Solving the Rubik's Cube Optimally is NP-complete

06/21/2017

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by   Erik D. Demaine, et al.

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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.