Exact counts of how many valid ways a small block can be filled at a given position of the official Eternity II board, under increasingly constrained rules: known-good numbers to check your solver's edge-matching and constraint code against.
deterministic — reproduces byte-for-byte
just research-subgridHow many distinct, edge-matching ways can you fill a small block of the official board? These counts are the ground truth to test a solver against: if your edge-matching and constraint code disagrees with the numbers below on a 2×2 corner, the bug is in your code, not the table.
The upright figures here are computed from the official piece set by this project's reproducible Rust generator (linked below), so anyone can rerun them. The handful of italic figures are counts too large to enumerate exactly in seconds (tens of billions to tens of trillions of fillings); those are sylvogel's published values, reproduced here for completeness and credited in the sources. Everything else is recomputed here from scratch.
Computed from the open research note + reproducible Rust generator on GitHub.
| Block | Empty | Fixed | Fixed + 4 hints | c0 | c1 | c2 | c3 |
|---|---|---|---|---|---|---|---|
| 2×2 — top-left corner | 1 312 | 1 307 | 1 291 | 339 | 386 | 276 | 290 |
| 2×2 — top-right corner | 1 312 | 1 307 | 1 291 | 339 | 386 | 276 | 290 |
| 2×2 — bottom-right corner | 1 312 | 1 307 | 1 291 | 339 | 386 | 276 | 290 |
| 2×2 — bottom-left corner | 1 312 | 1 307 | 1 291 | 339 | 386 | 276 | 290 |
| 2×2 — on a side | 73 003 | 72 365 | 69 531 | — | — | — | — |
| 2×2 — in the middle | 4 059 952 | 3 980 912 | 3 649 320 | — | — | — | — |
| 3×3 — top-left corner | 2 633 221 | 2 582 369 | 3 084 | 796 | 972 | 576 | 740 |
| 3×3 — top-right corner | 2 633 221 | 2 582 369 | 3 782 | 1 185 | 1 165 | 763 | 669 |
| 3×3 — bottom-right corner | 2 633 221 | 2 582 369 | 2 897 | 750 | 895 | 589 | 663 |
| 3×3 — bottom-left corner | 2 633 221 | 2 582 369 | 3 195 | 809 | 1 082 | 588 | 716 |
| 3×3 — on a side | 595 003 203 | 578 812 387 | 508 672 842 | — | — | — | — |
| 3×3 — in the middle | 126 907 480 216 | 121 499 303 928 | 99 388 402 312 | — | — | — | — |
| 4×4 — top-left corner | 29 775 113 571 | 28 535 440 805 | 33 845 721 | 8 635 576 | 10 899 481 | 6 220 091 | 8 090 573 |
| 4×4 — top-right corner | 29 775 113 571 | 28 535 440 805 | 41 511 370 | 13 443 985 | 12 896 007 | 8 118 548 | 7 052 830 |
| 4×4 — bottom-right corner | 29 775 113 571 | 28 535 440 805 | 32 538 628 | 8 504 049 | 9 770 922 | 6 488 724 | 7 774 933 |
| 4×4 — bottom-left corner | 29 775 113 571 | 28 535 440 805 | 33 293 752 | 8 630 175 | 11 261 057 | 6 026 551 | 7 375 969 |
| 4×4 — on a side | 26 868 130 739 598 | 25 404 038 618 010 | 19 435 737 986 966 | — | — | — | — |
| 3×3 — encircling the centre clue | — | 142 693 238 | 118 073 954 | — | — | — | — |
Values in italics are sylvogel's published figures, reproduced here for completeness: their counts are too large to enumerate exactly in seconds (tens of billions to tens of trillions of fillings), so this site does not independently recompute them. Every upright value is computed here from the official piece set.
Counting rules: each cell uses a piece of the class matching its border-facing sides (corner / edge / interior); board-rim edges are grey; block-boundary edges facing the interior are non-grey; shared internal edges match; pieces are distinct within the block.