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arXiv · 2609.19536

The smallest square tileable by pairwise incomparable integer rectangles

George M. Georgiou
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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.

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George M. Georgiou. 2026-09-17. The smallest square tileable by pairwise incomparable integer rectangles. https://arxiv.org/abs/2609.19536

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