Magic hypercube
In mathematics, a magic hypercube is the k-dimensional generalization of magic squares, magic cubes and magic tesseracts; that is, a number of integers arranged in an n × n × n × ... × n pattern such that the sum of the numbers on each pillar (along any axis) as well as the main space diagonals is equal to a single number, the so-called magic constant of the hypercube, denoted Mk(n). It can be shown that if a magic hypercube consists of the numbers 1, 2, ..., nk, then it has magic number
For n = 4, this sequence is 
.
Four-, five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by J. R. Hendricks.
Marian Trenkler proved the following theorem: A p-dimensional magic hypercube of order n exists if and only if p > 1 and n is different from 2 or p = 1. A construction of a magic hypercube follows from the proof.
The R programming language includes a module, library(magic), that will create magic hypercubes of any dimension (with n a multiple of 4).
Change to more modern conventions here-after (basically k ==> n and n ==> m)
It is customary to denote the dimension with the letter 'n' and the order of a hypercube with the letter 'm'.
Further: In this article the analytical number range [0..mn-1] is being used. For the regular number range [1..mn] you can add 1 to each number. This has absolutely no effect on the properties of the hypercube.
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