The square achievement game on n × n boards
Abstract
Two players alternately claim cells of an n × n grid; the first to hold all four corners of an axis-parallel square wins, and a full board without one is a draw. The first player has a winning strategy if and only if n ≥ 5, which answers a question of Erickson. For n ≥ 6 one strategy serves every board: a rule with no lookup table whose moves after the second reply are all forcing, built around the plus, a stone with the four cells at a common distance from it along the axes, a shape proposed for this game by Sirgedas in 2010. The rule is checked on the boards of size $6$ to $17$, and a simulation theorem carries one checked run on the 23 × 23 board to every larger board. The proofs are formalized in Lean 4, and the theorem for the rule states that the tree of its own moves is winning.
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BibTeX
@misc{reid2026square,
title = {{The square achievement game on $n \times n$ boards}},
author = {Sean Reid},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22410},
eprint = {2610.22410},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22410},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The square achievement game on n × n boards
AU - Sean Reid
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.22410
UR - https://arxiv.org/abs/2610.22410
AB - Two players alternately claim cells of an n × n grid; the first to hold all four corners of an axis-parallel square wins, and a full board without one is a draw. The first player has a winning strategy if and only if n ≥ 5, which answers a question of Erickson. For n ≥ 6 one strategy serves every board: a rule with no lookup table whose moves after the second reply are all forcing, built around the plus, a stone with the four cells at a common distance from it along the axes, a shape proposed for this game by Sirgedas in 2010. The rule is checked on the boards of size $6$ to $17$, and a simulation theorem carries one checked run on the 23 × 23 board to every larger board. The proofs are formalized in Lean 4, and the theorem for the rule states that the tree of its own moves is winning.
ER -