View publication
| Title | A Demigod's Number for the Rubik's Cube |
| Authors | Arturo Merino, Bernardo Subercaseaux |
| Publication date | 2026 |
| Abstract | It is by now well-known that any state of the 3 × 3 × 3 Rubik's Cube can be solved in at most 20 moves, a result often referred to as "God's Number". However, this result took Rokicki et al. around 35 CPU years to prove and is therefore very challenging to reproduce. We provide a novel approach to obtain a worse bound of 36 moves with high confidence, but that offers two main advantages: (i) it is easy to understand, reproduce, and verify, and (ii) our main idea generalizes to bounding the diameter of other vertex-transitive graphs by at most twice its true value, hence the name "demigod number". Our approach is based on the fact that, for vertex-transitive graphs, the diameter at most twice the average distance (of which we give a much simpler proof than in the literature). Then, by sampling uniformly random states and using a modern solver to obtain upper bounds on their distance, a standard concentration bound allows us to confidently state that the average distance is around 18.32 +- 0.18, from where the diameter is at most 36. |
| Pages | article 31 |
| Conference name | International Conference on Fun with Algorithms |
| Publisher | Springer-Verlag (Berlin/Heidelberg, Germany) |
| Reference URL |
|