Optimal solutions for the Rubik's Cube

Optimal solutions for the Rubik's Cube are solutions that are the shortest in some sense. There are two common ways to measure the length of a solution. The first is to count the number of quarter turns (90°). The second and more popular is to count the number of outer-layer twists, called "face turns". A move to turn an outer layer two quarter turns (2x90°) in the same direction would be counted as two moves in the quarter turn metric (QTM), but as one turn in the face metric (FTM, or HTM "Half Turn Metric"). It means that the length of an optimal solution in HTM ≤ the length of an optimal solution in QTM.

The maximal number of face turns needed to solve any instance of the Rubik's Cube is 20, and the maximal number of quarter turns is 26. These numbers are also the diameters of the corresponding Cayley graphs of the Rubik's Cube group. In STM (slice turn metric) the minimal number of turns is unknown, lower bound being 18 and upper bound being 20.

A randomly scrambled Rubik's Cube will most likely be optimally solvable in 18 moves (~ 67.0%), 17 moves (~ 26.7%), 19 moves (~ 3.4%), 16 moves (~ 2.6%) or 15 moves (~ 0.2%) in HTM. By the same token, it is estimated that there is approximately 1 configuration which needs 20 moves to be solved optimally in every 90 billion random scrambles. The exact number of configurations requiring 20 optimal moves to solve the cube is still unknown.

Move notation

To denote a sequence of moves on the 3×3×3 Rubik's Cube, this article uses "Singmaster notation", which was developed by David Singmaster.