When Can You Tile an Integer Rectangle with Integer Squares?

TL;DR

This paper provides a comprehensive characterization of when an integer rectangle can be perfectly tiled with integer-sided squares of at least size 2, including conditions for large and small dimensions and periodic behaviors.

Contribution

It establishes necessary and sufficient conditions for tiling rectangles with integer squares, including proofs for large rectangles and computational analysis for small ones.

Findings

Tiling is always possible for rectangles with both sides at least 10.

Behavior is periodic in one dimension when the other is small.

Small cases are fully characterized through exhaustive computational search.

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.