{
  "abstract": "In 2006, Alexander Soifer conjectured that one cannot fully cover a square of side length $> n$ with $n^2 + O(1)$ unit squares. For small $n \\in \\{2, 3\\}$, in 2009, Janusz Januszewski proved that it is impossible to fully cover a square of side length $> n$ with exactly $n^2 + 1$ unit squares. Recently in 2023, Baek and Lee proved that it is impossible to fully cover an equilateral triangle of side length $> n$ with exactly $n^2 + 1$ unit equilateral triangles whose sides are parallel to it. There were also some progress on a related problem: what is the largest square with side length $S(k)$ that can be fully covered by $k$ unit squares. However, there have been no improvement on the original conjecture. In this work, new tools have been developed that led to the proof of this conjecture for $n = 4$, with potential applications to related problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15876",
  "author_total": 1,
  "authors": [
    {
      "name": "Sira Sriswasdi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ed55aw2v9xge.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15876",
  "fulltext_url": null,
  "id": "7ed55aw2v9xge",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15876",
  "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"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "1704|Computational Geometry and Mesh Generation"
    ]
  ],
  "openalex": "W7213346291",
  "pdf_url": "https://arxiv.org/pdf/2609.15876",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ed55aw2v9xge.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15876",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15876"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the covering of $n + ε$ square with $n^2 + 1$ unit squares for $n \\geq 4$",
  "title_text": "On the covering of n + ε square with n^2 + 1 unit squares for n ≥ 4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/7ed55aw2v9xge",
  "venue": "arXiv",
  "versions": []
}