Sub-Optimal Multi-Phase Path Planning: A Method for Solving Rubik's Revenge

Rubik's Revenge, a 4x4x4 variant of the Rubik's puzzles, remains to date as an unsolved puzzle. That is to say, we do not have a method or successful categorization to optimally solve every one of its approximately $7.401 \times 10^{45}$ possible configurations. Rubik's Cube, Rubik's Revenge's predecessor (3x3x3), with its approximately $4.33 \times 10^{19}$ possible configurations, has only recently been completely solved by Rokicki et. al, further finding that any configuration requires no more than 20 moves. With the sheer dimension of Rubik's Revenge and its total configuration space, a brute-force method of finding all optimal solutions would be in vain. Similar to the methods used by Rokicki et. al on Rubik's Cube, in this paper we develop a method for solving arbitrary configurations of Rubik's Revenge in phases, using a combination of a powerful algorithm known as IDA* and a useful definition of distance in the cube space. While time-series results were not successfully gathered, it will be shown that this method far outweighs current human-solving methods and can be used to determine loose upper bounds for the cube space. Discussion will suggest that this method can also be applied to other puzzles with the proper transformations.