Rubik's Cube Group

The Rubik's Cube group is the group of all permutations of the facelets of a Rubik's Cube that can be reached by legal quarter and half turns of its faces, with group multiplication given by performing one sequence of moves after another. The group has order exactly 43,252,003,274,489,856,000, roughly 43 quintillion, reflecting the enormous number of distinct positions the puzzle can reach from a solved state, though this is only a small fraction of all conceivable facelet arrangements because physical turns cannot produce every permutation. The puzzle itself was invented in 1974 by the Hungarian architect Erno Rubik, and the group structure underlying it became a standard illustrative example in the teaching of group theory. In 2010 a team led by the American programmer Tomas Rokicki, using substantial donated computing time from Google, proved that every position of the cube can be solved in at most twenty moves, a bound popularly known as God's Number.