  The diameter of the Cayley graph of the Rubik's Cube group is the fewest
number of turns needed to solve the Cube from any initial configurations. 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$3 Cube, the diameter was
determined by Rokicki et al. to be 20 in the half-turn metric and 26 in the
quarter-turn metric. This study shows that a modified version of the coupon
collector's problem in probabilistic theory can predict the diameters correctly
for both 2$\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.
It is shown that the probabilistically estimated diameter is approximated by
$\ln N / \ln r + \ln N / r$, where $N$ is the number of configurations and $r$
is the branching ratio.
