Theoretical and Natural Science

- The Open Access Proceedings Series for Conferences


Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)

Series Vol. 5 , 25 May 2023


Open Access | Article

The Rubik’s Cubes in Group Theory

Jialun Yu * 1 , Wenxin Li 2
1 Wuhan Britain-China School, Wuhan, China
2 Yew Wah International Education Schools of Guangzhou, Guangzhou, China

* Author to whom correspondence should be addressed.

Theoretical and Natural Science, Vol. 5, 275-281
Published 25 May 2023. © 2023 The Author(s). Published by EWA Publishing
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Citation Jialun Yu, Wenxin Li. The Rubik’s Cubes in Group Theory. TNS (2023) Vol. 5: 275-281. DOI: 10.54254/2753-8818/5/20230454.

Abstract

This paper expounds the basic concept of group theory and its application in Rubik’s Cube transformation and restoration formula. The different states of the magic cube are regarded as the elements of the magic cube group, and the set generated by six basic operations is equivalent to the homomorphism of the magic cube group for analysis, from the mathematical characteristics of the permutation group to some practical examples. The collection of possible states of Rubik's Cube is a group, called Rubik's Cube Group, which can be analyzed with the knowledge of group theory. The essence of the magic cube group is the subgroup of the substitution group. There are six basic operations of the magic cube. The combination of basic operations can only produce even pairs of blocks to exchange positions or flip directions at the same time. Therefore, there are some restrictions on the transformation of the magic cube. Some practical examples give some ideas for creating the magic cube formula.

Keywords

Group theory, Permutation group, Homomorphism, Rubik’s Cubes.

References

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2. Cornock, C. (2015) “Teaching group theory using Rubik's Cubes,” International Journal of Mathematical Education in Science and Technology, 46(7), pp. 957–967. Available at: https://doi.org/10.1080/0020739x.2015.1070442.

3. Holt, D.F., Eick, B. and O'Brien, E.A. (2020) Handbook of Computational Group theory. S.l.: CRC PRESS.

4. “Introduction to group theory” (2010) Symmetries and Conservation Laws in Particle Physics, pp. 25–46. Available at: https://doi.org/10.1142/9781848167049_0002.

5. “Permutation groups: A complexity overview” (2003) Permutation Group Algorithms, pp. 48–54. Available at: https://doi.org/10.1017/cbo9780511546549.003.

6. Atkinson, M.D. (1975) “An algorithm for finding the blocks of a permutation group,” Mathematics of Computation, 29(131), pp. 911–913. Available at: https://doi.org/10.1090/s0025-5718-1975-0367030-3.

7. Okamoto, A. (no date) “Group theory visualized through the Rubik’s Cube.” Available at: https://doi.org/10.15760/honors.1001.

8. Chen, J. J. (2004). Group theory and the Rubik’s cube.

9. Cornell, C. (2018) Rubik's cube: How to solve a Rubik's Cube including Rubik's Cube algorithms. United States.

10. El-Sourani, N., Hauke, S. and Borschbach, M. (2010) “An evolutionary approach for solving the Rubik’s cube incorporating exact methods,” Applications of Evolutionary Computation, pp. 80–89. Available at: https://doi.org/10.1007/978-3-642-12239-2_9.

Data Availability

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

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Volume Title
Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)
ISBN (Print)
978-1-915371-53-9
ISBN (Online)
978-1-915371-54-6
Published Date
25 May 2023
Series
Theoretical and Natural Science
ISSN (Print)
2753-8818
ISSN (Online)
2753-8826
DOI
10.54254/2753-8818/5/20230454
Copyright
© 2023 The Author(s)
Open Access
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Copyright © 2023 EWA Publishing. Unless Otherwise Stated