The αβ-construction of magic hypercubes

Arroyo, Joshua; Holder, Leanne

doi:10.2140/involve.2022.15.393

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The $\alpha\beta$-construction of magic hypercubes

Joshua Arroyo and Leanne Holder

Vol. 15 (2022), No. 3, 393–410

DOI: 10.2140/involve.2022.15.393

Abstract

A magic hypercube of dimension $m$ and order $n$ is an $n^{m}$ array filled with $n^{m}$ distinct positive integers $1, 2, \dots, n^{m}$ such that the entries in each of the $m n^{m - 1}$ hyperrows and the $2^{m - 1}$ space diagonals sum to the same number, the magic sum. We present the $α β$ -construction which follows an algorithm which places entries into a hypercube by moving along permutations of the vector $⟨ α, 0, \dots, 0, β ⟩$ . Under certain restrictions, this construction creates odd-ordered magic hypercubes.

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Keywords

magic cube, magic hypercube

Mathematical Subject Classification

Primary: 05B30

Milestones

Received: 10 September 2020

Revised: 11 November 2021

Accepted: 25 November 2021

Published: 2 December 2022

Communicated by Vadim Ponomarenko

Authors

Joshua Arroyo
Department of Mathematics University of Florida Gainseville, FL United States

Leanne Holder
Department of Mathematics Rose–Hulman Institute of Technology Terre Haute, IN United States

Vol. 15, No. 3, 2022