A Demigod's Number for the Rubik's Cube

TL;DR

The paper introduces a simplified, reproducible method to bound the Rubik's Cube's diameter at 36 moves, generalizing to other graphs, using average distance sampling and concentration bounds.

Contribution

It presents an easy-to-understand, verifiable approach to bounding graph diameters, notably improving the simplicity over previous complex proofs for the Rubik's Cube.

Findings

Bound of 36 moves for Rubik's Cube diameter established

Method generalizes to other vertex-transitive graphs

Sampling and concentration bounds effectively estimate diameters

Abstract

It is well-known by now that any state of the $3\times 3 \times 3$ 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 average distance between vertices is at most half the diameter, and 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 $18.32 \pm 0.1$, from where the diameter is at most $36$.