Algebraic Modeling of Rubik’s Cube with Group Theory
Research Article
Open Access
CC BY

Algebraic Modeling of Rubik’s Cube with Group Theory

Chenyu Jia 1*
1 Xi'an Jiaotong Liverpool University
*Corresponding author: Chenyu.Jia23@student.xjtlu.edu.cn
Published on 16 September 2025
Volume Cover
TNS Vol.130
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-80590-289-8
ISBN (Online): 978-1-80590-290-4
Download Cover

Abstract

Transitioning from the practical manipulation of the Rubik’s Cube to its theoretical abstraction presents significant challenges, compounded by a scarcity of foundational resources facilitating this shift. To address this gap and establish a theoretical basis for abstract Rubik’s Cube analysis, this paper provides the fundamental methodology and key conclusions for constructing an algebraic model of the standard 3x3x3 Rubik’s Cube using group theory. Cube operations are defined using the definition of a group, and the concept of the Rubiks Cube group is formally introduced. Employing knowledge of permutation groups, it is demonstrated that any state of the cube can be represented by an element within the set denoted by the direct product of four special groups. Subsequently, by employing group actions, this paper successfully integrates the Rubik’s Cube group with the state space thereby completing the core algebraic modeling of the cube. Utilizing this model, the equivalence condition for solvable cube configurations: the sign of permutation in position of corner and edge are the same, and there was no single edge cube or corner cube artificially flipping. Finally, based on this equivalence, the number of solvable states for a standard 3x3x3 Rubik’s Cube is found.

Keywords:

Permutation group, Rubik's cube group, group action, direct product, reducible stat

View PDF
Jia,C. (2025). Algebraic Modeling of Rubik’s Cube with Group Theory. Theoretical and Natural Science,130,167-175.

References

[1]. Ananthanarayan, B., Das, S., Lahiri, A. et al. (2024) Group theory in physics: an introductionwith mathematics. Eur. Phys. J. Spec. Top., 233, 1443–1513.

[2]. Ramadevi, P. and Dubey, V. (2019) Group Theory for Physicists: With Applications. Cambridge University Press, Cambridge.

[3]. Fonseca, Q.M. (2021) Comput. Phys. Commun. 267, 108085.

[4]. Gururajan, M.P. (2019) Symmetry Relationships between Crystal Structures: Applicationsof Crystallographic Group Theory in Crystal Chemistry. Contemporary Physics, 60, 83–84.

[5]. Comock, C. (2015) Teaching group theory using Rubik’s Cubes. International Journal of Mathematical Education in Science and Technology, 46, 957–967.

[6]. Joyner, D. (2014) The man who found God’s number. The College Mathematics Journal, 258-266.

[7]. Sun, T. (2023) Commutators in the Rubik’s Cube Group. The American Mathematical Monthly, 131, 3–19.

[8]. Chen, J. (2007) Group theory and the Rubik’s cube. Harvard University.

References

[1]. Ananthanarayan, B., Das, S., Lahiri, A. et al. (2024) Group theory in physics: an introductionwith mathematics. Eur. Phys. J. Spec. Top., 233, 1443–1513.

[2]. Ramadevi, P. and Dubey, V. (2019) Group Theory for Physicists: With Applications. Cambridge University Press, Cambridge.

[3]. Fonseca, Q.M. (2021) Comput. Phys. Commun. 267, 108085.

[4]. Gururajan, M.P. (2019) Symmetry Relationships between Crystal Structures: Applicationsof Crystallographic Group Theory in Crystal Chemistry. Contemporary Physics, 60, 83–84.

[5]. Comock, C. (2015) Teaching group theory using Rubik’s Cubes. International Journal of Mathematical Education in Science and Technology, 46, 957–967.

[6]. Joyner, D. (2014) The man who found God’s number. The College Mathematics Journal, 258-266.

[7]. Sun, T. (2023) Commutators in the Rubik’s Cube Group. The American Mathematical Monthly, 131, 3–19.

[8]. Chen, J. (2007) Group theory and the Rubik’s cube. Harvard University.

Cite this article

Jia,C. (2025). Algebraic Modeling of Rubik’s Cube with Group Theory. Theoretical and Natural Science,130,167-175.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

About volume

Volume title: Proceedings of the 3rd International Conference on Applied Physics and Mathematical Modeling

ISBN: 978-1-80590-289-8(Print) / 978-1-80590-290-4(Online)
Editor: Marwan Omar, Shuxia Zhao
Conference website: https://2025.confapmm.org/
Conference date: 31 October 2025
Series: Theoretical and Natural Science
Volume number: Vol.130
ISSN: 2753-8818(Print) / 2753-8826(Online)

© 2024 by the author(s). Licensee EWA Publishing, Oxford, UK. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. Authors who publish this series agree to the following terms:

1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this series.

2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this series.

3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See Open access policy for details).

For authors
For editors
For reviewers

Cookies are used by this site.
Copyright © 2024 EWA Publishing, its licensors, and contributors. All rights reserved, including text and data mining, AI training, and similar technologies.