Skip to content
Accessible Unlicensed Requires Authentication Published by De Gruyter October 20, 2018

On the n × n × n Rubik's Cube

Stefano Bonzio, Andrea Loi and Luisa Peruzzi
From the journal Mathematica Slovaca

Abstract

We state and prove the “First law of Cubology”, i.e. the solvability criterion, for the n × n × n Rubik's Cube.

  1. (Communicated by Anatolij Dvurečenskij)

Acknowledgement

The work of the first author is supported by project GBP202/12/G061 of the Czech Science Foundation. The second author was supported by Prin 2015 – Real and Complex Manifolds; Geometry, Topology and Harmonic Analysis – Italy, by INdAM, by GNSAGA – Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni – and also by GESTA – Funded by Fondazione di Sardegna and Regione Autonoma della Sardegna.

References

[1] BANDELOW, C.: Inside Rubiks Cube and Beyond, Birkhäuser, 1982.Search in Google Scholar

[2] BONZIO, S.: Algebraic Structures from Quantum and Fuzzy Logics, PhD Thesis, Università di Cagliari, 2016.Search in Google Scholar

[3] BONZIO, S.—LOI, A.—PERUZZI, L.: The first law of cubology for the rubiks revenge, Math. Slovaca 67(3) (2017), 561–572.Search in Google Scholar

[4] CHEN, J.: Group Theory and the Rubiks Cube, Notes, 2004.Search in Google Scholar

[5] CZECH, B.—LARJO, K.—ROZALI, M.: Black holes as rubiks cubes, J. High Energy Phys. 2011(8) (2011), 143.Search in Google Scholar

[6] DEMAINE, E. D.—DEMAINE, M. L.—EISENSTAT, S.—LUBIW, A.—WINSLOW, A.: Algorithms for Solving Rubiks Cubes. In: Algorithms – ESA (C. Demetrescu and M. M. Halld orsson, eds.): 19th Annual European Symposium, 2011.Search in Google Scholar

[7] DIACONU, A.—LOUKHAOUKHA, K.: An improved secure image encryption algorithm based on Rubiks cube principle and digital chaotic cipher, Math. Probl. Eng. (2013).Search in Google Scholar

[8] FREY, A.—SINGMASTER, D.: Handbook of Cubik Math, Enslow Publishers, 1982.Search in Google Scholar

[9] JONES, M.—SHELTON, B.—WEAVERDYCK, M.: On god s number(s) for rubiks slide, College Math. J. 45(4) (2014), 267–275.Search in Google Scholar

[10] JOYNER, D.: Adventures in Group Theory, The Johns Hopkins University Press, 2008.Search in Google Scholar

[11] KOSNIOWSKI, C.: Conquer that Cube, Cambridge University Press, 1981.Search in Google Scholar

[12] KUNKLE, D.—COOPERMAN, G.: Twenty-six moves suffice for rubiks cube. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, 2007, pp. 235–242.Search in Google Scholar

[13] LARSEN, M. E.:Rubiks revenge: The group theoretical solution, Amer. Math. Monthly 92(6) (1985), 381–390.Search in Google Scholar

[14] LEE, C. L.—HUANG, M. C.: The Rubiks cube problem revisited: a statistical thermodynamic approach, Eur. Phys. J. B. 64(2) (2008), 257–261.Search in Google Scholar

[15] MILLER, J.: Move-Count Means with Cancellation and Word Selection Problems in Rubiks Cube Solution Approaches, PhD Thesis, Kent State University, 2012.Search in Google Scholar

[16] ROKICKI, T.: Towards god s number for Rubiks cube in the quarter-turn metric, College Math. J. 45(4) (2014), 242–242.Search in Google Scholar

[17] ROKICKI, T.—KOCIEMBA, H.—DAVIDSON, M.—DETHRIDGE, J.: The diameter of the Rubiks cube group is twenty, SIAM J. Discrete Math. 27(2) (2013), 1082–1105.Search in Google Scholar

[18] VOLTE, E.—PATARIN, J.—NACHEF, V.: Zero knowledge with Rubiks cubes and non-abelian groups. In: Cryptology and Network Security (M. Abdalla, C. Nita-Rotaru, and R. Dahab, eds.): 12th International Conference, 2013, pp. 74–91.Search in Google Scholar

Received: 2017-03-09
Accepted: 2017-08-15
Published Online: 2018-10-20
Published in Print: 2018-10-25

© 2018 Mathematical Institute Slovak Academy of Sciences