{
  "abstract": "Croft, Falconer and Guy ({Unsolved Problems in Geometry}, Problem~C5) exhibit a tiling of the $27\\times27$ square by eight pairwise incomparable integer rectangles and remark that it is not known whether $27$ is the smallest side length of a square that can be tiled by pairwise incomparable integer rectangles, no restriction being placed on the number of tiles. We show that it is: for every integer $n\\le26$ and every $k\\ge2$, the $n\\times n$ square admits no tiling by $k$ pairwise incomparable integer rectangles. The proof combines two structural reductions with an exhaustive search over the $167\\,538$ surviving candidate tile sets, carried out by two independently written programs. The complete software, build instructions and output logs are included as ancillary files.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19536",
  "author_total": 1,
  "authors": [
    {
      "name": "George M. Georgiou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/387kk7fr5krj8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19536",
  "fulltext_url": null,
  "id": "387kk7fr5krj8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19536",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2209|Optimization and Packing Problems"
    ]
  ],
  "openalex": "W7213587207",
  "pdf_url": "https://arxiv.org/pdf/2609.19536",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/387kk7fr5krj8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19536",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19536"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The smallest square tileable by pairwise incomparable integer rectangles",
  "title_text": "The smallest square tileable by pairwise incomparable integer rectangles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/387kk7fr5krj8",
  "venue": "arXiv",
  "versions": []
}