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A Demigod's Number for the Rubik's Cube

Arturo Merino, Bernardo Subercaseaux

TL;DR

The paper tackles the long-standing Rubik's Cube diameter problem (God's Number) by introducing a reproducible, probabilistic bound of $D \le 36$, derived from the mean distance $\mu$ in vertex-transitive graphs via the relation $D < 2\mu$. It estimates $\mu$ through uniform sampling of random cube states and upper-bounding single-state distances with a solver, then applying a Hoeffding-type concentration bound to translate the empirical mean $\widehat{\mu}$ into a high-confidence diameter bound. The key contributions are (i) a simple, verifiable method to bound the cube's diameter, (ii) a general demonstration that vertex-transitive graphs have diameters at most twice their mean distance, and (iii) practical steps to reduce sample requirements while maintaining rigorous confidence. The approach offers a meaningful, reproducible alternative to the heavy computation behind the original God’s Number proof and suggests applicability to other puzzles and vertex-transitive graphs in evaluating 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 $36$.

A Demigod's Number for the Rubik's Cube

TL;DR

The paper tackles the long-standing Rubik's Cube diameter problem (God's Number) by introducing a reproducible, probabilistic bound of , derived from the mean distance in vertex-transitive graphs via the relation . It estimates through uniform sampling of random cube states and upper-bounding single-state distances with a solver, then applying a Hoeffding-type concentration bound to translate the empirical mean into a high-confidence diameter bound. The key contributions are (i) a simple, verifiable method to bound the cube's diameter, (ii) a general demonstration that vertex-transitive graphs have diameters at most twice their mean distance, and (iii) practical steps to reduce sample requirements while maintaining rigorous confidence. The approach offers a meaningful, reproducible alternative to the heavy computation behind the original God’s Number proof and suggests applicability to other puzzles and vertex-transitive graphs in evaluating diameters.

Abstract

It is well-known by now that any state of the 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 , from where the diameter is at most .
Paper Structure (18 sections, 13 theorems, 34 equations, 16 figures, 1 table)

This paper contains 18 sections, 13 theorems, 34 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary
  3. Preliminaries
  4. The Relationship Between Diameter and Mean Distance
  5. The Demigod Number for the Rubik's Cube
  6. Randomly sampling cubes
  7. Experimental results
  8. Obtaining the Demigod's number
  9. Discussion
  10. Acknowledgments.
  11. The Beginner's Method and the "Human's Number"
  12. Step 1: The white cross
  13. Step 2: White corners
  14. Step 3: Edges of the second layer
  15. Step 4: The yellow cross
  16. ...and 3 more sections

Key Result

Theorem 1

Given a state $s$ of the Rubik's Cube, let $d(s)$ be the distance from $s$ to the solved state. Let $S$ be a set of states of the Rubik's cube sampled uniformly at random, and let $\widehat{\mu}_S = \frac{1}{|S|}\sum_{s \in S} d(s)$ be the random variable corresponding to the average distance betwee

Figures (16)

  • Figure 1: Illustration of the $3\times 3 \times 3$ Rubik's Cube.
  • Figure 2: Illustration of the Rubik's cube as a subgroup of $S_{54}$ (or even $S_{48}$ due to the centers being static).
  • Figure 3: A graph $G$ with $n = 9$, illustrating the proof of \ref{['lemma:clique']}.
  • Figure 4: Three invalid operations.
  • Figure 5: Histogram showing 500 000 samples. This plot aggregates how many moves the samples needed to be solved. No more than 20 moves were needed, and the empirical mean was 18.3189.
  • ...and 11 more figures

Theorems & Definitions (23)

  • Theorem 1
  • Lemma 1: Human's Number
  • Definition 1: Cayley Graph
  • Definition 2: Diameter
  • Definition 3: Mean distance
  • Definition 4: Graph automorphism
  • Lemma 2
  • Definition 5: Vertex Transitivity
  • Lemma 3
  • proof
  • ...and 13 more