{
 "title": "The 536 Ostomachion tilings",
 "source": "https://playnook.io/blog/ostomachion-536",
 "play": "https://playnook.io/games/ostomachion",
 "generated": "2026-09-01",
 "question": "In how many ways can the fourteen Ostomachion pieces be reassembled into the square? Counting tilings that differ only by turning or mirroring the whole square as one, the answer is 536.",
 "agreesWith": {
  "reference": "Bill Cutler, exhaustive computer search, November 2003 — reported in Reviel Netz & William Noel, The Archimedes Codex (2007), and in Netz, Acerbi & Wilson, Towards a Reconstruction of Archimedes' Stomachion, SCIAMVS 5 (2004)",
  "published": {
   "essentiallyDifferent": 536,
   "withTwinsDistinguished": 17152
  },
  "note": "Computed here from the piece set alone; the published figures were not used as an input. Getting BOTH numbers out of one run is what verifies the piece set — matching areas alone could be luck."
 },
 "method": {
  "geometry": "All piece corners lie on a 12x12 grid, so every coordinate is an integer and every test is exact — no floating point anywhere in the search.",
  "search": "Backtracking: take the first uncovered point of the square and try only placements that cover it. 7,072 candidate placements across the fourteen pieces.",
  "bookkeeping": "Coverage is tracked on a point lattice at 1/3 of the small unit. This is bookkeeping for the search, not the criterion: because the fourteen areas sum to exactly 144, any complete non-overlapping placement inside the square is necessarily a tiling.",
  "latticeSafety": "A lattice point sitting exactly on a piece edge would belong to two pieces or none, and the bookkeeping would be wrong. The offset is shifted until provably no point lies on any edge of any possible placement — verified at offset (1,7), 0 edge hits out of 1,296 points.",
  "verification": "400 of the found solutions were additionally re-checked with exact polygon intersection: 0 overlaps.",
  "folding": "Each tiling is reduced to a canonical form under the square's eight symmetries and the two twin swaps; distinct canonical forms are counted."
 },
 "run": {
  "date": "2026-09-01",
  "candidatePlacements": 7072,
  "searchNodes": 1640374,
  "seconds": 134.6,
  "rawTilings": 4288,
  "essentiallyDifferent": 536,
  "withTwinsDistinguished": 17152,
  "gridPoints": 1296,
  "gridOffset": [
   1,
   7
  ],
  "gridEdgeHits": 0,
  "exactlyVerifiedSample": 400,
  "overlapsFound": 0
 },
 "pieceFreedom": {
  "countedOver": 4288,
  "range": [
   16,
   72
  ],
  "note": "How many distinct positions each piece takes. Counted over all 4,288 arrangements. It is independent of area: the largest piece (area 24) is among the most constrained, while one of the two area-3 pieces is among the freest and its twin among the most pinned down.",
  "doNotCountOverThe536": "Counting over the 536 canonical representatives below gives 2 to 38. That is WRONG. The representatives are picked by a lexicographic minimum, which orients all of them the same way. Count over the 4,288 instead."
 },
 "pieces": [
  {
   "index": 0,
   "area": 12,
   "corners": [
    [
     2,
     10
    ],
    [
     0,
     12
    ],
    [
     0,
     0
    ]
   ]
  },
  {
   "index": 1,
   "area": 12,
   "corners": [
    [
     6,
     0
    ],
    [
     2,
     4
    ],
    [
     0,
     0
    ]
   ]
  },
  {
   "index": 2,
   "area": 12,
   "corners": [
    [
     4,
     8
    ],
    [
     2,
     10
    ],
    [
     0,
     0
    ]
   ]
  },
  {
   "index": 3,
   "area": 6,
   "corners": [
    [
     2,
     10
    ],
    [
     6,
     12
    ],
    [
     0,
     12
    ]
   ]
  },
  {
   "index": 4,
   "area": 6,
   "corners": [
    [
     2,
     4
    ],
    [
     4,
     2
    ],
    [
     4,
     8
    ]
   ]
  },
  {
   "index": 5,
   "area": 6,
   "corners": [
    [
     10,
     8
    ],
    [
     6,
     12
    ],
    [
     4,
     11
    ]
   ]
  },
  {
   "index": 6,
   "area": 12,
   "corners": [
    [
     10,
     8
    ],
    [
     12,
     12
    ],
    [
     6,
     12
    ]
   ]
  },
  {
   "index": 7,
   "area": 3,
   "corners": [
    [
     7,
     2
    ],
    [
     8,
     4
    ],
    [
     7,
     8
    ]
   ]
  },
  {
   "index": 8,
   "area": 6,
   "corners": [
    [
     7,
     8
    ],
    [
     8,
     4
    ],
    [
     10,
     8
    ]
   ]
  },
  {
   "index": 9,
   "area": 3,
   "corners": [
    [
     9,
     6
    ],
    [
     12,
     4
    ],
    [
     12,
     6
    ]
   ]
  },
  {
   "index": 10,
   "area": 9,
   "corners": [
    [
     9,
     6
    ],
    [
     12,
     6
    ],
    [
     12,
     12
    ]
   ]
  },
  {
   "index": 11,
   "area": 12,
   "corners": [
    [
     4,
     8
    ],
    [
     10,
     8
    ],
    [
     4,
     11
    ],
    [
     2,
     10
    ]
   ]
  },
  {
   "index": 12,
   "area": 24,
   "corners": [
    [
     6,
     0
    ],
    [
     12,
     0
    ],
    [
     12,
     4
    ],
    [
     9,
     6
    ]
   ]
  },
  {
   "index": 13,
   "area": 21,
   "corners": [
    [
     7,
     2
    ],
    [
     7,
     8
    ],
    [
     4,
     8
    ],
    [
     4,
     2
    ],
    [
     6,
     0
    ]
   ]
  }
 ],
 "areaSum": 144,
 "tilings": {
  "count": 536,
  "canonical": true,
  "coordinates": "Each tiling is fourteen polygons, given as [x,y] corner lists on the 12x12 grid, in the piece order of `pieces` above.",
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}