I want to shuffle the 3x3 cube randomly in quarter turn metric and determine the minimum number of moves of each position with some mathematical solution. I know that the minimum moves from the most shuffled position is 26 in quarter turn metric, but I want to know from any other position. Or know the level of difficulty of any position of the Rubik's cube. Perhaps not the minimum moves but some number that represents the level of difficulty from any position.
2026-03-25 10:58:59.1774436339
How to calculate the minimum number of moves for resolve the Rubik's cube (3x3) from any position or the level of difficulty?
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Tomas Rokicki and Morley Davidson found all the optimal solutions for a cube for quarter turn and half turn metrics. As you mentioned, the maximum number of turns is 20 for half turn metric and 26 for quarter turn metric.
Unfortunately I don't think there is an efficient way to get the guaranteed number of minimum moves, but you can get fairly close. The Kociemba method has a following where some people have made programs that will find close to or exactly optimal solutions.
Most of the solutions, as you can see in the distribution below, lie around 16-19. So even if the solution isn't guaranteed to be optimal, but still lies in that range, there is a good chance that it is optimal or very close to.