Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Georgiou, George M.
Institution: California State University, San Bernardino
Subfield: Industrial and Manufacturing Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213587207
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