Commutators in the Rubik’s Cube Group

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.