A Demigod's Number for the Rubik's Cube

2024-12-30

AI Summary

This paper addresses the problem of efficiently and verifiably estimating an upper bound (i.e., the *diameter*) on the number of moves required to solve the 3×3×3 Rubik’s Cube. We introduce the notion of the *semi-God’s number*—a tight upper bound approximately twice the true diameter. Our method exploits structural properties of vertex-transitive graphs: we uniformly sample cube states, compute their shortest-path distances to the solved state using modern solvers (e.g., Cube Explorer or IDA*), estimate the mean distance, and apply Chernoff bounds to rigorously derive a high-confidence diameter upper bound. This is the first work to leverage the theoretical relationship between average distance and diameter for reproducible, low-compute (a few CPU hours) diameter certification. We establish, with high confidence, that the cube’s diameter is at most 36, with an estimated average distance of ≈18.32 ± 0.1. The framework is generalizable to diameter estimation (within a factor of two) for any vertex-transitive graph.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

It is well-known by now that any state of the $3 imes 3 imes 3$ 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 $36$.

Problem

• Rubik's Cube
• Upper Bound Estimation
• Graph Theory

Innovation

• Half-God's Number
• Average Distance Estimation
• Random Sampling Method