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 2/logn). The upper bound comes from effectively parallelizing standard Θ(n 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). © 2011 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Demaine, E. D., Demaine, M. L., Eisenstat, S., Lubiw, A., & Winslow, A. (2011). Algorithms for solving rubik’s cubes. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6942 LNCS, pp. 689–700). https://doi.org/10.1007/978-3-642-23719-5_58
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