Algorithms for solving rubik's cubes
Author(s)Demaine, Erik D.; Demaine, Martin L.; Eisenstat, Sarah Charmian; Lubiw, Anna; Winslow, Andrew
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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).
19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011. Proceedings
DepartmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Algorithms – ESA 2011
Springer Berlin / Heidelberg
Demaine, Erik D. et al. “Algorithms for Solving Rubik’s Cubes.” Algorithms – ESA 2011. Ed. Camil Demetrescu & Magnús M. Halldórsson. LNCS Vol. 6942. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. 689–700.
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