# Analysis

> These do not try to solve the puzzle: they measure it, which is how the community knows where the walls are. Parity arguments give one-pass impossibility proofs; solution counting pins down how many full solutions exist to within a factor of two, without anyone having seen one.

- Canonical page (with interactive figures/demos): https://eternity2.dev/research/build/analysis/
- Updated: 2026-07-13

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These do not try to solve the puzzle: they measure it, which is how the community knows where the walls are. Parity arguments give one-pass impossibility proofs; solution counting pins down how many full solutions exist to within a factor of two, without anyone having seen one.

The pages below go technique by technique: what each is in a line, what it
actually reached on the real 16×16 board, where it stops, and the labs and
measurements to back it. For the whole territory at once, start with
[a map of every known approach](/research/build/approaches-map).

## Pages in this section

- [Parity arguments](https://eternity2.dev/research/build/analysis/parity-arguments) — Count anything on an edge-matching board twice, once from each side, and the totals must agree, giving impossibility proofs for the price of one pass. The 479 story shows both the power and the trap: a clean parity claim, true for every interior move, defeated through the sixty border edges nobody scores.
- [Solution counting: measuring what you cannot find](https://eternity2.dev/research/build/analysis/solution-counting) — Nobody has ever seen a full Eternity II solution, yet the community knows, to within a factor of two, how many exist. This page is the story and the craft of that number: exact censuses on small boards, the expectation formula and its twenty-year convergence on 14,702, and the culled-search estimates that were trusted only when four independent runs agreed.

## Related

- [A map of every known approach](https://eternity2.dev/research/build/approaches-map) — The survey the community keeps asking for and never finds: every family of attack tried on Eternity II, what each one actually reached, where it walls out, and a link to the deep page. One organising insight runs through all of them.
- [The techniques](https://eternity2.dev/research/build/techniques) — The technique shelf: the algorithms and pruning ideas that recur in every serious Eternity II solver, each with what it is, what it costs, and what it actually bought when measured on this puzzle.