1. The question
Two players alternately claim empty cells of square graph paper. Maker wants to occupy a particular six-cell shape called Snaky. Breaker only wants to prevent this from ever happening. Can Maker force the shape from an empty board without receiving an extra opening move?
The manuscript claims that Maker can win within twenty-one of its own moves, regardless of Breaker’s legal replies. The strategy works on the infinite square board and can be confined to a fixed finite region inside a -by- square.
The result settles the ordinary planar achievement question for this particular shape. Its main device is a finite collection of conditional winning positions that can be combined into larger ones. The endpoint of the construction has no initial Maker holdings, so it applies directly to the empty board.
2. Background and essential definitions
A polyomino is a shape made from unit square cells joined edge to edge. A six-cell polyomino is a hexomino. Snaky is the set of cells indexed by
Here a cell’s coordinates refer to its lower-left corner. In the small diagram below, # marks a target cell and . an unused location:
1 | ...## |
Translations, quarter turns and reflections are permitted. Scaling is not. Thus Maker may build the same shape in any location and any of its square-grid orientations.
The board is initially empty. Maker moves first, after which the players alternate, each claiming one previously unclaimed cell. Ownership is permanent. Maker wins as soon as it owns all six cells of an allowed copy; additional Maker cells do not invalidate that copy.
This is a weak achievement game, also called a Maker–Breaker game. Breaker has no independent objective of completing its own shape. In the strong game both players seek a copy, so the rules and possible outcomes differ.
A handicap of one means Maker gets two cells on the opening turn before Breaker’s first response. The theorem here gives no such advantage. Its move bound counts actual cells claimed by Maker, including replacement moves when a planned cell is already Maker-owned.
3. Development history and previous partial results
Polyomino achievement games were introduced in work associated with Frank Harary. Snaky became a difficult small example: its six cells are enough to create several interacting threats, yet its geometry resists simple short strategies.
Earlier work found winning strategies with extra initial resources. Harary, Harborth and Seemann obtained handicap-two strategies. Sieben proved a win in forty-one dimensions, showing that abundant extra directions can overcome the obstruction.
Another line examined pairing defenses. In such a defense, Breaker pairs cells in advance and responds to a Maker claim by taking its partner. If every target copy contains a full pair, this can prevent a win. Snaky was shown to admit no pairing defense, with a later computer-free proof by Csernenszky, Martin and Pluhár. But ruling out one class of defenses does not establish a Maker win against every possible strategy.
Halupczok and Schlage-Puchta proved that Snaky wins in three dimensions, loses on an -by- board, and has planar handicap number at most one. Ito and Miyagawa also presented a handicap-one construction.
Sieben’s work on proof trees gave a finite proof sequence for that handicap and developed a way to combine local winning situations. The ordinary empty planar-board problem remained distinct: Maker’s extra opening stone could not simply be removed from those arguments.
Computer encodings of finite positional games also studied Snaky and related targets. The new manuscript uses a finite certificate of sufficient conditions for the infinite weak game. It explicitly accommodates Breaker moves outside the displayed local regions, so its conclusion is not limited to an unmentioned finite search board.
4. What the new manuscripts claim
The main theorem states that one globally legal Maker strategy completes an allowed copy of Snaky within at most
against every legal Breaker continuation from the empty infinite board.
Since Maker moves first, its twenty-first move is the forty-first individual turn. There are at most twenty Breaker replies before that final Maker claim. Thus the theorem is not a claim of winning within twenty-one total alternating turns.
The same bound holds on a finite -by- board. More precisely, the strategy can keep all Maker claims inside a fixed region of 251 cells. The manuscript does not claim that either the move count or the board size is optimal.
The main certificate has 728 numbered conditions, including six elementary base conditions. Its final condition requires no pre-owned Maker cells, uses the 251-cell region, and has height twenty-one. Height is the bound on future actual Maker moves associated with a condition.
The paper also presents independent constructions with bounds of twenty-five and thirty-five Maker moves. They use different representations and openings, so they provide additional descriptions of the local tactics rather than improving the main numerical bound.
These assertions concern the ordinary weak game. They do not assert the same result for the strong game, biased play with several Breaker claims per turn, or an arbitrary smaller finite board.
5. Why the advance matters
Importance: 1/5. This is a focused contribution to positional games: it resolves a concrete longstanding achievement question for one small shape and gives a finite move bound.
The score measures relative mathematical scope across the collection, not how entertaining or difficult the problem is. Unlike a general theorem for all polyominoes or all Maker–Breaker games, the headline concerns Snaky under one precise rule set.
Its broader methodological interest is the composition of conditional strategies. A finite local condition can be strong enough to survive arbitrary earlier play and replies outside the region under discussion. Combining such conditions offers a way to express an infinite-board strategy through finite data, while retaining a clear accounting of moves and previously occupied cells.
The finite-board consequence is particularly concrete. The claimed infinite-board win does not require Maker to keep searching farther away; a bounded region already contains enough alternative continuations.
6. The main idea
Describe a position by what a tactic needs
A conditional winning position is recorded as a triple . The finite set contains cells Maker must already own. The finite envelope must contain no Breaker cell. The number bounds the further Maker moves needed to win.
There may be additional Maker cells anywhere, and Breaker may have cells outside . The condition makes no assumption about how the position arose. It applies whenever it is Maker’s turn and the stated holdings and clear region are available.
The elementary conditions start with five cells of a target already owned and the remaining cell free. One additional claim completes the shape.
Create several continuations that one reply cannot destroy
Suppose Maker has a pivot move after which several child conditions have their required cells. Their envelopes may overlap, but the construction arranges that every cell common to all envelopes is already Maker-owned.
Breaker can claim only one new cell. That cell cannot belong to every child envelope, because their common intersection is unavailable to Breaker. At least one envelope therefore remains clear, and Maker follows that child’s strategy.
The move bound is one for the pivot plus the largest bound among the children:
The bound uses the worst surviving continuation, so Maker does not need to predict Breaker’s choice. Repeating this composition builds a tree of sufficient conditions.
This generalizes the familiar idea of a double threat: if two different cells would each complete a target, Breaker cannot occupy both in one move. Here the alternatives can be multi-move strategies with different local envelopes, not merely immediate completions.
Keep the policy legal under previous ownership
A planned pivot may already belong to Maker because of earlier play. The tactic cannot count claiming it again as a legal move. The manuscript instead uses a fresh replacement claim, still charging one actual move, before proceeding with the same ownership guarantees.
If a target has already been completed, play stops. Otherwise the policy chooses among available cells and surviving child conditions by fixed rules. This turns the existential composition into one strategy selected before Breaker’s play, rather than a different strategy chosen after seeing the whole future.
On the infinite board a fresh cell always exists after finitely many moves. The finite-board argument keeps the policy within the stated envelope and accounts for available replacement cells there as well.
Assemble the finite certificate
Each nonbase condition refers backward to earlier conditions after translations, reflections or quarter turns. The final condition has , so the empty board meets its requirements.
The manuscript prints the exact coordinate data and describes reconstruction and checking programs. Their role is to express the finite identities used by the composition rule. The mathematical claim combines those identities with the rule’s game-theoretic interpretation; a list of coordinates alone would not explain why arbitrary outside replies are covered.
7. Scope, limitations, and open questions
The twenty-one-move figure is an upper bound. The paper does not determine the minimum number of Maker moves against optimal defense. Nor does it determine the smallest winning rectangular board or smallest winning cell region.
Rotations and reflections are allowed, and Maker may own extra cells. Changing either convention produces a different achievement problem. The move order and one-cell-per-turn rule also remain part of the theorem.
A handicap-one result gives Maker an extra initial claim; the present theorem starts from an empty board without that privilege. Conversely, the theorem says nothing about Breaker moving first or claiming several cells each turn.
The strong game is outside scope. A strategy that forces a Maker target while ignoring whether Breaker has already completed its own target need not settle the game where both players can win.
Finally, a win on the stated finite board does not contradict the earlier loss on an -by- board. The available space and the number of alternative local envelopes are part of what makes the construction work.
8. Sources and further reading
The manuscript gives the exact target, game rules, composition lemma and three finite constructions. Its main certificate supplies the twenty-one-move conclusion; the later constructions expose different local tactics.
Primary predecessors include Halupczok–Schlage-Puchta’s Achieving Snaky, Sieben’s higher-dimensional result, and Sieben’s proof-tree framework. Shaik and collaborators describe finite-game encodings that provide computational context, with different board and depth conventions.
Source manuscripts
- M1. OpenAI, Snaky in 21 Maker moves (September 25, 2026).
Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.