Unilateral and Equitransitive Tilings by Squares of Sizes 1, 2, 3, and 4

In this article we examine a problem stated in Grübaum and Shephard’s classic book on tiling theory, Tilings and Patterns, which asks the reader to find all unilateral and equitransitive tilings by squares of sizes $1,2,3,\dots ,n$ where $n\in \mathbb{N}$. The unilaterality condition requires that two squares of the same size in such a tiling may not intersect along an entire side, and the equitransitivity requires that two tiles of the same size in such a tiling must be in the same transitivity class with respect to the symmetry group of the tiling. These two conditions on tilings by squares allows for surprisingly interesting and complex tilings by squares. In this work we provide the answer to Grübaum and Shephard’s question for the case $n=4$, as well as developing methods and computer algorithms that may be extended to the cases where $n\ge 5$.