{
  "abstract": "Two players alternately claim cells of an $n \\times n$ 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 $n \\ge 5$, which answers a question of Erickson. For $n \\ge 6$ 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 $23 \\times 23$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22410",
  "author_total": 1,
  "authors": [
    {
      "name": "Sean Reid"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2rn403rzjyjc9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22410",
  "fulltext_url": null,
  "id": "2rn403rzjyjc9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22410",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222972875",
  "pdf_url": "https://arxiv.org/pdf/2610.22410",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2rn403rzjyjc9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22410",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22410"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The square achievement game on $n \\times n$ boards",
  "title_text": "The square achievement game on n × n boards",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/2rn403rzjyjc9",
  "venue": "arXiv",
  "versions": []
}