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Algorithms for Solving Rubik’s Cubes

2011; Springer Science+Business Media; Linguagem: Inglês

10.1007/978-3-642-23719-5_58

1611-3349

Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Anna Lubiw, Andrew Winslow,

Advanced Graph Theory Research

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).