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On the covering of n + ε square with n2 + 1 unit squares for n ≥ 4

  • Sira Sriswasdi
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Abstract

In 2006, Alexander Soifer conjectured that one cannot fully cover a square of side length > n with n2 + O(1) unit squares. For small n ∈ {2, 3}, in 2009, Janusz Januszewski proved that it is impossible to fully cover a square of side length > n with exactly n2 + 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 n2 + 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.

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Cite

BibTeXRIS
BibTeX
@misc{sriswasdi2026covering,
  title = {{On the covering of $n + ε$ square with $n^2 + 1$ unit squares for $n \geq 4$}},
  author = {Sira Sriswasdi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15876},
  eprint = {2609.15876},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15876},
}
RIS
TY  - UNPB
TI  - On the covering of n + ε square with n^2 + 1 unit squares for n ≥ 4
AU  - Sira Sriswasdi
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15876
UR  - https://arxiv.org/abs/2609.15876
AB  - 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 ∈ {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.
ER  -