1 · The essentials
Two players take turns claiming unused squares on an initially empty square grid. The first player, called Maker, wants to collect the six squares of a shape called Snaky. The other player, called Breaker, tries to stop this by claiming squares first. Maker can force a win within 21 of its own moves, no matter which legal moves Breaker chooses. The shape can be shifted, turned or flipped and still count as a win. The main result has a Lean formalization, while the manuscript also states that the same move limit works on a 17-by-17 square and that Maker can keep all its choices within a fixed set of 251 squares.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine two people taking turns placing differently coloured stickers on blank squares of graph paper: one wants their stickers to contain a particular six-square picture, while the other tries to occupy useful squares first. The result says that the picture-maker has a way to succeed using at most 21 stickers, even when the other person chooses freely where to interfere.
Why it matters
The repository summary describes this as settling the Snaky achievement problem: Maker can force the desired shape rather than merely hope that Breaker leaves an opening. The explicit move limit strengthens that conclusion by saying how quickly Maker can succeed. The finite-board statements also show that an unlimited playing area is not necessary for the stated guarantee.
2 · Going further
The game starts on an empty infinite square board. Maker moves first, and each player claims one previously unclaimed cell per turn. Maker's target is the Snaky hexomino, a six-cell shape, with translations, rotations and reflections all accepted. The main result is a strategy guaranteeing completion within 21 actual Maker moves against arbitrary legal Breaker play; the count is of Maker's moves, not the combined moves of both players.
The manuscript also states that the same bound holds on a square. More specifically, it states that Maker can confine its claims to a fixed 251-cell board. These are restrictions on the space needed for Maker's strategy, alongside the bound on how many cells Maker must claim before winning.
The supplied abstract states the guarantees but does not describe the strategy or the main steps of its argument. It also does not give the arrangement of the six Snaky cells or of the fixed 251-cell board, so those details cannot be reconstructed from this excerpt. Only one manuscript is supplied; no companion-manuscript relationship is described. The main result has a Lean formalization, but the supplied material does not specify whether that formalization also covers both finite-board statements.
Useful notions
- Square grids and cells
- Alternating turns and legal moves
- Shifting, rotating and reflecting a shape
- A strategy that works against every legal response
What was known before / what is new
Before
The repository summary identifies the question as the Snaky achievement problem and describes the result as settling it. The supplied text gives no earlier bounds, prior strategies or historical account, so no more specific account of what was previously known is available here.
New
The central guarantee is that Maker can achieve Snaky within 21 of its own moves against arbitrary legal Breaker play on the initially empty infinite square board. The manuscript further states the same bound on a square and confinement of Maker's claims to a fixed 251-cell board. The main result has a Lean formalization.
Limits and assumptions
The guarantee assumes an initially empty board, Maker moving first, and exactly one free cell claimed by each player per turn; shifted, rotated and reflected copies of Snaky are accepted. The supplied material does not say that 21 is the smallest possible bound or that 251 cells are necessary. The main result has a Lean formalization, but its detailed scope is not supplied, so formal verification of every finite-board claim should not be assumed. No remaining open questions are specified.
Mini glossary
- Maker–Breaker game —
- Here, a two-player claiming game in which Maker seeks a specified shape and Breaker tries to prevent it.
- Hexomino —
- A shape consisting of six square cells, as Snaky does.
- Translation —
- A shift of a shape to a different location without changing its orientation.
- Reflection —
- A flip of a shape to its mirror image.
- Lean formalization —
- A mathematical result expressed and checked within the Lean proof system.
Keywords : Snaky · Maker–Breaker game · Lean formalization
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Snaky in 21 Maker moves
25/09/2026We 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 square; in fact, Maker can confine its claims to a fixed 251-cell board.