The computation of the number of unequivalent states of the Rubik's Revenge

TL;DR

This paper calculates the number of distinct states of the Rubik's Revenge considering legal and mechanical constraints, and introduces a new method to determine the probability of solving it after random assembly.

Contribution

It provides a novel approach to counting Rubik's Revenge states using group theory and introduces a new method for computing solving probabilities.

Findings

Count of inequivalent Rubik's Revenge states with constraints

Probability of solving after random assembly

New computational method for state enumeration

Abstract

After having translated the problem of solving the Rubik's Revenge in terms of group actions, we use the result of the structure of the legal transformations (Larsen) to count the states of the Rubik's Revenge modulo legal and indistinguishable transformations, in the presence and absence of mechanical constraints. We also find by a new method the computation made by Bonzio, Loi and Peruzzi of the probability of being able to solve the Rubik's Revenge after having assembled it randomly, once again in the presence and absence of mechanical constraints.