Groups of rotating squares

2014/04/22 by Ravi Montenegro, Montenegro, Ravi, David A. Huckaby +3
Computer Science · Engineering · Mathematics · #05E18 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Mathematics and Applications #Matrix Theory and Algorithms #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1404.5455

openalex publication_date 2014/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper discusses the permutations that are generated by rotating k × k blocks of squares in a union of overlapping k × (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 × A_n-m, and the subgroup of all even permutations in S_m × S_n-m, with exceptions when k = 2 and k = 3.

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