Groups of Rotating Squares

Abstract

This paper discusses the permutations that are generated by rotating $k\times k$ blocks of squares in a union of overlapping $k\times (k+1)$ rectangles. It is found that the single-rotation parity constraints effectively determine the group of accessible permutations. If there are $m$ squares, and the space is partitioned as a checkerboard with $m$ squares shaded and $n–m$ squares unshaded, then the four possible cases are $A_n$, $S_n$, $A_m\times A_{n-m}$, and the subgroup of all even permutations in $S_m\times S_{n-m}$, with exceptions when $k=2$ and $k=3$.