When Can You Tile an Integer Rectangle with Integer Squares?

• Published in 2023
• Added on 2023-10-09

This paper characterizes when an $m \times n$ rectangle, where $m$ and $n$ are integers, can be tiled (exactly packed) by squares where each has an integer side length of at least 2. In particular, we prove that tiling is always possible when both $m$ and $n$ are sufficiently large (at least 10). When one dimension $m$ is small, the behavior is eventually periodic in $n$ with period 1, 2, or 3. When both dimensions $m,n$ are small, the behavior is determined computationally by an exhaustive search.

Links

• http://arxiv.org/abs/2308.15317v1
• http://arxiv.org/pdf/2308.15317v1

Other information

BibTeX entry

@article{WhenCanYouTileanIntegerRectanglewithIntegerSquares,
	key = {WhenCanYouTileanIntegerRectanglewithIntegerSquares},
	type = {article},
	title = {When Can You Tile an Integer Rectangle with Integer Squares?},
	author = {MIT CompGeom Group and Zachary Abel and Hugo A. Akitaya and Erik D. Demaine and Adam C. Hesterberg and Jayson Lynch},
	abstract = {This paper characterizes when an {\$}m \times n{\$} rectangle, where {\$}m{\$} and {\$}n{\$}
are integers, can be tiled (exactly packed) by squares where each has an
integer side length of at least 2. In particular, we prove that tiling is
always possible when both {\$}m{\$} and {\$}n{\$} are sufficiently large (at least 10).
When one dimension {\$}m{\$} is small, the behavior is eventually periodic in {\$}n{\$}
with period 1, 2, or 3. When both dimensions {\$}m,n{\$} are small, the behavior is
determined computationally by an exhaustive search.},
	comment = {},
	date_added = {2023-10-09},
	date_published = {2023-10-09},
	urls = {http://arxiv.org/abs/2308.15317v1,http://arxiv.org/pdf/2308.15317v1},
	collections = {easily-explained,fun-maths-facts,geometry,integerology},
	url = {http://arxiv.org/abs/2308.15317v1 http://arxiv.org/pdf/2308.15317v1},
	year = 2023,
	urldate = {2023-10-09},
	archivePrefix = {arXiv},
	eprint = {2308.15317},
	primaryClass = {cs.CG}
}