Algorithms for solving rubik's cubes

• Main Authors: Demaine, Erik D. (Contributor), Demaine, Martin L. (Contributor), Eisenstat, Sarah Charmian (Contributor), Lubiw, Anna (Author), Winslow, Andrew (Author)
• Other Authors: Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory (Contributor), Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor)
• Format: Article
• Language: English
• Published: Springer Berlin / Heidelberg, 2012-10-10T15:43:42Z.
• Subjects: Article
• Online Access: Get fulltext

19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011. Proceedings
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 [superscript 2]/logn). The upper bound comes from effectively parallelizing standard Θ(n [superscript 2]) 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 NP-hard in an n ×n ×1 Rubik's Cube when the positions and colors of some cubies are ignored (not used in determining whether the cube is solved).