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

Determine the exact diameter of the Cayley graph of the 5×5×5 Rubik’s Cube group in the half-turn metric, defined as the minimal number of allowed moves (90° and 180° 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 half-turn metric, which remains unknown; current work provides only probabilistic estimates (58 turns).

Background

For the 5×5×5 Cube, the paper computes early-step exact counts and then applies the probabilistic coupon collector-based method to estimate diameters. In the half-turn metric, it predicts a diameter of 58.

The authors explicitly note that the exact diameter is still unknown, highlighting the computational infeasibility of brute-force determination for such large configuration spaces.