Algorithms for Solving Rubik's Cubes

The Rubik’s Cube is perhaps the world’s most famous and iconic puzzle, well-known to have a rich underlying mathematical structure (group theory). In this paper, we show that the Rubik’s Cube also has a rich underlying algorithmic structure. Specifically, we show that the n×n×n Rubik’s Cube, as well as the n×n×1 variant, has a “God’s Number” (diameter of the configuration space) of Θ(n/ logn). The upper bound comes from effectively parallelizing standard Θ(n) solution algorithms, while the lower bound follows from a counting argument. The upper bound gives an asymptotically optimal algorithm for solving a general Rubik’s Cube in the worst case. Given a specific starting state, we show how to find the shortest solution in an n×O(1)×O(1) Rubik’s Cube. Finally, we show that finding this optimal solution becomes NPhard in an n×n×1 Rubik’s Cube when the positions and colors of some of the cubies are ignored (not used in determining whether the cube is solved).