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Solving Small Rubik’s Cubes as Slowly as Possible

Authors Jenny Quan , Noah Kim , Bernardo Subercaseaux , John Mackey

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LIPIcs.FUN.2026.38.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Hamilton connectivity
  • Rubik’s Cube
  • Finite group theory

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Abstract

We study, for different Rubik’s puzzles, whether from any starting state one can solve the puzzle as slowly as possible, visiting every reachable state exactly once before reaching the solved configuration. This question corresponds to the existence of Hamiltonian paths (ending in the solved state) in the Cayley graphs associated with these puzzles. A major conjecture attributed to Lovász is that every Cayley graph has a Hamiltonian path. An even stronger version of the conjecture, considered by Dupuis and Wagon (2015) and Gregor et al. (2024), is that every Cayley graph of degree at least 3 is either bipartite and has Hamiltonian paths between any pair of vertices on opposite parts, or is non-bipartite and has Hamiltonian paths between any pair of vertices. Our study of slowly solving Rubik’s puzzles amounts to studying this Strong Lovász Conjecture in their respective Cayley graphs. We first verify the Strong Lovász Conjecture computationally for small Rubik’s puzzles like the 1 × 2 × 3 or 1 × 3 × 3 cuboids, which have under 200 states. This approach, however, becomes infeasible for the 2 × 2 × 2, which has over 3.6 million states. Our main result is then showing that the Strong Lovász Conjecture holds for the 2 × 2 × 2 cube, using a careful graph-theoretic construction based on the subgroup induced by the R and U turns.

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Jenny Quan, Noah Kim, Bernardo Subercaseaux, and John Mackey. Solving Small Rubik’s Cubes as Slowly as Possible. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.38

Author Details

Jenny Quan
  • Carnegie Mellon University, Pittsburgh, PA, USA
Noah Kim
  • Carnegie Mellon University, Pittsburgh, PA, USA
Bernardo Subercaseaux
  • Carnegie Mellon University, Pittsburgh, PA, USA
John Mackey
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Subercaseaux, Bernardo: supported by NSF grant DMS-2434625.

Supplementary Materials

References

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