The Legal Rubik's Cube Group

Authors

Wes Schirmer

Senior Project Advisor

Stephanie Treneer

Document Type

Project

Publication Date

Spring 2025

Keywords

Rubik's Cube, Group theory, toys, math

Abstract

This project describes the group structure found in the popular children’s toy, the 3x3 Rubik’s Cube. We begin with an introduction to notation specific to the Rubik’s Cube, leading to the first fundamental theorem of cube theory which describes a unique state of the cube. We then build up the illegal cube group which encompasses all possible combinations of states, including ones not achievable with the physical limits of the basic moves. This group is then reduced to the legal cube group described in the second fundamental theorem of cube theory. Investigating the legal group we find the center, its order, how many moves of prime order it has, and how far from solved a move can be.

[An accessible MathML version of the paper is included with this page as a supplementary file.]

Department

Mathematics

Recommended Citation

Schirmer, Wes, "The Legal Rubik's Cube Group" (2025). WWU Honors College Senior Projects. 993.
https://cedar.wwu.edu/wwu_honors/993

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Language

English

Format

application/pdf