count of500·512 coin/flip leaves:512 double-headed eligible leaves plus499 fair eligible leaves."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.14",
      "contentHash": "7d798936fe041889ad78e37abadeeabe6ca27b3757394d7f4338d831568e943f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          257
        ],
        "answerPdfPages": [
          131,
          132
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "7d798936fe041889ad78e37abadeeabe6ca27b3757394d7f4338d831568e943f",
        "method": "Enumerated eight three-type patterns under each weather state;exact posterior192/199."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.15",
      "contentHash": "6eef6ef021ff630dde31a471e556981cb9d52eae6843096ed2bb8727b1f40a08",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          257
        ],
        "answerPdfPages": [
          132
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed.",
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "6eef6ef021ff630dde31a471e556981cb9d52eae6843096ed2bb8727b1f40a08",
        "method": "Exact disk containment via triangle inequality;uniform area ratio1²/4²."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.16",
      "contentHash": "7ee080ad8f1ac5832c5d0b8c61f4d935bf96b080a42589180ce471ef9b8b8985",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          257
        ],
        "answerPdfPages": [
          132
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "7ee080ad8f1ac5832c5d0b8c61f4d935bf96b080a42589180ce471ef9b8b8985",
        "method": "Exact four-leaf attendance model;posterior24/29."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.17",
      "contentHash": "7e80151dde6094ec8ad94187380f55783505b73a4619ca2bee06b34e10316105",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          257
        ],
        "answerPdfPages": [
          132
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "7e80151dde6094ec8ad94187380f55783505b73a4619ca2bee06b34e10316105",
        "method": "Enumerated all64 six-game words,retaining only first attainment of4 wins on game6;condition200/729,Royals160/729."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.18",
      "contentHash": "144eadf5439b0f4c47f1533d20109a6be5e5405763ce05418cd8d8e0356051ca",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          257
        ],
        "answerPdfPages": [
          133
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed.",
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "144eadf5439b0f4c47f1533d20109a6be5e5405763ce05418cd8d8e0356051ca",
        "method": "Exact rational polygon clipping gives overlap area31/2 and contained rectangle area2;ratio4/31."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.19",
      "contentHash": "c00a63b883b0ede10207a374fd6d361b6c2719a47d93d550784996a47a019080",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          257
        ],
        "answerPdfPages": [
          133
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed."
      },
      "math": {
        "status": "passed",
        "contentHash": "c00a63b883b0ede10207a374fd6d361b6c2719a47d93d550784996a47a019080",
        "method": "Enumerated all38760 choices of six tail positions;5005 have no adjacent pair."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.20",
      "contentHash": "f89ff5958596364f94312c60e4c04d308fdbb287dd51ad9cf25e2bc5ed570800",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          257
        ],
        "answerPdfPages": [
          133,
          134
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed.",
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "f89ff5958596364f94312c60e4c04d308fdbb287dd51ad9cf25e2bc5ed570800",
        "method": "Exact rational binomial likelihoods for every k0–10;minimum posterior≥9/10 occurs at6."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "11.21",
      "contentHash": "449f6d731eefefed3f5e97dc4886fb30f3b40593937cac4d112192d462d41da1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          257
        ],
        "answerPdfPages": [
          134
        ],
        "method": "Original256–257/manual131–134 scans visually compared;lost die pips restored as six;all tables/geometric diagrams reconstructed.",
        "chapter11FreshFidelity": "All original242–257/manual125–134 scans compared; existing source proofs/tables/figures and explicit model repairs retained; mobile labels/table spacing restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "449f6d731eefefed3f5e97dc4886fb30f3b40593937cac4d112192d462d41da1",
        "method": "Independent circular-gap model:condition largest spacing≥1/2 has probability3/4;all on arc1/3 has probability1/3. Complement ratio1−(1/3)/(3/4)=5/9. Exact polygon area scales4/9."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.1",
      "contentHash": "92811e5a6779fd719af61eb5e8b440cee8843470ecf7b91ce6deafde7682ae1f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          258,
          260
        ],
        "answerPdfPages": [],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson."
      },
      "math": {
        "status": "passed",
        "contentHash": "92811e5a6779fd719af61eb5e8b440cee8843470ecf7b91ce6deafde7682ae1f",
        "method": "Exhaustive subset complements and path-word counts through n10;factorial symmetry holds symbolically."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.2",
      "contentHash": "efdbaeb35df40f4733277ec2f48a694ae8c61c5b367e8131e430d3beb3a20830",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          259,
          261,
          262
        ],
        "answerPdfPages": [],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson."
      },
      "math": {
        "status": "passed",
        "contentHash": "efdbaeb35df40f4733277ec2f48a694ae8c61c5b367e8131e430d3beb3a20830",
        "method": "Partitioned every(r+1)-subset by membership of distinguished Katie through n9;exact Pascal values through n60."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.3",
      "contentHash": "f2cd8be0942aaefba97583339daf2ef81502c83821298511139d04da66abb3a0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          259,
          262
        ],
        "answerPdfPages": [],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson."
      },
      "math": {
        "status": "passed",
        "contentHash": "f2cd8be0942aaefba97583339daf2ef81502c83821298511139d04da66abb3a0",
        "method": "Enumerated all2^n x/y words through n12;coefficient histogram matches every binomial coefficient."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.4",
      "contentHash": "49d84a7e2d0c51c9c6321660714cb7b8126dc5f75cbc50a3208b2e0eb9bd3c0a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          259,
          263,
          264
        ],
        "answerPdfPages": [],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson."
      },
      "math": {
        "status": "passed",
        "contentHash": "49d84a7e2d0c51c9c6321660714cb7b8126dc5f75cbc50a3208b2e0eb9bd3c0a",
        "method": "Enumerated subsets/paths through n12 and exact row sums through n60."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.5",
      "contentHash": "5a316f7cd727d72b03881b109bf99c45d393dc5d267bbf669632f1ff84081bdf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          259,
          264,
          265
        ],
        "answerPdfPages": [],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson."
      },
      "math": {
        "status": "passed",
        "contentHash": "5a316f7cd727d72b03881b109bf99c45d393dc5d267bbf669632f1ff84081bdf",
        "method": "Counted even subsets through n12;toggle a fixed member bijects even and odd subsets."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.6",
      "contentHash": "e9aa6f3ff466089800fd79d90eb4ffa6955374ed48d7d5a6b183fcadd5c02c0d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          259,
          265,
          266,
          267,
          268
        ],
        "answerPdfPages": [],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson.",
        "chapter12FreshFidelity": "All original258–279 and manual135–147 scans compared; all three proof methods and complete sidebar practice retained; source explicit examples and native captions restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "e9aa6f3ff466089800fd79d90eb4ffa6955374ed48d7d5a6b183fcadd5c02c0d",
        "method": "Partitioned all subsets and path words through n10 by largest member/last right-step row;all case counts C(i,k)."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.2.1",
      "contentHash": "2ad2bf0d178ab8d5e6b0ae85ee2843213831cf167f60082cfc340ac1113718b5",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          268
        ],
        "answerPdfPages": [
          135
        ],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson."
      },
      "math": {
        "status": "passed",
        "contentHash": "2ad2bf0d178ab8d5e6b0ae85ee2843213831cf167f60082cfc340ac1113718b5",
        "method": "Even/odd subset sign cancellation through n12;exact sums through n60."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.2.2",
      "contentHash": "9de65515645967c844d4f535951552b126f81b6e6b06bf195ade4f95d0c42695",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          268
        ],
        "answerPdfPages": [
          135
        ],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson."
      },
      "math": {
        "status": "passed",
        "contentHash": "9de65515645967c844d4f535951552b126f81b6e6b06bf195ade4f95d0c42695",
        "method": "Enumerated all marked subsets through n9;boundary k0 separately verified."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.2.3",
      "contentHash": "19fde10fc96f31f7cc599fd0e8361ee75e15c132f02ba8c451f5e1fc7737363c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          268
        ],
        "answerPdfPages": [
          135,
          136
        ],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson."
      },
      "math": {
        "status": "passed",
        "contentHash": "19fde10fc96f31f7cc599fd0e8361ee75e15c132f02ba8c451f5e1fc7737363c",
        "method": "Enumerated all three-state words through n8 and grouped by non-absent positions."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.2.4",
      "contentHash": "7a63e06c4d7f4f80e3aaf5175f1c6cd8319ce161df89485950f895b272f85891",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          268
        ],
        "answerPdfPages": [
          136
        ],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson."
      },
      "math": {
        "status": "passed",
        "contentHash": "7a63e06c4d7f4f80e3aaf5175f1c6cd8319ce161df89485950f895b272f85891",
        "method": "Enumerated nested subset pairs through n8;common multinomial factorial expression."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.2.5",
      "contentHash": "988661bbc4c8e4039f1b6c958228683971ce4cfdafd4367a8a4d5ed896d210bd",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          268
        ],
        "answerPdfPages": [
          136,
          137
        ],
        "method": "Original258–268/manual135–137 scans visually compared;all requested proof methods retained;two hockey-stick path diagrams reconstructed;noninteger-binomial sidebar preserved in lesson."
      },
      "math": {
        "status": "passed",
        "contentHash": "988661bbc4c8e4039f1b6c958228683971ce4cfdafd4367a8a4d5ed896d210bd",
        "method": "Enumerated odd subsets with a marked member through n12;exact weighted sums through n60."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.7",
      "contentHash": "be4d13d2b57493017ec5ee9308b996fce6d27a2cec1f4b6a6492584cf5968e52",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          269,
          270
        ],
        "answerPdfPages": [],
        "method": "Original269–276/manual137–140 scans visually compared;all subparts,alternate methods,exploration tables,false-conjecture checks and induction boundaries retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "be4d13d2b57493017ec5ee9308b996fce6d27a2cec1f4b6a6492584cf5968e52",
        "method": "Exhaustive subset grouping for m,n≤6 and exact convolution for m,n,r≤20,including impossible sizes."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.8",
      "contentHash": "3450f065b50cda9d77536f0ba7597dfd9885956967d1d4486aebd0bf8cc89c8b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          269,
          270,
          271
        ],
        "answerPdfPages": [],
        "method": "Original269–276/manual137–140 scans visually compared;all subparts,alternate methods,exploration tables,false-conjecture checks and induction boundaries retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "3450f065b50cda9d77536f0ba7597dfd9885956967d1d4486aebd0bf8cc89c8b",
        "method": "Exact rational-free convolution through m,n,r20;independent subset partition by selected first-group count."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.9",
      "contentHash": "6bbf2152bdedf63aca640c8b517b406d2bc5ad45380e6d8737232f8f2b494608",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          269,
          271,
          272
        ],
        "answerPdfPages": [],
        "method": "Original269–276/manual137–140 scans visually compared;all subparts,alternate methods,exploration tables,false-conjecture checks and induction boundaries retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "6bbf2152bdedf63aca640c8b517b406d2bc5ad45380e6d8737232f8f2b494608",
        "method": "Enumerated all marked subsets of11 people;11264;independent complementary-index pairing."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.10",
      "contentHash": "b72cf40fc6ce82991ec8fc75b713859d9d8de7947fd4224369ddd1737abd1f4a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          269,
          272,
          273,
          274,
          275
        ],
        "answerPdfPages": [],
        "method": "Original269–276/manual137–140 scans visually compared;all subparts,alternate methods,exploration tables,false-conjecture checks and induction boundaries retained.",
        "chapter12FreshFidelity": "All original258–279 and manual135–147 scans compared; all three proof methods and complete sidebar practice retained; source explicit examples and native captions restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "b72cf40fc6ce82991ec8fc75b713859d9d8de7947fd4224369ddd1737abd1f4a",
        "method": "Exact Fraction sum for every1≤a≤b≤60 and original19/99 case;independent hockey-stick representation."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.3.1",
      "contentHash": "a84820df591fcdb86320744c69db0f4cc5a8f7d17485a7418ff30b441519aecf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          275
        ],
        "answerPdfPages": [
          137
        ],
        "method": "Original269–276/manual137–140 scans visually compared;all subparts,alternate methods,exploration tables,false-conjecture checks and induction boundaries retained.",
        "chapter12FreshFidelity": "All original258–279 and manual135–147 scans compared; all three proof methods and complete sidebar practice retained; source explicit examples and native captions restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "a84820df591fcdb86320744c69db0f4cc5a8f7d17485a7418ff30b441519aecf",
        "method": "Enumerated all unordered pairs of edges through n9;classified by3 or4 endpoints with counts3C(n,3)+3C(n,4)=3C(n+1,4)."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.3.2",
      "contentHash": "8950e1fbee8859ff10b2cef1cfceed6d04b93f029b8ae04774e7ead344819632",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          276
        ],
        "answerPdfPages": [
          138
        ],
        "method": "Original269–276/manual137–140 scans visually compared;all subparts,alternate methods,exploration tables,false-conjecture checks and induction boundaries retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "8950e1fbee8859ff10b2cef1cfceed6d04b93f029b8ae04774e7ead344819632",
        "method": "Enumerated all permutations up to size8 and classified by exact initial fixed-prefix length;matches(n−k)(n−k)!."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.3.3",
      "contentHash": "18d49ea39f167c639d66bbdeec63e1942a0f38e8c56695b44c41bfdec7723c25",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          276
        ],
        "answerPdfPages": [
          138
        ],
        "method": "Original269–276/manual137–140 scans visually compared;all subparts,alternate methods,exploration tables,false-conjecture checks and induction boundaries retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "18d49ea39f167c639d66bbdeec63e1942a0f38e8c56695b44c41bfdec7723c25",
        "method": "Exact integer weighted sum and independent complementary coefficient pairing give7340032."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.3.4",
      "contentHash": "da57d7544c7089fa0359268107d31307069704f1ae5c09b1fbdeb0b0df56672e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          276
        ],
        "answerPdfPages": [
          138,
          139
        ],
        "method": "Original269–276/manual137–140 scans visually compared;all subparts,alternate methods,exploration tables,false-conjecture checks and induction boundaries retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "da57d7544c7089fa0359268107d31307069704f1ae5c09b1fbdeb0b0df56672e",
        "method": "Exact enumeration of14-choose10 committees classified by first-group size;zero terms and empty-range cases checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.3.5",
      "contentHash": "3516dc912a715bdfd0b9c939c243ff650dced40e354e00ceabe1ea8b121c97ae",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          276
        ],
        "answerPdfPages": [
          139,
          140
        ],
        "method": "Original269–276/manual137–140 scans visually compared;all subparts,alternate methods,exploration tables,false-conjecture checks and induction boundaries retained.",
        "chapter12FreshFidelity": "All original258–279 and manual135–147 scans compared; all three proof methods and complete sidebar practice retained; source explicit examples and native captions restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "3516dc912a715bdfd0b9c939c243ff650dced40e354e00ceabe1ea8b121c97ae",
        "method": "Exact termwise binomial/falling-factorial equality for1≤a≤b≤60 and independent hockey-stick sum."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.11",
      "contentHash": "fc25a376769e024dcfd9f4188c09d8d70836e15137342fb7e26da8bc2802a188",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          277
        ],
        "answerPdfPages": [
          140
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "fc25a376769e024dcfd9f4188c09d8d70836e15137342fb7e26da8bc2802a188",
        "method": "Exact convolutions for0≤m,n≤40;independent subset split."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.12",
      "contentHash": "c5f3cfc846dacedb531573218124b115cf3c5edc515181c733890a0b5f8efe9c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          277
        ],
        "answerPdfPages": [
          140
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "c5f3cfc846dacedb531573218124b115cf3c5edc515181c733890a0b5f8efe9c",
        "method": "Exact convolution through n80;zero term and n1 boundary checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.13",
      "contentHash": "b79ae4ea32808011c97ab048f9897cd35a9946501929f98d3072229997998e6b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          277
        ],
        "answerPdfPages": [
          140,
          141
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "b79ae4ea32808011c97ab048f9897cd35a9946501929f98d3072229997998e6b",
        "method": "Enumerated committees with a distinguished nonmember through n9;factorial cancellation and endpoint separately checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.14",
      "contentHash": "b7b578ab9d4e441c02ed09c30b105943808468c3bfdf0169b2d013ecb2b58f6c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          277,
          278
        ],
        "answerPdfPages": [
          141
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "b7b578ab9d4e441c02ed09c30b105943808468c3bfdf0169b2d013ecb2b58f6c",
        "method": "Enumerated all ordered triples for n0–12 and classified by number of distinct values."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.15",
      "contentHash": "57d86c39a43c064b890368425e53f691638e19692a932728b23bd5078a24ac64",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          278
        ],
        "answerPdfPages": [
          141,
          142
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "57d86c39a43c064b890368425e53f691638e19692a932728b23bd5078a24ac64",
        "method": "Enumerated all label pairs for1≤p,q≤8 and exact polynomial equality through p,q60."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.16",
      "contentHash": "32512242f4d723c57efdb72ab0520851ddc5a6555a01165a0905ec5e677227fa",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          278
        ],
        "answerPdfPages": [
          142
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "32512242f4d723c57efdb72ab0520851ddc5a6555a01165a0905ec5e677227fa",
        "method": "Exact lower-half row sums for n0–100;complement pairing and central overlap checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.17",
      "contentHash": "0e7e853f9d173f7c4c1e889e4b001b8f2db5b328bea5118eaf609e7169c8aa33",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          278
        ],
        "answerPdfPages": [
          143,
          144
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "0e7e853f9d173f7c4c1e889e4b001b8f2db5b328bea5118eaf609e7169c8aa33",
        "method": "Exact Fraction exponent accumulation over every nonempty subset through n12;each final exponent1/n."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.18",
      "contentHash": "bf57c8fdaabb5511fc1c12632fdf41a283d5862fc089c8294dfad65d7d691251",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          278
        ],
        "answerPdfPages": [
          144
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "bf57c8fdaabb5511fc1c12632fdf41a283d5862fc089c8294dfad65d7d691251",
        "method": "Exact weighted even-index sums for positive even n through100;independent three-state marked-word coefficient computation."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.19",
      "contentHash": "30c17261ba67d025cc70f92fd5bdbc2cc8c0537c7982051df87fca225feea8dd",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          278
        ],
        "answerPdfPages": [
          145
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "30c17261ba67d025cc70f92fd5bdbc2cc8c0537c7982051df87fca225feea8dd",
        "method": "Exact integer binomial sum equals−2^49;independent Gaussian-integer multiplication of(1+i)^99."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.20",
      "contentHash": "619841e08eb8aa485e1e33a6234a56880f3f082694ea7aa53e63249676c3fedc",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          278
        ],
        "answerPdfPages": [
          146
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "619841e08eb8aa485e1e33a6234a56880f3f082694ea7aa53e63249676c3fedc",
        "method": "Enumerated all n-person subsets of2n+1 people through n8,classified by singleton-couple count;matches every summand."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.21",
      "contentHash": "a8e2f3ece4d87361221c3616dbf8d2ee8dc01bb331800240639c6d4e015ec61e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          278
        ],
        "answerPdfPages": [
          146
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "a8e2f3ece4d87361221c3616dbf8d2ee8dc01bb331800240639c6d4e015ec61e",
        "method": "Exact sums for n1–150 and integer recurrence for coefficients modulo3;piecewise filter formula checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "12.22",
      "contentHash": "64a5b6c3b84406663582f48c46df256eead232e9eaa231cc8245e5c05eecf0b6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          279
        ],
        "answerPdfPages": [
          146,
          147
        ],
        "method": "Original276–279/manual140–147 visually compared;all alternate proof methods and Stirling subparts/example partitions retained;domain boundaries explicit.",
        "chapter12FreshFidelity": "All original258–279 and manual135–147 scans compared; all three proof methods and complete sidebar practice retained; source explicit examples and native captions restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "64a5b6c3b84406663582f48c46df256eead232e9eaa231cc8245e5c05eecf0b6",
        "method": "Enumerated restricted-growth strings through n8;compared block counts,recurrence,and exact PIE formula for every k;all six S4,3 partitions retained."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.1",
      "contentHash": "1003d9d340538aab1785aea2230a6a7b89b0bb25ffd8087f6f08fd7244bb2c3e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          281,
          282,
          283,
          284,
          285
        ],
        "answerPdfPages": [],
        "method": "Original280–292/manual148–150 visually compared;all subparts,progressive grids,state transition chains,full probability tables,and cycle cases retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "1003d9d340538aab1785aea2230a6a7b89b0bb25ffd8087f6f08fd7244bb2c3e",
        "method": "Independently enumerated all terminated game words and compared exact forward-state DP for each of four cases;terminal mass1."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.2",
      "contentHash": "7af6e16d1b425af5567e5714548d9f60d3af4db771d01b99dd4f8b28602590d9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          281,
          286,
          287,
          288
        ],
        "answerPdfPages": [],
        "method": "Original280–292/manual148–150 visually compared;all subparts,progressive grids,state transition chains,full probability tables,and cycle cases retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "7af6e16d1b425af5567e5714548d9f60d3af4db771d01b99dd4f8b28602590d9",
        "method": "Enumerated all3^6 labeled-edge walks on tetrahedron and cube;183 returns each. Exact distance-state tables through6 compared with vertex distributions."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.3",
      "contentHash": "06f9a6b549a9a6fe9425b887da1a42f9ee4d9d064bf732b732e57e4bf8ea7a2a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          281,
          288,
          289,
          290,
          291
        ],
        "answerPdfPages": [],
        "method": "Original280–292/manual148–150 visually compared;all subparts,progressive grids,state transition chains,full probability tables,and cycle cases retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "06f9a6b549a9a6fe9425b887da1a42f9ee4d9d064bf732b732e57e4bf8ea7a2a",
        "method": "Enumerated all2^n line-walk words through n16 and exact endpoint distributions through n100;expected distances and parity formulas match. Exact barrier harmonic equations checked for a1–6,b−1…−6."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.2.1",
      "contentHash": "6a2231c6e3eba5164077049d4fdf3531b5824687dcaeed7b3ba540bce7090d0e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          291
        ],
        "answerPdfPages": [
          148
        ],
        "method": "Original280–292/manual148–150 visually compared;all subparts,progressive grids,state transition chains,full probability tables,and cycle cases retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "6a2231c6e3eba5164077049d4fdf3531b5824687dcaeed7b3ba540bce7090d0e",
        "method": "Enumerated every cube edge walk until first hit or depth5;48 terminal paths."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.2.2",
      "contentHash": "b6c6f864a46f3f61bee6ef67b8ff3495ec22164251ee786eff981f8e93920003",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          291
        ],
        "answerPdfPages": [
          148
        ],
        "method": "Original280–292/manual148–150 visually compared;all subparts,progressive grids,state transition chains,full probability tables,and cycle cases retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "b6c6f864a46f3f61bee6ef67b8ff3495ec22164251ee786eff981f8e93920003",
        "method": "Enumerated all2187 seven-step cube walks;18 visit all eight vertices without repetition."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.2.3",
      "contentHash": "a884a0c8460c2af900a7e54669e1c766d6df1be337930bdcbe5a39eaae438290",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          291
        ],
        "answerPdfPages": [
          148,
          149
        ],
        "method": "Original280–292/manual148–150 visually compared;all subparts,progressive grids,state transition chains,full probability tables,and cycle cases retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "a884a0c8460c2af900a7e54669e1c766d6df1be337930bdcbe5a39eaae438290",
        "method": "Exact variable-probability state DP and independently enumerated all128 preassigned seven-game words;Sox win mass1/2."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.2.4",
      "contentHash": "67c1adb7cadc8add0e77b9b0bb6c6a4bfe5766d2cba3a9a7dc93b1d15b558a47",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          291,
          292
        ],
        "answerPdfPages": [
          149,
          150
        ],
        "method": "Original280–292/manual148–150 visually compared;all subparts,progressive grids,state transition chains,full probability tables,and cycle cases retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "67c1adb7cadc8add0e77b9b0bb6c6a4bfe5766d2cba3a9a7dc93b1d15b558a47",
        "method": "Enumerated all4^6 simultaneous moves and all valid destination permutations;80 successes with cycle-type counts16,24,8,32."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.4",
      "contentHash": "3d7a034c4d6e360782ec4c6afcde03cc6d2757aede6e80159de348ff990abb7a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          292,
          293
        ],
        "answerPdfPages": [],
        "method": "Original292–299/manual151–153 visually compared;all subparts,alternate geometric/renewal methods,full craps/winner tables,and both run automata retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "3d7a034c4d6e360782ec4c6afcde03cc6d2757aede6e80159de348ff990abb7a",
        "method": "Exact rational three-transient-state absorption system gives9/13;independent two-point renewal sum."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.5",
      "contentHash": "48693316a8cc6c9df099dd281cdd46ecbb9f8c9c71b62a2d8c7c873bc1c8b41c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          292,
          294
        ],
        "answerPdfPages": [],
        "method": "Original292–299/manual151–153 visually compared;all subparts,alternate geometric/renewal methods,full craps/winner tables,and both run automata retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "48693316a8cc6c9df099dd281cdd46ecbb9f8c9c71b62a2d8c7c873bc1c8b41c",
        "method": "Exact renewal linear system and independent tail-sumΣP(T>k)=Σ(5/6)^k=6;source die pips visually restored as six."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.6",
      "contentHash": "84a88cca947329395c8e7f0212605db808a0798682ad105f618b1babfc83716c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          292,
          295,
          296
        ],
        "answerPdfPages": [],
        "method": "Original292–299/manual151–153 visually compared;all subparts,alternate geometric/renewal methods,full craps/winner tables,and both run automata retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "84a88cca947329395c8e7f0212605db808a0798682ad105f618b1babfc83716c",
        "method": "Exact six-state rational expectation system with terminal cost1 gives121/30;checked all displayed intermediate equations."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.7",
      "contentHash": "0e883243e5a921b24e40964455b00c07e92aee416e750027965f7736a459e7a2",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          292,
          296,
          297
        ],
        "answerPdfPages": [],
        "method": "Original292–299/manual151–153 visually compared;all subparts,alternate geometric/renewal methods,full craps/winner tables,and both run automata retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "0e883243e5a921b24e40964455b00c07e92aee416e750027965f7736a459e7a2",
        "method": "Exact birth-death expectation system and independent eight-labeled-ball-configuration system both give10."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.8",
      "contentHash": "6e291e30729cb38fe8dfbd3756b318f40338f5fdac495719c00bc27edf530adf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          292,
          297,
          298
        ],
        "answerPdfPages": [],
        "method": "Original292–299/manual151–153 visually compared;all subparts,alternate geometric/renewal methods,full craps/winner tables,and both run automata retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "6e291e30729cb38fe8dfbd3756b318f40338f5fdac495719c00bc27edf530adf",
        "method": "Enumerated all36 ordered rolls,solved each point’s exact absorbing equation,and combined disjoint initial branches;244/495."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.3.1",
      "contentHash": "4bc81b9e1d711f9c0478258189dbcc8de1d323516ffb0cfd41288a3911054297",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          298
        ],
        "answerPdfPages": [
          151
        ],
        "method": "Original292–299/manual151–153 visually compared;all subparts,alternate geometric/renewal methods,full craps/winner tables,and both run automata retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "4bc81b9e1d711f9c0478258189dbcc8de1d323516ffb0cfd41288a3911054297",
        "method": "Exact three-turn Markov absorption system and geometric complete-round decomposition;25/91."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.3.2",
      "contentHash": "4710f870bd7fdf13d51ec707717bbdcc93668216d8e0f6eda9d16d7d26f184a7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          298,
          299
        ],
        "answerPdfPages": [
          151
        ],
        "method": "Original292–299/manual151–153 visually compared;all subparts,alternate geometric/renewal methods,full craps/winner tables,and both run automata retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "4710f870bd7fdf13d51ec707717bbdcc93668216d8e0f6eda9d16d7d26f184a7",
        "method": "Exact independent geometric home-run distributions and paired-swing renewal equations;Doug1/5,tie2/5."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.3.3",
      "contentHash": "41e411d4645651cd5c219141b204cb9f30603c66f758c6ddb6b014c4b54ebbea",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          299
        ],
        "answerPdfPages": [
          151
        ],
        "method": "Original292–299/manual151–153 visually compared;all subparts,alternate geometric/renewal methods,full craps/winner tables,and both run automata retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "41e411d4645651cd5c219141b204cb9f30603c66f758c6ddb6b014c4b54ebbea",
        "method": "Exact three-transient-state point-by-point expectation system gives50/13;independent two-point block geometric mean."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.3.4",
      "contentHash": "80303660fae51f134694261516417769e27725bca44d8051d03ad4ba17283790",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          299
        ],
        "answerPdfPages": [
          151,
          152
        ],
        "method": "Original292–299/manual151–153 visually compared;all subparts,alternate geometric/renewal methods,full craps/winner tables,and both run automata retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "80303660fae51f134694261516417769e27725bca44d8051d03ad4ba17283790",
        "method": "Exact original seven-transient-state suffix automaton and independent two-state compressed equations both give3/34."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.3.5",
      "contentHash": "92190aabec02936829eb5695797efc10cda5d7c1f6c194eb46cc8b441ad52146",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          299
        ],
        "answerPdfPages": [
          152,
          153
        ],
        "method": "Original292–299/manual151–153 visually compared;all subparts,alternate geometric/renewal methods,full craps/winner tables,and both run automata retained.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "92190aabec02936829eb5695797efc10cda5d7c1f6c194eb46cc8b441ad52146",
        "method": "Exact two-player winner-state transition repeated through6;independent closed-form alternating difference gives364/729."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.9",
      "contentHash": "a67da3b963ee2bc69ea773ac4edf11c121359bb20c85899dba52915740233df4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          299,
          300,
          301,
          302
        ],
        "answerPdfPages": [],
        "method": "Original299–304,378–382/manual153–155 visually compared;all subparts,case tables,Nim response families,Chomp cookies,and Wythoff table retained;duplicated PDF block skipped canonically.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "a67da3b963ee2bc69ea773ac4edf11c121359bb20c85899dba52915740233df4",
        "method": "Full minimax DP for1≤k≤12,n0–500 matches multiples of k+1;all17 source states retained."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.10",
      "contentHash": "6503b2f9e50ec21203597926b31286efcb81b71552c43dc3414abcf1312f9198",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          299,
          303,
          304
        ],
        "answerPdfPages": [],
        "method": "Original299–304,378–382/manual153–155 visually compared;all subparts,case tables,Nim response families,Chomp cookies,and Wythoff table retained;duplicated PDF block skipped canonically.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "6503b2f9e50ec21203597926b31286efcb81b71552c43dc3414abcf1312f9198",
        "method": "Memoized exhaustive legal-move minimax for all0≤a,b≤80;matches every source table category and boundary."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.11",
      "contentHash": "f38877026e581c51b3165c35523e25d3e5e9fa8df9c645596af21761552b6db0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          299,
          378,
          379,
          380
        ],
        "answerPdfPages": [],
        "method": "Original299–304,378–382/manual153–155 visually compared;all subparts,case tables,Nim response families,Chomp cookies,and Wythoff table retained;duplicated PDF block skipped canonically.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "f38877026e581c51b3165c35523e25d3e5e9fa8df9c645596af21761552b6db0",
        "method": "Exhaustive three-pile minimax through sizes25;all even/odd n states and every move-response family checked for k1–30."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.12",
      "contentHash": "a77b4768f854db94172fea8a4fc2630c4ad2b68e8bc15663c05805adf6b6deb0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          300,
          380,
          381
        ],
        "answerPdfPages": [],
        "method": "Original299–304,378–382/manual153–155 visually compared;all subparts,case tables,Nim response families,Chomp cookies,and Wythoff table retained;duplicated PDF block skipped canonically.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "a77b4768f854db94172fea8a4fc2630c4ad2b68e8bc15663c05805adf6b6deb0",
        "method": "Exact memoized Chomp minimax over monotone shapes inside5×7 verifies the initial board wins and every nontrivial rectangle through5×7 wins;792 shapes including empty counted."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.4.1",
      "contentHash": "cfdd04a147750b4221f1ee2aae32c19d764c2abb8e76d37fb45b5f66a00ee376",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          381
        ],
        "answerPdfPages": [
          153,
          154
        ],
        "method": "Original299–304,378–382/manual153–155 visually compared;all subparts,case tables,Nim response families,Chomp cookies,and Wythoff table retained;duplicated PDF block skipped canonically.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "cfdd04a147750b4221f1ee2aae32c19d764c2abb8e76d37fb45b5f66a00ee376",
        "method": "Exhaustive move DP through n2000 for all five rule sets;residue/parity classifications and every source small case agree."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.4.2",
      "contentHash": "71f2c42c1d4e7bb28cfcddb0d54917f9e785ab9226cdbaa338e64745f54dbded",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          382
        ],
        "answerPdfPages": [
          154
        ],
        "method": "Original299–304,378–382/manual153–155 visually compared;all subparts,case tables,Nim response families,Chomp cookies,and Wythoff table retained;duplicated PDF block skipped canonically."
      },
      "math": {
        "status": "passed",
        "contentHash": "71f2c42c1d4e7bb28cfcddb0d54917f9e785ab9226cdbaa338e64745f54dbded",
        "method": "Exact backward minimax for current values1–1000 and all multipliers2–9;all intervals and Larry’s openings checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.4.3",
      "contentHash": "37d9756514f08fc9255080b42b0824e58152952a93ed28b82377b06d56e0b10d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          382
        ],
        "answerPdfPages": [
          154
        ],
        "method": "Original299–304,378–382/manual153–155 visually compared;all subparts,case tables,Nim response families,Chomp cookies,and Wythoff table retained;duplicated PDF block skipped canonically."
      },
      "math": {
        "status": "passed",
        "contentHash": "37d9756514f08fc9255080b42b0824e58152952a93ed28b82377b06d56e0b10d",
        "method": "Memoized complete legal-split minimax for both pile sizes1–60;matches both-odd losing exactly."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.4.4",
      "contentHash": "0e8d6eabc7253f000a85a1cb747a08f11bf108d0213cce8b49dccff1e4fd83a1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          382
        ],
        "answerPdfPages": [
          154
        ],
        "method": "Original299–304,378–382/manual153–155 visually compared;all subparts,case tables,Nim response families,Chomp cookies,and Wythoff table retained;duplicated PDF block skipped canonically.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "0e8d6eabc7253f000a85a1cb747a08f11bf108d0213cce8b49dccff1e4fd83a1",
        "method": "Complete three-pile minimax and verification of every source move-response pair;Nim xor independently agrees."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.4.5",
      "contentHash": "0b50f14920ede48fbc6c7f4992ca06ba0e5d525106306273a8707c1a4a6d5e88",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          382
        ],
        "answerPdfPages": [
          154
        ],
        "method": "Original299–304,378–382/manual153–155 visually compared;all subparts,case tables,Nim response families,Chomp cookies,and Wythoff table retained;duplicated PDF block skipped canonically."
      },
      "math": {
        "status": "passed",
        "contentHash": "0b50f14920ede48fbc6c7f4992ca06ba0e5d525106306273a8707c1a4a6d5e88",
        "method": "Geometric reflection invariant checked for paired diagonals in regular even polygons;finite-game parity argument establishes first-player win."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.4.6",
      "contentHash": "412bdcc04676f6a6456e3efda0afef307a7aabf0d9321298f1db24118a3d91ac",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          382
        ],
        "answerPdfPages": [
          155
        ],
        "method": "Original299–304,378–382/manual153–155 visually compared;all subparts,case tables,Nim response families,Chomp cookies,and Wythoff table retained;duplicated PDF block skipped canonically.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "412bdcc04676f6a6456e3efda0afef307a7aabf0d9321298f1db24118a3d91ac",
        "method": "Exact two-pile Wythoff minimax through100;least-unused pairs compared with exact integer-floor golden-ratio formula through10000;all move classifications checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.13",
      "contentHash": "7aa8574803fa38f70201545b9bf056fe5799cc6999006dcd8626521cdfed21ce",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          383
        ],
        "answerPdfPages": [
          155
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "7aa8574803fa38f70201545b9bf056fe5799cc6999006dcd8626521cdfed21ce",
        "method": "Exact suffix automaton with absorbing HH/TH agrees with direct first-two-flip event;probability1/4."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.14",
      "contentHash": "a071e9db25d7af05b5f22a29c4b547b1218b9c94d6b8eadf08ec002edf970ac8",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          383
        ],
        "answerPdfPages": [
          155,
          156
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "a071e9db25d7af05b5f22a29c4b547b1218b9c94d6b8eadf08ec002edf970ac8",
        "method": "Exact26-transient-state rational solver on the full27-cube adjacency graph gives21 at every corner;independent four-type reduction agrees."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.15",
      "contentHash": "8a25bf6c929c823723c975f256557de05aefafba5cbc4b4d7d2d3439bbf25f5a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          383
        ],
        "answerPdfPages": [
          156
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "8a25bf6c929c823723c975f256557de05aefafba5cbc4b4d7d2d3439bbf25f5a",
        "method": "Exact two-state expectation equations and independent sum of geometric waits2+4=6."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.16",
      "contentHash": "ddebe026c7cbfde79bfdbebc3611720c0aa340f534e478a89a784caf025b6860",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          383
        ],
        "answerPdfPages": [
          156,
          157
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "ddebe026c7cbfde79bfdbebc3611720c0aa340f534e478a89a784caf025b6860",
        "method": "Exact point-by-point rational self-loop system yields75/13;independent thinning of scored points by mean3/2 agrees."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.17",
      "contentHash": "8338c387c89dbbb8cb549729f6e1440d428ec4c7a54e0885193cc91b1ef5f732",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          383
        ],
        "answerPdfPages": [
          157,
          158
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "8338c387c89dbbb8cb549729f6e1440d428ec4c7a54e0885193cc91b1ef5f732",
        "method": "Exact Markov reward system counts transition entries to0,not initial occupancy;solution[4,4,3,2,1]."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.18",
      "contentHash": "e63cf585a24a9a456f546dfe4bf72c30219bfe44a02b49df7e581f3011d298d6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          383
        ],
        "answerPdfPages": [
          158
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "e63cf585a24a9a456f546dfe4bf72c30219bfe44a02b49df7e581f3011d298d6",
        "method": "Exact forward probability DP with boundary forcing and independent enumeration of all35 complete paths weighted by their actual branch choices;C mass11/32."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.19",
      "contentHash": "90fb66cfce6ef7097aba799ac18ec20285e6d696a1e3cb7c7fefed54c1db4319",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          384
        ],
        "answerPdfPages": [
          158,
          159
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "90fb66cfce6ef7097aba799ac18ec20285e6d696a1e3cb7c7fefed54c1db4319",
        "method": "Exact Bellman values for0–3 remaining strikes and bounded pot states plus analytic large-pot inequality;policy success37/64 and expectation925/8."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.20",
      "contentHash": "cf10b80f5a554df0b111528b34544acec49d8b565e3b9fcaac223c1229ce7668",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          384
        ],
        "answerPdfPages": [
          159,
          160,
          161
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "cf10b80f5a554df0b111528b34544acec49d8b565e3b9fcaac223c1229ce7668",
        "method": "Independent recursive enumeration of all allowed step words gives83,with last-step counts27,28,28;compared full three-state DP grid."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.21",
      "contentHash": "531f0319f1470069cb583db5d35cef6f38fea4fbd1e5253bade3fd77a4714fd9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          384
        ],
        "answerPdfPages": [
          161
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "531f0319f1470069cb583db5d35cef6f38fea4fbd1e5253bade3fd77a4714fd9",
        "method": "Exact three-position probability transitions through2004 steps match the closed rational formula;independent two-state reduction."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.22",
      "contentHash": "0a35c91ca9ad1017c6b68238d93cf712b3159669c5ce72979d1b88e43fa702b0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          384
        ],
        "answerPdfPages": [
          161,
          162
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "0a35c91ca9ad1017c6b68238d93cf712b3159669c5ce72979d1b88e43fa702b0",
        "method": "Exact descending-state DP for several targets through start2003 and independent staged hitting product;denominator2224222."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.23",
      "contentHash": "3887d58a71e82d9e1b5e56e515239f51e22d40ea9b0a1d41f2cac2684e0d9a49",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          384
        ],
        "answerPdfPages": [
          162,
          163
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "3887d58a71e82d9e1b5e56e515239f51e22d40ea9b0a1d41f2cac2684e0d9a49",
        "method": "Exact seven-state transition enumeration and rational absorbing system gives5/8;all rows and complementary symmetry checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.24",
      "contentHash": "7827364a35a6a2b797bbca6969093bd807611e2f8c0c4bf6fa88d875986a2f5a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          384
        ],
        "answerPdfPages": [
          163
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "7827364a35a6a2b797bbca6969093bd807611e2f8c0c4bf6fa88d875986a2f5a",
        "method": "Exact27-transient-state rational absorption-time system gives196;verified quadratic at every bankroll0–28 and independent difference recurrence."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.25",
      "contentHash": "efe1168f3c32a6034e5735e35db86ee866e52d819f2c350b2c684b8dd0b51758",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          384,
          385
        ],
        "answerPdfPages": [
          163,
          164
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "efe1168f3c32a6034e5735e35db86ee866e52d819f2c350b2c684b8dd0b51758",
        "method": "Exhaustive legal-move minimax for up to5 piles of sizes0–8 agrees with A/B parity;all winning-move and losing-move categories checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "13.26",
      "contentHash": "d492848005869226f1bc623cf65ec5fa1d83a0a9dd885260367f7b421286636e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          385
        ],
        "answerPdfPages": [
          164
        ],
        "method": "Original383–385/manual155–164 visually compared;all alternate methods,full state/triple/transition/strategy tables and native cube/city diagrams retained;initial-visit convention explicit.",
        "chapter13FreshFidelity": "All canonical original280–304,378–385/manual148–164 scans compared; complete source proofs/automata/tables retained; readable pannable state diagrams and missing source moth diagram restored."
      },
      "math": {
        "status": "passed",
        "contentHash": "d492848005869226f1bc623cf65ec5fa1d83a0a9dd885260367f7b421286636e",
        "method": "Memoized exhaustive Nim through5 piles of sizes0–7 agrees with xor;verified constructed target a xor X is legal and zeroes xor for every nonzero-xor state."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.1",
      "contentHash": "982c92b74c042896906ae44f1dd8a246fb629be77b277162d3c6e03082163aa0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          387,
          388,
          389
        ],
        "answerPdfPages": [],
        "method": "Original386–392/manual165–167 visually compared;all urn/monomial/coefficient tables,source dice pips,alternate counting methods,and unique-factorization dice proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "982c92b74c042896906ae44f1dd8a246fb629be77b277162d3c6e03082163aa0",
        "method": "Enumerated all3·3·4 quantity triples and exact coefficient convolution;eight total-four choices."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.2",
      "contentHash": "6308e53a77b2ad6148a42d67e71239a826b9af7dffd1d3d848ad2d7798dd4ca7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          387,
          389,
          390
        ],
        "answerPdfPages": [],
        "method": "Original386–392/manual165–167 visually compared;all urn/monomial/coefficient tables,source dice pips,alternate counting methods,and unique-factorization dice proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "6308e53a77b2ad6148a42d67e71239a826b9af7dffd1d3d848ad2d7798dd4ca7",
        "method": "Enumerated all35 labeled4-subsets of7 balls;group counts match all eight weighted monomials and the full convolution."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.3",
      "contentHash": "bb8813af7ef5e6fed3a41701c732640c8ae1a653b2ae22627fe4508b62127318",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          387,
          391
        ],
        "answerPdfPages": [],
        "method": "Original386–392/manual165–167 visually compared;all urn/monomial/coefficient tables,source dice pips,alternate counting methods,and unique-factorization dice proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "bb8813af7ef5e6fed3a41701c732640c8ae1a653b2ae22627fe4508b62127318",
        "method": "Enumerated all216 ordered three-die rolls;25 have sum9;full convolution coefficients independently match."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.4",
      "contentHash": "8efa5c930b379032c5e19ec0732def2945bac424f5edfde27b41e10c5024e5c0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          387,
          391,
          392
        ],
        "answerPdfPages": [],
        "method": "Original386–392/manual165–167 visually compared;all urn/monomial/coefficient tables,source dice pips,alternate counting methods,and unique-factorization dice proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "8efa5c930b379032c5e19ec0732def2945bac424f5edfde27b41e10c5024e5c0",
        "method": "Enumerated all36 labeled-face pairs;sum histogram and coefficient convolution agree;17 successes,weighted total240."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.2.1",
      "contentHash": "97c9c2f57dcf7a47834e851a3cdbdd15f217b16b95a2a42de66ce2e567e345ea",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          392
        ],
        "answerPdfPages": [
          165
        ],
        "method": "Original386–392/manual165–167 visually compared;all urn/monomial/coefficient tables,source dice pips,alternate counting methods,and unique-factorization dice proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "97c9c2f57dcf7a47834e851a3cdbdd15f217b16b95a2a42de66ce2e567e345ea",
        "method": "Enumerated all18 named tip triples;five total6 and full polynomial coefficients match."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.2.2",
      "contentHash": "f25d74402f21717a1d6943d7145c9b815ec5df05c5ea29ae90dd04ad06ce89df",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          392
        ],
        "answerPdfPages": [
          165
        ],
        "method": "Original386–392/manual165–167 visually compared;all urn/monomial/coefficient tables,source dice pips,alternate counting methods,and unique-factorization dice proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "f25d74402f21717a1d6943d7145c9b815ec5df05c5ea29ae90dd04ad06ce89df",
        "method": "Enumerated all5^4 bounded assignments;20 satisfy parity and sum;coefficient convolution independently checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.2.3",
      "contentHash": "0a60a06c312be55abc6170c1919a3ec518644234132ef1867232854b38340a71",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          392
        ],
        "answerPdfPages": [
          165,
          166
        ],
        "method": "Original386–392/manual165–167 visually compared;all urn/monomial/coefficient tables,source dice pips,alternate counting methods,and unique-factorization dice proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "0a60a06c312be55abc6170c1919a3ec518644234132ef1867232854b38340a71",
        "method": "Enumerated all625 ordered four-friend outcomes;85 sum0;counts by zero results are1,0,24,24,36."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.2.4",
      "contentHash": "60aa658cf732aa789dfe252d0568e48d7d92f06366329e9eaa9d15e16a30cb46",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          392
        ],
        "answerPdfPages": [
          166,
          167
        ],
        "method": "Original386–392/manual165–167 visually compared;all urn/monomial/coefficient tables,source dice pips,alternate counting methods,and unique-factorization dice proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "60aa658cf732aa789dfe252d0568e48d7d92f06366329e9eaa9d15e16a30cb46",
        "method": "Enumerated all36 Sicherman face pairs and matched the ordinary histogram at every sum2–12;exact polynomial factor multiplication checked;allocation argument proves uniqueness."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.5",
      "contentHash": "30489a20bab9650903fd23d2e773ade1d006ec214cb9bee8066d4f190646be56",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          393
        ],
        "answerPdfPages": [],
        "method": "Original393–395/manual167–168 visually compared;baseline factorization,all shifted coefficient cases,and generating-function Vandermonde proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "30489a20bab9650903fd23d2e773ade1d006ec214cb9bee8066d4f190646be56",
        "method": "Enumerated all15-slot six-subsets;5005;independent product of three Pascal-row polynomials gives the same coefficient."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.6",
      "contentHash": "adc3fc27e9f473bf373ca0fe31abbdebea39c9aa65eb32851022b7e3dc5b3830",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          393,
          394
        ],
        "answerPdfPages": [],
        "method": "Original393–395/manual167–168 visually compared;baseline factorization,all shifted coefficient cases,and generating-function Vandermonde proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "adc3fc27e9f473bf373ca0fe31abbdebea39c9aa65eb32851022b7e3dc5b3830",
        "method": "Exact binomial-weighted Curtis cases and independent24-fold polynomial convolution;4800008."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.3.1",
      "contentHash": "48cb7bd2b421f511fa1140f428d62777efcacfdf1d01aacbdffb7d3bd37f90d9",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          394
        ],
        "answerPdfPages": [
          167
        ],
        "method": "Original393–395/manual167–168 visually compared;baseline factorization,all shifted coefficient cases,and generating-function Vandermonde proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "48cb7bd2b421f511fa1140f428d62777efcacfdf1d01aacbdffb7d3bd37f90d9",
        "method": "Enumerated all15-slot six-subsets and grouped by the three hot-dog topping counts;coefficient5005."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.3.2",
      "contentHash": "312a5459c97e22cc880879e8c875869765ca29822ecb4b95616c2de88daee54c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          394
        ],
        "answerPdfPages": [
          167
        ],
        "method": "Original393–395/manual167–168 visually compared;baseline factorization,all shifted coefficient cases,and generating-function Vandermonde proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "312a5459c97e22cc880879e8c875869765ca29822ecb4b95616c2de88daee54c",
        "method": "Enumerated all2^14 binary participation patterns;1218 yield20 flowers;feasible-size cases independently checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.3.3",
      "contentHash": "155c038ac67eb0a60ba7a10a6177d58a5c39deae46d0b55d0062f3a50c48932b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          394
        ],
        "answerPdfPages": [
          167
        ],
        "method": "Original393–395/manual167–168 visually compared;baseline factorization,all shifted coefficient cases,and generating-function Vandermonde proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "155c038ac67eb0a60ba7a10a6177d58a5c39deae46d0b55d0062f3a50c48932b",
        "method": "Independent shifted-binomial sum and exact29-fold contribution-polynomial convolution give63471720."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.3.4",
      "contentHash": "247e0db1edc7961cfd3cc7371e3289f90b14d58ab92826798f9e21cb28c678ab",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          395
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        "answerPdfPages": [
          167
        ],
        "method": "Original393–395/manual167–168 visually compared;baseline factorization,all shifted coefficient cases,and generating-function Vandermonde proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "247e0db1edc7961cfd3cc7371e3289f90b14d58ab92826798f9e21cb28c678ab",
        "method": "Enumerated all2^13 yes/no assignments with Chief weight3;1287 have weighted yes count8."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.3.5",
      "contentHash": "e2425f73c19e62504fbf081dfca653a88c1edbf2c60e8d8777251e9996550095",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          395
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        "answerPdfPages": [
          167,
          168
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        "method": "Original393–395/manual167–168 visually compared;baseline factorization,all shifted coefficient cases,and generating-function Vandermonde proof retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "e2425f73c19e62504fbf081dfca653a88c1edbf2c60e8d8777251e9996550095",
        "method": "Exact coefficient convolution for all m,n≤30 and every degree through m+n+2;compared closed binomial coefficients."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.7",
      "contentHash": "d7a54277c4004fa8fe0e746f8eed513ae6371dfefa4cdf1f07c4e43c6438b94e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          395,
          396,
          397
        ],
        "answerPdfPages": [],
        "method": "Original395–402/manual168–170 visually compared;full coefficient patterns,induction,formal/analytic distinction,upper-bound terms,and both candy-bijection directions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "d7a54277c4004fa8fe0e746f8eed513ae6371dfefa4cdf1f07c4e43c6438b94e",
        "method": "Enumerated all496 triples summing30 and truncated exact convolutions;coefficient pattern C(k+2,2)checked through k100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.8",
      "contentHash": "66f3e6fb2f323643e49670908f8ff93800cbd0760cd577393a3ec98873d6c836",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          397,
          398,
          399
        ],
        "answerPdfPages": [],
        "method": "Original395–402/manual168–170 visually compared;full coefficient patterns,induction,formal/analytic distinction,upper-bound terms,and both candy-bijection directions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "66f3e6fb2f323643e49670908f8ff93800cbd0760cd577393a3ec98873d6c836",
        "method": "Exact truncated convolution through n12,k100 matches C(n+k−1,n−1);independent weak-composition enumeration for small parameters."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.9",
      "contentHash": "a0ff07aaedaf53974c8ddfd5bd63757c907268eb40bc471a49d8e732345ac881",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          399
        ],
        "answerPdfPages": [],
        "method": "Original395–402/manual168–170 visually compared;full coefficient patterns,induction,formal/analytic distinction,upper-bound terms,and both candy-bijection directions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "a0ff07aaedaf53974c8ddfd5bd63757c907268eb40bc471a49d8e732345ac881",
        "method": "Enumerated all5^5 bounded assignments;121 sum15;independent complement-to5 weak-composition check and coefficient PIE."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.10",
      "contentHash": "566dafdd0ebada2a8e812e2e070a6a793a86224ec667d2b0e6256d06ccc24dbf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          400,
          401
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        "answerPdfPages": [],
        "method": "Original395–402/manual168–170 visually compared;full coefficient patterns,induction,formal/analytic distinction,upper-bound terms,and both candy-bijection directions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "566dafdd0ebada2a8e812e2e070a6a793a86224ec667d2b0e6256d06ccc24dbf",
        "method": "Enumerated all4950 allowed candy allocations and all4950 positive triples summing101;explicit forward/inverse maps agree."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.11",
      "contentHash": "de41a68cd8341fbfbb2a607780849961da249ed831290cca25b878c4ccb37fc0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          401,
          402
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        "answerPdfPages": [],
        "method": "Original395–402/manual168–170 visually compared;full coefficient patterns,induction,formal/analytic distinction,upper-bound terms,and both candy-bijection directions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "de41a68cd8341fbfbb2a607780849961da249ed831290cca25b878c4ccb37fc0",
        "method": "Enumerated all nonnegative Chelsea multiples3 and positive odd Barbara/Jenna amounts totaling50;117;independent truncated convolution."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.4.1",
      "contentHash": "fbcc49c5ae76d9089c5bf930f6bb6d405278ab26c6972c025fdf4bd4350f2717",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          402
        ],
        "answerPdfPages": [
          168
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        "method": "Original395–402/manual168–170 visually compared;full coefficient patterns,induction,formal/analytic distinction,upper-bound terms,and both candy-bijection directions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "fbcc49c5ae76d9089c5bf930f6bb6d405278ab26c6972c025fdf4bd4350f2717",
        "method": "Exact truncated geometric-power convolutions for m1–10,n1–8,k0–100;divisibility and coefficient formula match."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.4.2",
      "contentHash": "655db2a82d27ed05754eae6236bb1168abfdfc34c2e62f2e1dbda5dd7de7a555",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          402
        ],
        "answerPdfPages": [
          168,
          169
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        "method": "Original395–402/manual168–170 visually compared;full coefficient patterns,induction,formal/analytic distinction,upper-bound terms,and both candy-bijection directions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "655db2a82d27ed05754eae6236bb1168abfdfc34c2e62f2e1dbda5dd7de7a555",
        "method": "Enumerated all constrained five-child allocations of100 and compared exact truncated polynomial product;4756."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.4.3",
      "contentHash": "748de1da6abc29fdcf345fec05ee4c4a4962a031ddac7568b8de3d392a738ed3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          402
        ],
        "answerPdfPages": [
          169
        ],
        "method": "Original395–402/manual168–170 visually compared;full coefficient patterns,induction,formal/analytic distinction,upper-bound terms,and both candy-bijection directions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "748de1da6abc29fdcf345fec05ee4c4a4962a031ddac7568b8de3d392a738ed3",
        "method": "Exhaustive forward/inverse check on all4950 positive triples and all4950 valid allocations."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "14.4.4",
      "contentHash": "01c92302ab0a73d76a59316a8de0e704a93b08a6a2fafe0766cd621c0bc9e164",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          402
        ],
        "answerPdfPages": [
          169,
          170
        ],
        "method": "Original395–402/manual168–170 visually compared;full coefficient patterns,induction,formal/analytic distinction,upper-bound terms,and both candy-bijection directions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "01c92302ab0a73d76a59316a8de0e704a93b08a6a2fafe0766cd621c0bc9e164",
        "method": "Independent bounded-sum dynamic programming and exact coefficient convolution for four0–6 and three0–10 variables;112567."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.4.5",
      "contentHash": "f6af40c24919614e4bb436f21eec0454ac18fa7e201f4d07005e5a726b0ce347",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          402
        ],
        "answerPdfPages": [
          170
        ],
        "method": "Original395–402/manual168–170 visually compared;full coefficient patterns,induction,formal/analytic distinction,upper-bound terms,and both candy-bijection directions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "f6af40c24919614e4bb436f21eec0454ac18fa7e201f4d07005e5a726b0ce347",
        "method": "Enumerated all100 unordered-pair/Sergio-number outcomes;40 successes;sum polynomial independently verified."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.12",
      "contentHash": "ffb193aadf2394379859804394c1ead3059a3cf386ba176f25e59175561298c8",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          402,
          403
        ],
        "answerPdfPages": [],
        "method": "Original402–413/manual170–172 visually compared;complete4/5/6 lists,all statistic regrouping tables,both restricted-partition identities,and triangle bijection retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "ffb193aadf2394379859804394c1ead3059a3cf386ba176f25e59175561298c8",
        "method": "Recursively enumerated descending positive partitions;counts5,7,11;each list unique and sums to its target."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.13",
      "contentHash": "bb1827a3ed7b577d08995da2ee9a171309e41498bd4c583c4e480bf452ed4a91",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          403,
          404
        ],
        "answerPdfPages": [],
        "method": "Original402–413/manual170–172 visually compared;complete4/5/6 lists,all statistic regrouping tables,both restricted-partition identities,and triangle bijection retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "bb1827a3ed7b577d08995da2ee9a171309e41498bd4c583c4e480bf452ed4a91",
        "method": "Exact partition enumeration through n20 and independent multiplicity-product coefficient DP through n100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.14",
      "contentHash": "4fd0b67cdd015fbf6ea0d5bc956b23913246ec8e9a32b0a6d39bfca9af7aa97a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          405,
          406
        ],
        "answerPdfPages": [],
        "method": "Original402–413/manual170–172 visually compared;complete4/5/6 lists,all statistic regrouping tables,both restricted-partition identities,and triangle bijection retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "4fd0b67cdd015fbf6ea0d5bc956b23913246ec8e9a32b0a6d39bfca9af7aa97a",
        "method": "Enumerated both restricted families through n20 and compared product coefficients through n100;independent odd-multiplicity binary splitting map checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.15",
      "contentHash": "792d8eb409519d6aa060b81917f44fd3cf131b866b2cb362d63a3ac7320e90f4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          405,
          407,
          408
        ],
        "answerPdfPages": [],
        "method": "Original402–413/manual170–172 visually compared;complete4/5/6 lists,all statistic regrouping tables,both restricted-partition identities,and triangle bijection retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "792d8eb409519d6aa060b81917f44fd3cf131b866b2cb362d63a3ac7320e90f4",
        "method": "Exact restricted-partition enumeration through n20 and independent multiplicity-product coefficients through n100;source n7 lists both recovered."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.16",
      "contentHash": "5dd0ee02be44b0bd055b5836af6d071fcedfdf0c6e56241c9c26e6fc69590828",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          405,
          408,
          409,
          410,
          411,
          412
        ],
        "answerPdfPages": [],
        "method": "Original402–413/manual170–172 visually compared;complete4/5/6 lists,all statistic regrouping tables,both restricted-partition identities,and triangle bijection retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "5dd0ee02be44b0bd055b5836af6d071fcedfdf0c6e56241c9c26e6fc69590828",
        "method": "Enumerated all partitions through n25;total ones equals total distinct values and sum of earlier partition counts;all three source regrouping tables retained."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.5.1",
      "contentHash": "ef91aaac53c7481d3ddf546a9c25521c6bc32b49dffa780bb44b501e01b2fab1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          412
        ],
        "answerPdfPages": [
          170
        ],
        "method": "Original402–413/manual170–172 visually compared;complete4/5/6 lists,all statistic regrouping tables,both restricted-partition identities,and triangle bijection retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "ef91aaac53c7481d3ddf546a9c25521c6bc32b49dffa780bb44b501e01b2fab1",
        "method": "Enumerated denomination-count triples through amount100 and compared truncated product coefficients."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.5.2",
      "contentHash": "d80b1e918f7f82ea7693f753996afe2abcf15e22486c21099975d6dd0a7eb8ff",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          412
        ],
        "answerPdfPages": [
          170,
          171
        ],
        "method": "Original402–413/manual170–172 visually compared;complete4/5/6 lists,all statistic regrouping tables,both restricted-partition identities,and triangle bijection retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "d80b1e918f7f82ea7693f753996afe2abcf15e22486c21099975d6dd0a7eb8ff",
        "method": "Enumerated even-part partitions through n25;rescaling bijection and product coefficients match."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.5.3",
      "contentHash": "32a8f351e3d549646bdc6b717560e1a0ae886b7fa8d4fdfbdc58bf1a16301327",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          412
        ],
        "answerPdfPages": [
          171
        ],
        "method": "Original402–413/manual170–172 visually compared;complete4/5/6 lists,all statistic regrouping tables,both restricted-partition identities,and triangle bijection retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "32a8f351e3d549646bdc6b717560e1a0ae886b7fa8d4fdfbdc58bf1a16301327",
        "method": "Enumerated both restricted families through n20 and exact product coefficients through n100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.5.4",
      "contentHash": "5750caa167381deeddcba69c618f43d78d5ce680e3d8e17d41870ffe7f7fc94a",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          412,
          413
        ],
        "answerPdfPages": [
          171
        ],
        "method": "Original402–413/manual170–172 visually compared;complete4/5/6 lists,all statistic regrouping tables,both restricted-partition identities,and triangle bijection retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "5750caa167381deeddcba69c618f43d78d5ce680e3d8e17d41870ffe7f7fc94a",
        "method": "Independently enumerated compositions with parts1–6 through total16 and compared recurrence coefficients through total100."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.5.5",
      "contentHash": "c851826bc17439c9e9c5c697d25c8e52642d78241a084a5215d688c3ca0e48b1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          413
        ],
        "answerPdfPages": [
          171,
          172
        ],
        "method": "Original402–413/manual170–172 visually compared;complete4/5/6 lists,all statistic regrouping tables,both restricted-partition identities,and triangle bijection retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "c851826bc17439c9e9c5c697d25c8e52642d78241a084a5215d688c3ca0e48b1",
        "method": "Enumerated all sorted positive integer triangles through perimeter100 and matched coefficients of x³/[(1−x²)(1−x³)(1−x⁴)];explicit Ferrers/pairing inverse checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.17",
      "contentHash": "0d3967eefeba31f9b4a6e9087ca64eabe17f506176f43b415802021ea4f518ea",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          413,
          414,
          415
        ],
        "answerPdfPages": [],
        "method": "Original413–415 visually compared;coefficient boundary cases,root definitions,sign,partial-fraction system,index shift,and Binet form retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "0d3967eefeba31f9b4a6e9087ca64eabe17f506176f43b415802021ea4f518ea",
        "method": "Exact formal coefficient cancellation through degree200;only the degree1 coefficient remains1."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.18",
      "contentHash": "d2c56708063bcb3f68bf2577c15239575a8b0902129e2dc98a644f3d3db7d3bc",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          413,
          414,
          415
        ],
        "answerPdfPages": [],
        "method": "Original413–415 visually compared;coefficient boundary cases,root definitions,sign,partial-fraction system,index shift,and Binet form retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "d2c56708063bcb3f68bf2577c15239575a8b0902129e2dc98a644f3d3db7d3bc",
        "method": "Exact arithmetic in Q(√5)compares the two root powers with the integer Fibonacci recurrence through n200;formal coefficients and partial-fraction identities checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.19",
      "contentHash": "384a9a0e469985678dede7167928c2a6e953195e2124a338552217d36efd0bed",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          416
        ],
        "answerPdfPages": [
          172,
          173
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "384a9a0e469985678dede7167928c2a6e953195e2124a338552217d36efd0bed",
        "method": "Independent exact truncated polynomial/geometric convolutions verify all six values."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.20",
      "contentHash": "d32dd424e4a364b24ca2c8e591f82db4fbe6c76dd6b3e07ae11fed699bc4abc3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          416
        ],
        "answerPdfPages": [
          173
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "d32dd424e4a364b24ca2c8e591f82db4fbe6c76dd6b3e07ae11fed699bc4abc3",
        "method": "Enumerated all positive triples summing25;276;compared shifted stars-and-bars coefficient."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.21",
      "contentHash": "490ee3f5f8e747680f98b74091c5cec50bb0fcb0289f09c38e4b73d810a326a7",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          416
        ],
        "answerPdfPages": [
          173
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "490ee3f5f8e747680f98b74091c5cec50bb0fcb0289f09c38e4b73d810a326a7",
        "method": "Enumerated all four-child nonnegative allocations of8 with first≥2;84."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.22",
      "contentHash": "2404f02843193d3f33ad848ae72eac20d890b52c0e2eda276eb2db6635d4de07",
      "source": {
        "status": "source-assumption-explicit",
        "textbookPdfPages": [
          416
        ],
        "answerPdfPages": [
          173
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "2404f02843193d3f33ad848ae72eac20d890b52c0e2eda276eb2db6635d4de07",
        "method": "Enumerated all32 subsets of the five coin types;finite polynomial coefficients match. Separately checked unlimited-type convolution for the alternative reading."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.23",
      "contentHash": "1f4df7e14039a6ca2fb057c219e5183540142f461f58443f5e539aafb3f276a8",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          416
        ],
        "answerPdfPages": [
          173,
          174
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "1f4df7e14039a6ca2fb057c219e5183540142f461f58443f5e539aafb3f276a8",
        "method": "Enumerated all odd Sara/you amounts,Krishna multiples3,and theft0–2 consistent with net30;136;independent Laurent-factor cancellation."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.24",
      "contentHash": "d462795bdb3380f88a804e16edbc02604d488a9a25da6cd5fe7aa47057cd8f83",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          417
        ],
        "answerPdfPages": [
          174
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "d462795bdb3380f88a804e16edbc02604d488a9a25da6cd5fe7aa47057cd8f83",
        "method": "Exact finite polynomial convolution and independent lower-half binomial sum;1048576."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.25",
      "contentHash": "7d0999a370867dfcdf5248093b6f96e65047af4fba2afffa6b78a92bff19273d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          417
        ],
        "answerPdfPages": [
          174,
          175
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "7d0999a370867dfcdf5248093b6f96e65047af4fba2afffa6b78a92bff19273d",
        "method": "Enumerated all three-hero amount assignments at totals37 and87;55 and62;compared exact rational-GF coefficients."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.26",
      "contentHash": "ad840f220ebdf00fc6613e42e1f085541f370faa0e9662a6aea9c8345e204d15",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          417
        ],
        "answerPdfPages": [
          175
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "ad840f220ebdf00fc6613e42e1f085541f370faa0e9662a6aea9c8345e204d15",
        "method": "Enumerated all12^4 positive bounded quadruples;994 sum30;coefficient PIE agrees."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.27",
      "contentHash": "ef043eb754d49705dbeded6d784b9dbddb36b99064d23201a509d7c8368074b1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          417
        ],
        "answerPdfPages": [
          175,
          176
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "ef043eb754d49705dbeded6d784b9dbddb36b99064d23201a509d7c8368074b1",
        "method": "Independently counted exponent triples with total2006 and even y+z exponent;1008016;all surviving multinomial coefficients nonzero."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.28",
      "contentHash": "3288a696bc0bb7fd352d8a2396ebdaf1a70ca85fb7d754684414ba75db99c3bf",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          417
        ],
        "answerPdfPages": [
          176
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "3288a696bc0bb7fd352d8a2396ebdaf1a70ca85fb7d754684414ba75db99c3bf",
        "method": "Exact finite-sum formula with a vanishing geometric tail verifies limit3/4;generating-function coefficient shift independently checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.29",
      "contentHash": "e586ff2e37e6705838d93483381cc82da68d9ab1fd2cd46d78a73ff03c3aa740",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          417
        ],
        "answerPdfPages": [
          176,
          177
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "e586ff2e37e6705838d93483381cc82da68d9ab1fd2cd46d78a73ff03c3aa740",
        "method": "Exact coefficient convolution through n30 and all degrees0–3n+2;independent binary1/2 component representation checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.30",
      "contentHash": "ade072d01fa8cabd786105fed595e076754d899ff53009f51ba1549edb8ba6fc",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          417
        ],
        "answerPdfPages": [
          177,
          178
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "ade072d01fa8cabd786105fed595e076754d899ff53009f51ba1549edb8ba6fc",
        "method": "Exact rational coefficient products and n-fold geometric convolutions through n12,k60;generalized falling-product definition matches signed binomial values."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.31",
      "contentHash": "589becaa2f3d21abe11cf9fa07f95083c0fe536d5a6698273d9c926fd2361c86",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          418
        ],
        "answerPdfPages": [
          179,
          180,
          181
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "589becaa2f3d21abe11cf9fa07f95083c0fe536d5a6698273d9c926fd2361c86",
        "method": "Enumerated every equation’s nonnegative solutions through k150,checked disjointness and union size k;independent truncated rational-series sums match the telescoping identity."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "14.32",
      "contentHash": "77ca965102026892227bd6f2671fa477dd0bc05fbb2a043060554c1d29497cd0",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          418
        ],
        "answerPdfPages": [
          182,
          183
        ],
        "method": "Original416–418/manual172–183 visually compared;all multipart coefficients,hero cases,alternate counting proofs,induction/telescoping details,and Catalan derivation retained;14.22inventory assumption explicit."
      },
      "math": {
        "status": "passed",
        "contentHash": "77ca965102026892227bd6f2671fa477dd0bc05fbb2a043060554c1d29497cd0",
        "method": "Exact Catalan recurrence and half-binomial coefficient formula compared through n100;formal quadratic convolution and branch constant checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.1",
      "contentHash": "0421c85254827a5581be464531750609367d1c6cf45df3df61c41c18669254a2",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          422,
          423
        ],
        "answerPdfPages": [],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "0421c85254827a5581be464531750609367d1c6cf45df3df61c41c18669254a2",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.2",
      "contentHash": "aa455461dbb8587aa9de65dffcf7e21eabefbb0f1b5c39b4c9ccc9aa8968bf35",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          422,
          424
        ],
        "answerPdfPages": [],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "aa455461dbb8587aa9de65dffcf7e21eabefbb0f1b5c39b4c9ccc9aa8968bf35",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.3",
      "contentHash": "f68550a65d189a8e96c04721a00b75431661d2af52646478b52cb6a31d597dac",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          422,
          424
        ],
        "answerPdfPages": [],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "f68550a65d189a8e96c04721a00b75431661d2af52646478b52cb6a31d597dac",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.4",
      "contentHash": "58e7c586d4f8774750567cdf3cd90e28326f960486480e8bd4e6aa8eaf4566be",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          422,
          424,
          425
        ],
        "answerPdfPages": [],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "58e7c586d4f8774750567cdf3cd90e28326f960486480e8bd4e6aa8eaf4566be",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.5",
      "contentHash": "83f43650ce1a7545323643067efe1a2111c3eb7729650f1122bb401ca1955bb6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          422,
          425,
          426
        ],
        "answerPdfPages": [],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated.",
        "restoredAttribution": {
          "name": "USAMO",
          "pdfPages": [
            422,
            425
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        }
      },
      "math": {
        "status": "passed",
        "contentHash": "83f43650ce1a7545323643067efe1a2111c3eb7729650f1122bb401ca1955bb6",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked.",
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          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.6",
      "contentHash": "6be94e6719275b9a793b746bc1081388241094759bc3d68560dbf414ca7b7dc2",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          422,
          426,
          427,
          428,
          429
        ],
        "answerPdfPages": [],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "6be94e6719275b9a793b746bc1081388241094759bc3d68560dbf414ca7b7dc2",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.3.1",
      "contentHash": "3b71a0f42237221792cab2ca93e2e5e27c3ce8a306b4d76b327ce445c50f3109",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          429
        ],
        "answerPdfPages": [
          184
        ],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "3b71a0f42237221792cab2ca93e2e5e27c3ce8a306b4d76b327ce445c50f3109",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.3.2",
      "contentHash": "351bfaef9c3dbbdaf4654eaa58f6a4bc825adad231f7f64bdd76ebc7461046ff",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          429
        ],
        "answerPdfPages": [
          184
        ],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "351bfaef9c3dbbdaf4654eaa58f6a4bc825adad231f7f64bdd76ebc7461046ff",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.3.3",
      "contentHash": "8027c5b511edb2a5f134af9f7f5e92a3db7d5150be8405db4f2695800bfb414e",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          429
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        "answerPdfPages": [
          184
        ],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "8027c5b511edb2a5f134af9f7f5e92a3db7d5150be8405db4f2695800bfb414e",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.3.4",
      "contentHash": "e4b7a3ef39832fbc1ce69eaeb1a06d9930722d26a78b7ff2f91ccab27268c2d3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          429
        ],
        "answerPdfPages": [
          184
        ],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "e4b7a3ef39832fbc1ce69eaeb1a06d9930722d26a78b7ff2f91ccab27268c2d3",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.3.5",
      "contentHash": "356d4dd3bfbabf92fa9ab8475e44b79fa4af01e9e28e4972886506df4cd4bf13",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          429
        ],
        "answerPdfPages": [
          184,
          185
        ],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "356d4dd3bfbabf92fa9ab8475e44b79fa4af01e9e28e4972886506df4cd4bf13",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.3.6",
      "contentHash": "fb1464f6fcc10e81470f4ccb3bf409914e5bcd7451e5e5cf819e415c31b8b9c1",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          429
        ],
        "answerPdfPages": [
          185
        ],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "fb1464f6fcc10e81470f4ccb3bf409914e5bcd7451e5e5cf819e415c31b8b9c1",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.3.7",
      "contentHash": "067e369f461b645f7ebd1e8a2c2e42c9ff1ee90f1c0c7cffd9d66819cda3673d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
          185
        ],
        "method": "Original419–429/manual184–185 visually compared;all thirteen numbered records and subparts,graphs,algorithm,failed shortcut,extremal construction,parity proof retained. Small-order implicit assumptions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "067e369f461b645f7ebd1e8a2c2e42c9ff1ee90f1c0c7cffd9d66819cda3673d",
        "method": "Independent finite graph enumeration and incidence/parity proofs; source assumptions checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.7",
      "contentHash": "0670f9f05edc76122d62982e3a93630543964aad1920fc2f43a61ce3b31b30f6",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          431
        ],
        "answerPdfPages": [],
        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "0670f9f05edc76122d62982e3a93630543964aad1920fc2f43a61ce3b31b30f6",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.8",
      "contentHash": "626f95bc9b77e70fd0f2572f88a2a569721d92cd896a36d0638c7ada2996b114",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          432,
          433
        ],
        "answerPdfPages": [],
        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "626f95bc9b77e70fd0f2572f88a2a569721d92cd896a36d0638c7ada2996b114",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.9",
      "contentHash": "720358f1caecaf1505a4ce59331e218cbd9983190ad4b8ec5da4637431b4f3ed",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [],
        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained.",
        "restoredAttribution": {
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          "pdfPages": [
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            433
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        }
      },
      "math": {
        "status": "passed",
        "contentHash": "720358f1caecaf1505a4ce59331e218cbd9983190ad4b8ec5da4637431b4f3ed",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification.",
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        }
      },
      "grading": {
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        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.10",
      "contentHash": "59265a29163b8ec80b4b62560ba93d06fce2a7ad3f6771764a3d5d8f6f5e2544",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          435,
          436
        ],
        "answerPdfPages": [],
        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "59265a29163b8ec80b4b62560ba93d06fce2a7ad3f6771764a3d5d8f6f5e2544",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.4.1",
      "contentHash": "40d04c98523859424558750d9cbe99a1c78cf8f9471ea78ae8c154206ca4b717",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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          186
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        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "40d04c98523859424558750d9cbe99a1c78cf8f9471ea78ae8c154206ca4b717",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.4.2",
      "contentHash": "f4f508a824f1a53f70093518e0965c940ba7b9971a2460ef88cf00db090eaf66",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "f4f508a824f1a53f70093518e0965c940ba7b9971a2460ef88cf00db090eaf66",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.4.3",
      "contentHash": "c8b4749313e4df8ca7a79350999a40a872839d01bf6e4dcd2f2c5ba544079d42",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "c8b4749313e4df8ca7a79350999a40a872839d01bf6e4dcd2f2c5ba544079d42",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.4.4",
      "contentHash": "160e38476dd013be14ed90bbaf9dd460ee6ee0384b874ab777d85f3bbd0120fd",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "160e38476dd013be14ed90bbaf9dd460ee6ee0384b874ab777d85f3bbd0120fd",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.4.5",
      "contentHash": "c4f4e0f8c3ec34c14aa4a72f5936ab062dbd1dd37449e2e3016a4a8ffa52bc44",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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          187
        ],
        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "c4f4e0f8c3ec34c14aa4a72f5936ab062dbd1dd37449e2e3016a4a8ffa52bc44",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "15.4.6",
      "contentHash": "b9a7560de69325e8e583b9293015f042f0e2c5e78564935cfee136ec6e75201f",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "b9a7560de69325e8e583b9293015f042f0e2c5e78564935cfee136ec6e75201f",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.4.7",
      "contentHash": "89002463847873e97a917e11cc457cad29d8380f187af45111e8792d5ec5fd16",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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          437
        ],
        "answerPdfPages": [
          187
        ],
        "method": "Original430–437/manual185–187 visually compared;all11 records and all proof parts,tree shapes,component and nap diagrams,book-specific definitions retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "89002463847873e97a917e11cc457cad29d8380f187af45111e8792d5ec5fd16",
        "method": "Independent finite graph enumeration,BFS cycle/bipartition checks,and canonical tree classification."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
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        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction."
      },
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        "mode": "written",
        "automaticScore": false,
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    },
    {
      "id": "15.26",
      "contentHash": "a157ec731784d4f4a6ff67ad9bbb817143fa076debba7f70bec3b3e51500bd1b",
      "source": {
        "status": "passed",
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          450
        ],
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          193
        ],
        "method": "Original448–450/manual192–196 visually compared;all16review/challenge records and subparts,source proofs,domino inventory/native graph,knight odd-degree diagram retained. Ordered length-two path and Dirac n≥3 conventions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "a157ec731784d4f4a6ff67ad9bbb817143fa076debba7f70bec3b3e51500bd1b",
        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
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    },
    {
      "id": "15.27",
      "contentHash": "9eecb778482cf037607dff88bd236f9a10dd056d4e9737d61654bd389dab9867",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          450
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          194
        ],
        "method": "Original448–450/manual192–196 visually compared;all16review/challenge records and subparts,source proofs,domino inventory/native graph,knight odd-degree diagram retained. Ordered length-two path and Dirac n≥3 conventions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "9eecb778482cf037607dff88bd236f9a10dd056d4e9737d61654bd389dab9867",
        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction."
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      "grading": {
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        "automaticScore": false,
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    },
    {
      "id": "15.28",
      "contentHash": "f9c7d9605716af51b771ca8f68e61820fb51fd2f7720da0c133deb18b63f5543",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          450
        ],
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          194
        ],
        "method": "Original448–450/manual192–196 visually compared;all16review/challenge records and subparts,source proofs,domino inventory/native graph,knight odd-degree diagram retained. Ordered length-two path and Dirac n≥3 conventions explicitly stated."
      },
      "math": {
        "status": "passed",
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        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction."
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    },
    {
      "id": "15.29",
      "contentHash": "50f7e64a90565fae42bfffb52f3f03eaa5740782788a206b222ac07ed0c25c81",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
          450
        ],
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          194
        ],
        "method": "Original448–450/manual192–196 visually compared;all16review/challenge records and subparts,source proofs,domino inventory/native graph,knight odd-degree diagram retained. Ordered length-two path and Dirac n≥3 conventions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "50f7e64a90565fae42bfffb52f3f03eaa5740782788a206b222ac07ed0c25c81",
        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction."
      },
      "grading": {
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        "automaticScore": false,
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    },
    {
      "id": "15.30",
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "method": "Original448–450/manual192–196 visually compared;all16review/challenge records and subparts,source proofs,domino inventory/native graph,knight odd-degree diagram retained. Ordered length-two path and Dirac n≥3 conventions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "3d7d9946c4e80ee3b676cda35536a069307640cb508794320ea800b6574d6aeb",
        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction."
      },
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        "automaticScore": false,
        "sourceKeyPreserved": true
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      "id": "15.31",
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      "source": {
        "status": "passed",
        "textbookPdfPages": [
          450
        ],
        "answerPdfPages": [
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        ],
        "method": "Original448–450/manual192–196 visually compared;all16review/challenge records and subparts,source proofs,domino inventory/native graph,knight odd-degree diagram retained. Ordered length-two path and Dirac n≥3 conventions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "2ca1e73185f3f00679293bcd95d91274c77811a71d46159c47cd7c660fee57aa",
        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction."
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        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.32",
      "contentHash": "c5d4792310ed91b963b119bcd11764e8893c42bde3da0711a783d51b03e77866",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
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          195
        ],
        "method": "Original448–450/manual192–196 visually compared;all16review/challenge records and subparts,source proofs,domino inventory/native graph,knight odd-degree diagram retained. Ordered length-two path and Dirac n≥3 conventions explicitly stated.",
        "restoredAttribution": {
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        }
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        "status": "passed",
        "contentHash": "c5d4792310ed91b963b119bcd11764e8893c42bde3da0711a783d51b03e77866",
        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction.",
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        "automaticScore": false,
        "sourceKeyPreserved": true
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    {
      "id": "15.33",
      "contentHash": "9d7b15a13ce0c8e42b90b2f347ed9d329378adeb4616bf975490ca9d02136b08",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original448–450/manual192–196 visually compared;all16review/challenge records and subparts,source proofs,domino inventory/native graph,knight odd-degree diagram retained. Ordered length-two path and Dirac n≥3 conventions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "9d7b15a13ce0c8e42b90b2f347ed9d329378adeb4616bf975490ca9d02136b08",
        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction."
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        "automaticScore": false,
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    },
    {
      "id": "15.34",
      "contentHash": "714ca2c964dcadd54493624e32f51aed10b25d6d28d915dc908d606d00aa9649",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
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        ],
        "method": "Original448–450/manual192–196 visually compared;all16review/challenge records and subparts,source proofs,domino inventory/native graph,knight odd-degree diagram retained. Ordered length-two path and Dirac n≥3 conventions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "714ca2c964dcadd54493624e32f51aed10b25d6d28d915dc908d606d00aa9649",
        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction."
      },
      "grading": {
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        "automaticScore": false,
        "sourceKeyPreserved": true
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    },
    {
      "id": "15.35",
      "contentHash": "0b0e025806d2aad89c3b2a74ec23731667b913e88e3d83dee0c5689372b1c894",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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          196
        ],
        "method": "Original448–450/manual192–196 visually compared;all16review/challenge records and subparts,source proofs,domino inventory/native graph,knight odd-degree diagram retained. Ordered length-two path and Dirac n≥3 conventions explicitly stated."
      },
      "math": {
        "status": "passed",
        "contentHash": "0b0e025806d2aad89c3b2a74ec23731667b913e88e3d83dee0c5689372b1c894",
        "method": "Independent exhaustive finite graph checks,exact incidence and parity proofs,explicit Eulerian certificates and gap-reducing Hamiltonian construction."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.1",
      "contentHash": "53337a458a44a04df8978c01a274a2f679b416e55774abcd3715316ddd83625d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
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        ],
        "method": "Original451/manual197–198 visually compared;allfive statements,all case work,tables and follow-ups retained. Independent-engine assumption made explicit;16.2 supplied nontrivial progression and missing literal equality reconciled.",
        "restoredAttribution": {
          "name": "HMMT",
          "pdfPages": [
            451
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "53337a458a44a04df8978c01a274a2f679b416e55774abcd3715316ddd83625d",
        "method": "Enumerated every failure subset for3 and5 engines;exact polynomial coefficients and factorization verified;odd-engine symmetry.",
        "attributionOnlyAmendment": {
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        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.2",
      "contentHash": "278ed456dc4dee8d4fd00e1e1337cf83d6a13bc775a12f683c70ec15ce5640a4",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original451/manual197–198 visually compared;allfive statements,all case work,tables and follow-ups retained. Independent-engine assumption made explicit;16.2 supplied nontrivial progression and missing literal equality reconciled.",
        "restoredAttribution": {
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          "pdfPages": [
            451
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "278ed456dc4dee8d4fd00e1e1337cf83d6a13bc775a12f683c70ec15ce5640a4",
        "method": "Exact full sum distributions for every n2–2004 independently checked for symmetry and strict lower-half increase;support and reflection progression checked.",
        "attributionOnlyAmendment": {
          "priorContentHash": "f8c642739044de934690b27df585e516e91bec0a6eb788e5766678511a461b3e",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.3",
      "contentHash": "bd23e71e92d137d675fb0dc2dc8d58f8836aba7b20f3f60911e92b89bb8972d3",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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          198
        ],
        "method": "Original451/manual197–198 visually compared;allfive statements,all case work,tables and follow-ups retained. Independent-engine assumption made explicit;16.2 supplied nontrivial progression and missing literal equality reconciled.",
        "restoredAttribution": {
          "name": "AIME",
          "pdfPages": [
            451
          ]
        }
      },
      "math": {
        "status": "passed",
        "contentHash": "bd23e71e92d137d675fb0dc2dc8d58f8836aba7b20f3f60911e92b89bb8972d3",
        "method": "Exhaustively enumerated all positive black/white count quadruples totaling25;two unordered box configurations and common white probability1/25.",
        "attributionOnlyAmendment": {
          "priorContentHash": "16fd1e9c9b176a41d8bb8a3e1aef6305b8e3c3b1a3e45a00a923ddceb8441a53",
          "reason": "Only source credit line restored;mathematical content unchanged."
        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.4",
      "contentHash": "6ea03808659cdd6d3c8082fac54de932f99d58f352b9a922101d04dc49b3dc40",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original451/manual197–198 visually compared;allfive statements,all case work,tables and follow-ups retained. Independent-engine assumption made explicit;16.2 supplied nontrivial progression and missing literal equality reconciled."
      },
      "math": {
        "status": "passed",
        "contentHash": "6ea03808659cdd6d3c8082fac54de932f99d58f352b9a922101d04dc49b3dc40",
        "method": "Enumerated all36 labeled-face pairs;every integer sum1–12 occurs exactly3 times."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.5",
      "contentHash": "01f2805645fec7f86b1831046e18aa897a9366cd1524af3c4906e44bf53c599b",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original451/manual197–198 visually compared;allfive statements,all case work,tables and follow-ups retained. Independent-engine assumption made explicit;16.2 supplied nontrivial progression and missing literal equality reconciled."
      },
      "math": {
        "status": "passed",
        "contentHash": "01f2805645fec7f86b1831046e18aa897a9366cd1524af3c4906e44bf53c599b",
        "method": "Independent adversarial dynamic programming over all count states reachable from(13,13,13,13)gives a minimum of13 correct guesses;potential maximum-count proof handles arbitrary suit counts and ties."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.6",
      "contentHash": "b3de118568d8c4ed27766ab67c7a7b1d012a85212fbe9b2a0c7ba4ebcd77114d",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original452/manual198–201 visually compared;allfive statements,simultaneous-room table,three line families and exterior-point bijection,bag factorization,composition tables/recurrences,and even-parity binomial sum retained.",
        "restoredAttribution": {
          "name": "HMMT",
          "pdfPages": [
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        }
      },
      "math": {
        "status": "passed",
        "contentHash": "b3de118568d8c4ed27766ab67c7a7b1d012a85212fbe9b2a0c7ba4ebcd77114d",
        "method": "Independent synchronous1000-room simulation through60minutes,conservation check each minute,and exact even/odd-state formulas.",
        "attributionOnlyAmendment": {
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        }
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.7",
      "contentHash": "6f5489adec9b78217879ccaaca5d0628ae9b14c17bad0f9bb231d4eee18c4740",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original452/manual198–201 visually compared;allfive statements,simultaneous-room table,three line families and exterior-point bijection,bag factorization,composition tables/recurrences,and even-parity binomial sum retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "6f5489adec9b78217879ccaaca5d0628ae9b14c17bad0f9bb231d4eee18c4740",
        "method": "Enumerated all4³ starts and26 direction vectors,deduplicated exact point sets;48 axis,24 plane,4 space lines. Exterior endpoint involution independently checked."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.8",
      "contentHash": "3bf05daf2882b57664b6528118fb9f65ab5bb7eb7500e56bf76b4719b097f673",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        "answerPdfPages": [
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          200
        ],
        "method": "Original452/manual198–201 visually compared;allfive statements,simultaneous-room table,three line families and exterior-point bijection,bag factorization,composition tables/recurrences,and even-parity binomial sum retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "3bf05daf2882b57664b6528118fb9f65ab5bb7eb7500e56bf76b4719b097f673",
        "method": "Exhaustively enumerated all sorted positive four-slip multisets with labels≤7 and all unordered bag-pairs;only the ordinary and stated nonordinary pairs have the target histogram."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.9",
      "contentHash": "6f309d3fabbc4df42024f778c7b1515dfef4e4a510afec1f54df45c44cda7b2c",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original452/manual198–201 visually compared;allfive statements,simultaneous-room table,three line families and exterior-point bijection,bag factorization,composition tables/recurrences,and even-parity binomial sum retained."
      },
      "math": {
        "status": "passed",
        "contentHash": "6f309d3fabbc4df42024f778c7b1515dfef4e4a510afec1f54df45c44cda7b2c",
        "method": "Independent explicit ordered-composition enumeration and recurrence checks through k20,target g(k+2). Full source small-case sets compared."
      },
      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.10",
      "contentHash": "6469a584c16147ef2d3d81b2150940be47809e2f32af5efc7a2886d70945b2cb",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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          201
        ],
        "method": "Original452/manual198–201 visually compared;allfive statements,simultaneous-room table,three line families and exterior-point bijection,bag factorization,composition tables/recurrences,and even-parity binomial sum retained.",
        "restoredAttribution": {
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          "pdfPages": [
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        "contentHash": "6469a584c16147ef2d3d81b2150940be47809e2f32af5efc7a2886d70945b2cb",
        "method": "Exact Fraction two-state parity chain through50 tosses and separate binomial subset sum match(1+3^−50)/2.",
        "attributionOnlyAmendment": {
          "priorContentHash": "d3605e16c2377eb62f2fc3cb9ada3c855948eaace67b43ca209bb55a0cdc398d",
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        }
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      "grading": {
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        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.11",
      "contentHash": "e31c7e0501f8d4425a68f6d7f91ed4043a7cea7a05265dd7c24a394e7b1585dd",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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        ],
        "method": "Original452–453/manual201–205 visually compared;all6statements,subset size cases,two tournament recurrences,pirate backward cases,full contest probability derivation,score rejection and existence,and expected-win ratio retained.16.14 unstated exchangeability model explicitly reconciled.",
        "restoredAttribution": {
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          "pdfPages": [
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        "contentHash": "e31c7e0501f8d4425a68f6d7f91ed4043a7cea7a05265dd7c24a394e7b1585dd",
        "method": "Exhaustively tested all32767 nonempty subsets;counts by size15,78,165,126,21;spacing map independently checked.",
        "attributionOnlyAmendment": {
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          "reason": "Only source credit line restored;mathematical content unchanged."
        }
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      "grading": {
        "mode": "written",
        "automaticScore": false,
        "sourceKeyPreserved": true
      }
    },
    {
      "id": "16.12",
      "contentHash": "aba11176c16d9d1e95ef0ca5689d79e8554df87edd2d1401bdb08849a5fb7554",
      "source": {
        "status": "passed",
        "textbookPdfPages": [
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        ],
        "answerPdfPages": [
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          202
        ],
        "method": "Original452–453/manual201–205 visually compared;all6statements,subset size cases,two tournament recurrences,pirate backward cases,full contest probability derivation,score rejection and existence,and expected-win ratio retained.16.14 unstated exchangeability model explicitly reconciled.",
        "restoredAttribution": {
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          "pdfPages": [
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      "math": {
        "status": "passed",
        "contentHash": "aba11176c16d9d1e95ef0ca5689d79e8554df87edd2d1401bdb08849a5fb7554",
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