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A generalization of Trenkler’s magic cubes formula

Livinus U. Uko
  • Corresponding author
  • School of Science and Technology, Georgia Gwinnett College, 1000 University Center Ln, Lawrenceville, GA 30043, USA
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/ Terry L. Barron
  • School of Science and Technology, Georgia Gwinnett College, 1000 University Center Ln, Lawrenceville, GA 30043, USA
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Published Online: 2018-01-11 | DOI: https://doi.org/10.1515/rmm-2017-0019

Abstract

A Magic Cube of order p is a p×p×p cubical array with non-repeated entries from the set {1, 2, . . . , p3} such that all rows, columns, pillars and space diagonals have the same sum. In this paper, we show that a formula introduced in The Mathematical Gazette 84(2000), by M. Trenkler, for generating odd order magic cubes is a special case of a more general class of formulas. We derive sufficient conditions for the formulas in the new class to generate magic cubes, and we refer to the resulting class as regular magic cubes. We illustrate these ideas by deriving three new formulas that generate magic cubes of odd order that differ from each other and from the magic cubes generated with Trenkler’s rule.

Keywords: Magic cube; regular magic cube; magic cube formula; Trenkler’s formula

References

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About the article

Published Online: 2018-01-11

Published in Print: 2017-12-20


Citation Information: Recreational Mathematics Magazine, Volume 4, Issue 8, Pages 39–45, ISSN (Online) 2182-1976, DOI: https://doi.org/10.1515/rmm-2017-0019.

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© 2018. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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