$n \times n \times n$ Rubik's Cubes and God's Number

TL;DR

This paper extends the mathematical understanding of $n imes n imes n$ Rubik's Cubes by establishing solvability criteria, calculating configuration counts, and deriving bounds for God's Number through group theory and combinatorics.

Contribution

It provides necessary and sufficient solvability conditions for $n imes n imes n$ Rubik's Cubes and computes key group theoretical properties and configuration counts.

Findings

Established solvability criteria for $n imes n imes n$ cubes.

Calculated the order of the cube group and configuration counts.

Derived a lower bound for God's Number based on group theory.

Abstract

The Rubik's Cube is the most popular puzzle in the world. Two of its studied aspects are God's Number, the minimum number of turns necessary to solve any state, and the first law of cubology, a solvability criterion. We modify previous statements of the first law of cubology for $n \times n \times n$ Rubik's Cubes, and prove necessary and sufficient solvability conditions. We compute the order of the Rubik's Cube group and the number of distinct configurations of the $n \times n \times n$ Rubik's Cube. Finally, we derive a lower bound for God's Number using the group theoretical results and a counting argument.