Result 187, Combinatorics

Snaky in 21 Maker moves

Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins.

Lean formalization Proof

The bigger picture

Why it matters

Can a player build a prescribed six-square pattern while an opponent tries to block it? For the pattern called Snaky, the manuscript reports a guaranteed win within 21 moves by the player building it.

What changes?

The manuscript reports settling the Snaky achievement problem in a precise setting: an initially empty infinite square board, with Maker moving first and each player claiming one free cell per turn. Maker wins by owning all six cells of Snaky, a shape made of six connected squares; translations, rotations and reflections count. Against any legal Breaker play, Maker can win within 21 of its own moves. The same bound holds on a 17-by-17 square and on a fixed 251-cell board.

What does that help mathematicians do?

The claim rules out a successful blocking strategy for Breaker under these rules: even an opponent choosing each move in response to Maker cannot prevent completion beyond the stated deadline. The finite-board bounds also show that success does not require access to arbitrarily distant cells. Researchers gain explicit limits on both the time and space sufficient for a forced win, not a claim that 21 moves is optimal.

Are there practical applications?

The immediate value is foundational for combinatorial games, where researchers study what can be forced despite an adversary's choices. Here, a shape-building question receives concrete move and board-size bounds. Those bounds provide a specific target for further study of shorter winning strategies or smaller sufficient boards; the supplied material describes no practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Snaky in 21 Maker moves

September 25, 2026 84 pages

We prove that Maker can achieve the Snaky hexomino within 21 actual Maker moves against arbitrary legal Breaker play on the initially empty infinite square board. The same bound holds on a 17×1717\times17 square; in fact, Maker can confine its claims to a fixed 251-cell board.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Snaky-in-21-Maker-moves-September-25-2026,
  author = {{OpenAI}},
  title = {{Snaky in 21 Maker moves}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Snaky-in-21-Maker-moves-September-25-2026/article.pdf}{OAI:Snaky-in-21-Maker-moves-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/187.md.

Snaky in 21 Maker moves

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

In the Snaky Maker–Breaker game, the players alternately claim cells of Z2\mathbb Z^2, and Maker seeks a translated, rotated, or reflected copy of the six-cell Snaky shape. The formalization gives a legal strategy that wins within 2121 actual Maker moves against every legal Breaker play from the empty infinite board. The selected theorem checks disjointness and the move counts throughout play; it does not impose the paper's smaller finite-board restriction.

The linked supplements reconstruct the older 3535-move appendix's finite recursive certificate and prove four conditional winning templates from partial positions, assuming the required Maker cells are present and Breaker avoids the corresponding finite envelope.

Comparator links

Result Comparator statement
Reconstruction of the 35-move Snaky certificate SnakyCertificate.lean
Conditional winning templates from finite partial positions SnakyConditional.lean
Legal Snaky winning strategy within 21 Maker moves SnakyTwentyOne.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.