Commutators in the Rubik’s Cube Group

Abstract

Since the Rubik’s Cube was introduced in the 1970s, mathematicians and puzzle enthusiasts have studied the Rubik’s Cube group, i.e., the group of all ≈4.3×1019 solvable positions of the Rubik’s Cube. Group-theoretic ideas have found their way into practical methods for solving the Rubik’s Cube, and perhaps the most notable of these is the commutator. It is well-known that the commutator subgroup of the Rubik’s Cube group has index 2 and consists of the positions reachable by an even number of quarter turns. A longstanding open problem, first posed in 2004, asks whether every element of the commutator subgroup is itself a commutator. We answer this in the affirmative and sketch a generalization to the n×n×n Rubik’s Cube, for all n≥2.