Exact diameter of the 5×5×5 Rubik’s Cube in the quarter-turn metric

Determine the exact diameter of the Cayley graph of the 5×5×5 Rubik’s Cube group in the quarter-turn metric, defined as the minimal number of allowed moves (only 90° turns on exterior and interior layers for each axis and their inverses) required to solve the Cube from any configuration. This establishes the precise “God’s number” for the 5×5×5 Cube in the quarter-turn metric, which remains unknown; current work provides only probabilistic estimates (68 turns).

Background

In the quarter-turn metric for the 5×5×5 Cube, the authors predict a diameter of 68 based on branching ratios and cumulative counts derived via a modified coupon collector framework.

They explicitly state that the exact diameter is not known, underscoring the gap between probabilistic prediction and definitive determination for higher-order Cubes.