<abstract xmlns="http://www.w3.org/1999/xhtml"><p> A Magic Cube of order p is a p×p×p cubical array with non-repeated entries from the set {1, 2, . . . , p<sup>3</sup>} 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.</p></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.

A generalization of Trenkler’s magic cubes formula

Recreational Mathematics Magazine

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"A generalization of Trenkler\u2019s magic cubes formula"}

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...

A generalization of Trenkler’s magic cubes formula

Published Online: Jan 11, 2018

Page range: 39 - 45

DOI: https://doi.org/10.1515/rmm-2017-0019

Keywords
Magic cube, regular magic cube, magic cube formula, Trenkler’s formula

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

A generalization of Trenkler’s magic cubes formula

Published Online: Jan 11, 2018

Page range: 39 - 45

DOI: https://doi.org/10.1515/rmm-2017-0019

KeywordsMagic cube, regular magic cube, magic cube formula, Trenkler’s formula

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Magic cube, regular magic cube, magic cube formula, Trenkler’s formula