• DOI:10.1007/978-3-642-23719-5_58
• Corpus ID: 664306

Algorithms for Solving Rubik's Cubes

@article{Demaine2011AlgorithmsFS,
  title={Algorithms for Solving Rubik's Cubes},
  author={Erik D. Demaine and Martin L. Demaine and Sarah Eisenstat and Anna Lubiw and Andrew Winslow},
  journal={ArXiv},
  year={2011},
  volume={abs/1106.5736}
}

• E. Demaine, M. Demaine, +2 authors Andrew Winslow
• Published 28 June 2011
• Computer Science, Mathematics
• ArXiv

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(n2/ log n). The upper bound comes from effectively parallelizing standard Θ(n2) 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).