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Solving the Rubik’s cube with deep reinforcement learning and search

Nature Machine Intelligence volume 1, pages 356–363 (2019)Cite this article

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Abstract

The Rubik’s cube is a prototypical combinatorial puzzle that has a large state space with a single goal state. The goal state is unlikely to be accessed using sequences of randomly generated moves, posing unique challenges for machine learning. We solve the Rubik’s cube with DeepCubeA, a deep reinforcement learning approach that learns how to solve increasingly difficult states in reverse from the goal state without any specific domain knowledge. DeepCubeA solves 100% of all test configurations, finding a shortest path to the goal state 60.3% of the time. DeepCubeA generalizes to other combinatorial puzzles and is able to solve the 15 puzzle, 24 puzzle, 35 puzzle, 48 puzzle, Lights Out and Sokoban, finding a shortest path in the majority of verifiable cases.

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Fig. 1: Visualization of scrambled states and goal states.
Fig. 2: The performance of DeepCubeA versus PDBs when solving the Rubik’s cube with BWAS.
Fig. 3: The performance of DeepCubeA.
Fig. 4: An example of symmetric solutions that DeepCubeA finds to symmetric states.

Data availability

The environments for all puzzles presented in this paper, code to generate labelled training data and initial states used to test DeepCubeA are available through a Code Ocean compute capsule (https://doi.org/10.24433/CO.4958495.v1)44.

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Acknowledgements

The authors thank D.L. Flores for useful suggestions regarding the DeepCubeA server and T. Rokicki for useful suggestions and help with the optimal Rubik’s cube solver.

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Author notes

  1. These authors contributed equally: Forest Agostinelli, Stephen McAleer, Alexander Shmakov.

Authors and Affiliations

  1. Department of Computer Science, University of California Irvine, Irvine, CA, USA

    Forest Agostinelli, Alexander Shmakov & Pierre Baldi

  2. Department of Statistics, University of California Irvine, Irvine, CA, USA

    Stephen McAleer & Pierre Baldi

Authors
  1. Forest Agostinelli
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  2. Stephen McAleer
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Contributions

P.B. designed and directed the project. F.A., S.M. and A.S. contributed equally to the development and testing of DeepCubeA. All authors contributed to writing and editing the paper.

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Correspondence to Pierre Baldi.

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The authors declare no competing interests.

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Agostinelli, F., McAleer, S., Shmakov, A. et al. Solving the Rubik’s cube with deep reinforcement learning and search. Nat Mach Intell 1, 356–363 (2019). https://doi.org/10.1038/s42256-019-0070-z

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