**Reference**:- Erik D. Demaine, Sarah Eisenstat, and Mikhail Rudoy, “Solving the Rubik's Cube Optimally is NP-complete”, in
*Proceedings of the 35th International Symposium on Theoretical Aspects of Computer Science (STACS 2018)*, Caen, France, February 28–March 3, 2018. **Abstract**:-
In this paper, we prove that optimally solving an
*n*×*n*×*n*Rubik's Cube is NP-complete by reducing from the Hamiltonian Cycle problem in square grid graphs. This improves the previous result that optimally solving an*n*×*n*×*n*Rubik's Cube with missing stickers is NP-complete. We prove this result first for the simpler case of the Rubik's Square—an*n*×*n*× 1 generalization of the Rubik's Cube—and then proceed with a similar but more complicated proof for the Rubik's Cube case. Our results hold both when the goal is make the sides monochromatic and when the goal is to put each sticker into a specific location. **Comments**:- The full version of this paper is available as arXiv:1706.06708.
**Length**:- The paper is 13 pages.
**Availability**:- The paper is available in PDF (658k).
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**Related papers**:- Rubik_ESA2011 (Algorithms for Solving Rubik's Cubes)

See also other papers by Erik Demaine.

Last updated January 18, 2021 by Erik Demaine.