Probabilistic estimates of the diameters of the Rubik's Cube groups

Abstract: The diameter of the Cayley graph of the Rubik's Cube group is the fewest number of turns needed to solve the Cube from the hardest initial configuration. For the 2$\times$2$\times$2 Cube, the diameter is 11 in the half-turn metric, 14 in the quarter-turn metric, 19 in the semi-quarter-turn metric, and 10 in the bi-quarter-turn metric. For the 3$\times$3$\times$3 Cube, the diameter was determined by Rokicki et al. to be 20 in the half-turn metric and 26 in the quarter-turn metric. This study shows that a modified version of the coupon collector's problem in probability theory can predict the diameters correctly for both 2$\times$2$\times$2 and 3$\times$3$\times$3 Cubes insofar as the quarter-turn metric is adopted. In the half-turn metric, the diameters are overestimated by one and two, respectively, for the 2$\times$2$\times$2 and 3$\times$3$\times$3 Cubes, whereas for the 2$\times$2$\times$2 Cube in the semi-quarter-turn and bi-quarter-turn metrics, they are overestimated by two and underestimated by one, respectively. Invoking the same probabilistic logic, the diameters of the 4$\times$4$\times$4 and 5$\times$5$\times$5 Cubes are predicted to be 48 (41) and 68 (58) in the quarter-turn (half-turn) metric, whose precise determinations are far beyond reach of classical supercomputing. The probabilistically estimated diameter is shown to obey the approximate formula of $\ln N / \ln r + \ln N / r$, where $N$ is the number of configurations and $r$ is the branching ratio.