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Snaky in 21 Maker moves

On an endless grid, the first player can always build the six-square Snaky shape within 21 moves.

1 paper September 25, 2026 Explained by AI

Two players take turns claiming squares of an endless grid. Maker wants six squares forming the Snaky shape; Breaker only blocks. Whether Maker can win on a flat board without a head start was long undecided. This paper proves Maker wins within 21 of its own moves.

First step
—
Significance
Solid
It settles a long-open case of polyomino achievement games, with a checked strategy.
Trust
Formal proof listed
Field
Combinatorics
1 paper

The explanation

Draw a row of four squares on graph paper. Step up one square from the last, then add one more square to the right. That six-square shape is called Snaky. It can be turned and flipped. Counting carefully, there are eight different versions of it on the grid.

Now play a game on an endless grid. Two players, Maker and Breaker, take turns claiming one empty square each. Maker goes first. Maker wins by owning all six squares of some copy of Snaky, in any of its turned or flipped forms. Breaker only tries to stop Maker. Breaker does not need a shape of its own. This is called a Maker–Breaker game.

Earlier work settled Snaky only in other settings. Sieben showed in 2004 that Maker wins in 41 dimensions. Halupczok and Schlage-Puchta showed in 2007 that Maker wins in three dimensions, but loses on an 8 by 8 board. They also showed Maker wins in the plane with a handicap of one. That means Maker places two squares on its first turn. In 2008, Sieben described the ordinary flat game as undecided.

This paper proves that Maker wins on the empty flat grid, with no handicap. Maker can always complete Snaky within 21 of its own moves. Breaker may play anywhere, by any legal rule. In whole turns, Maker wins by the 41st turn of the game. The paper says the same bound holds on a finite 17 by 17 board. It does not claim that 21 is the fewest moves possible.

The difficulty is that the board never ends. Breaker can play far away, or right next to Maker's squares. A computer cannot search every game on an endless board. The paper turns the strategy into a finite record, called a certificate. It lists 728 numbered winning positions, each built from simpler ones. Each position names squares Maker must own, a region Breaker must avoid, and a move budget. The final position needs no Maker squares, a region of 251 squares, and 21 moves.

The key step is simple to state. Maker claims a square that keeps several winning plans alive at once. Breaker's next single square cannot block all of them. So at least one plan always survives. Squares Breaker claims outside the region do not matter.

How sure is this? A computer proof checker called Lean has checked a formal version of the main claim. The Lean scope note says it gives a legal strategy that wins within 21 Maker moves on the endless board. It checks against every legal Breaker play. The smaller finite-board statement is outside that check. No specialist review of the paper has been published.

What it doesn't show. This does not show that 21 moves is the fewest possible, and it says nothing about the game where both players race for Snaky.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Solid. Lean has checked the 21-move strategy on the endless board; the finite-board claim is outside the check, and no specialist review is published.

Words used

Snaky
A six-square shape: a row of four squares, then a step up and one more square.
Maker–Breaker game
A game where one player tries to build a shape and the other only tries to block it.
Polyomino
A shape made of squares joined edge to edge.
Handicap
Extra squares given to Maker at the start; a handicap of one means two squares on the first turn.
Certificate
A finite list of positions and moves that records a full winning strategy.

The original paper

Lean notes

All quoted lines (14)
  1. Maker moves first, and thereafter Maker and Breaker alternate, each claiming one previously unclaimed cell.
    Snaky in 21 Maker moves Back to text
  2. Snaky is the six-cell target S={(0,0),(1,0),(2,0),(3,0),(3,1),(4,1)}⊂Z2.S=\{(0,0),(1,0),(2,0),(3,0),(3,1),(4,1)\}\subset\Z^2.
    Snaky in 21 Maker moves Back to text
  3. Thus reflections and quarter turns are permitted, and scaling is not.
    Snaky in 21 Maker moves Back to text
  4. His account records handicap-two strategies of Harary, Harborth and Seemann and the nonexistence of a pairing defense for Snaky, and proves victory in 4141 dimensions.
    Snaky in 21 Maker moves Back to text
  5. Halupczok and Schlage-Puchta proved that Snaky wins in three dimensions, loses on an 8×88\times8 board, and has planar handicap number at most one.
    Snaky in 21 Maker moves Back to text
  6. A handicap of one allows Maker two stones on the opening turn, before Breaker's first reply.
    Snaky in 21 Maker moves Back to text
  7. Sieben subsequently gave a finite proof sequence for that handicap and described the ordinary planar problem as undecided at the time.
    Snaky in 21 Maker moves Back to text
  8. In this game, there is one globally legal Maker policy that completes an allowed copy of SS within at most 2121 actual Maker claims against every legal continuation of Breaker.
    Snaky in 21 Maker moves Back to text
  9. It corresponds to a win no later than the 4141st individual turn of the alternating game.
    Snaky in 21 Maker moves Back to text
  10. Corollary … also gives the same bound on a finite 17×1717\times17 board.
    Snaky in 21 Maker moves Back to text
  11. We do not assert optimal move counts or board sizes, or a result for the strong game in which both players seek a copy.
    Snaky in 21 Maker moves Back to text
  12. Section … specifies 728728 numbered conditions, including the six bases, by exact integer coordinates and backward references.
    Snaky in 21 Maker moves Back to text
  13. The final one has A=∅A=\varnothing, an envelope of 251251 cells, and height 2121.
    Snaky in 21 Maker moves Back to text
  14. Consequently one new Breaker cell cannot lie in every child envelope, and at least one continuation remains available.
    Snaky in 21 Maker moves Back to text

Other sources

  1. openai/math, Lean scope note for family 187 (2026) The formalization gives a legal strategy that wins within 2121 actual Maker moves against every legal Breaker play from the empty infinite board.
  2. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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