The smallest square tileable by pairwise incomparable integer rectangles

@misc{georgiou2026smallest,
  title = {{The smallest square tileable by pairwise incomparable integer rectangles}},
  author = {George M. Georgiou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19536},
  eprint = {2609.19536},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19536},
}

TY  - UNPB
TI  - The smallest square tileable by pairwise incomparable integer rectangles
AU  - George M. Georgiou
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.19536
UR  - https://arxiv.org/abs/2609.19536
AB  - Croft, Falconer and Guy ({Unsolved Problems in Geometry}, Problem~C5) exhibit a tiling of the 27×27 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≤26 and every k≥2, the n× 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.
ER  -