Exact diameter of the 4×4×4 Rubik’s Cube in the half-turn metric

Determine the exact diameter of the Cayley graph of the 4×4×4 Rubik’s Cube group in the half-turn metric, defined as the minimal number of allowed moves (90° and 180° turns on any of the three layer indices for each axis and their inverses) required to solve the Cube from any configuration. This establishes the precise “God’s number” for the 4×4×4 Cube in the half-turn metric, which remains unknown; current work provides only probabilistic estimates (41 turns).

Background

The diameter of a Rubik’s Cube group is the fewest number of turns needed to solve the Cube from any configuration (God’s number). For 2×2×2 and 3×3×3 Cubes, the exact diameters are known in several metrics. This paper develops a probabilistic estimation framework and applies it to larger Cubes.

For the 4×4×4 Cube in the half-turn metric, the authors compute branching ratios and cumulative configuration counts and predict a diameter of 41 using a modified coupon collector argument. However, they explicitly note that the true diameter is not yet determined and brute-force computation appears infeasible at present.