Groups of rotating squares

TL;DR

This paper characterizes the permutation groups generated by rotating k-by-k blocks within overlapping rectangles, revealing how parity constraints influence the accessible permutations and identifying specific group structures.

Contribution

It provides a detailed analysis of the permutation groups generated by rotations in overlapping rectangles, including parity effects and special cases for k=2 and k=3.

Findings

The generated groups are typically A_n, S_n, A_m × A_{n-m}, or even permutations subgroup.

Parity constraints from rotations determine the permutation group structure.

Special cases occur when k=2 or k=3, affecting the group classification.

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 $n$ 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 $k = 3$.