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Additional info for The Mathematical Intelligencer, Volume 32, 2010
By consid ering dozens, thousands, or even millions of such sequences, he has found that, in practice, nearly optimal solutions are found very quickly. Given an input which is the initial cube position denoted by a, his algorithm is given as Algorithm 1. The algorithm can either run to completion, or it can be terminated by the user or when a solution of a desired length is attained. In Kociemba’s algorithm, d2 is a table lookup that takes a position in H and returns the distance to the identity ele ment (e) using moves in A.
Colors, Moves and the Size of Cube Space The Rubik’s cube appears as a stack of 27 smaller cubes (cubies), with each visible face of the cubies colored one of six colors. Of these 27 cubies, seven form a fixed frame around which the other 20 move. The seven that form the fixed frame are the center cubies on each face and the central cubie. Each move on the cube consists of grabbing the nine cubies that form a full face of the larger cube and rotating them as a group 90 or 180 degrees around a central axis Ó 2009 Springer Science+Business Media, LLC, Volume 32, Number 1, 2010 33 shared by the main cube and the nine cubies.
Since for lower values of m our set solver only gives us an upper bound on the set distance, in many cases the true distance of all these positions is less than the calculated upper bound. By solving these explicitly using a single position cube solver, and proving they do not require as many moves as our set solver found, we can frequently reduce our bound on the distance for the set by 1. To facilitate this, if the count of unsolved positions in one of the sets falls below 65,536 at the top of the loop, we print each of these positions to a log file.