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

### Joshua Arroyo and Leanne Holder

Vol. 15 (2022), No. 3, 393–410
##### 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 $\alpha \beta$-construction which follows an algorithm which places entries into a hypercube by moving along permutations of the vector $⟨\alpha ,0,\dots ,0,\beta ⟩$. Under certain restrictions, this construction creates odd-ordered magic hypercubes.

##### Keywords
magic cube, magic hypercube
Primary: 05B30