Algorithms for solving rubik's cubes
Name
Demaine-Algoritms for solving.pdf
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Author(s)
Demaine, Erik D.
Demaine, Martin L.
Eisenstat, Sarah Charmian
Lubiw, Anna
Winslow, Andrew
Date Issued
August 2011
Journal
Algorithms – ESA 2011
Publisher
Springer Berlin / Heidelberg
Citation
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.
Version
Author's final manuscript
Abstract
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).
Description
19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011. Proceedings
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/978-3-642-23719-5_58
