Groups of Rotating Squares

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 \(k = 3\).