{
  "abstract": "We determine the smallest square container for eight axis-parallel squares of prescribed side lengths. The optimal side length is the minimum of the costs of 23 explicit packing strategies, and each cost is a maximum of sums of side lengths. No minimum of fewer convex functions represents it. Three strategies suffice for seven squares; the jump to 23 for eight squares witnesses the beginning of a combinatorial explosion. We prove the formula by combining the constructions with geometric lower bounds derived from disjoint coordinate projections, cornering arguments, and a weighted area comparison.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28504",
  "author_total": 1,
  "authors": [
    {
      "name": "Matthew Badger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d1ae6r181sjsy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28504",
  "fulltext_url": null,
  "id": "d1ae6r181sjsy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28504",
  "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"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2209|Optimization and Packing Problems"
    ]
  ],
  "openalex": "W7223148352",
  "pdf_url": "https://arxiv.org/pdf/2610.28504",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d1ae6r181sjsy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28504",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28504"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An exact formula for optimal axis-parallel packing of eight squares",
  "title_text": "An exact formula for optimal axis-parallel packing of eight squares",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/d1ae6r181sjsy",
  "venue": "arXiv",
  "versions": []
}