Editing Magic hypercube (section)

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In [[mathematics]], a '''magic hypercube''' is the [[dimension|''k''-dimensional]] generalization of [[magic square]]s and [[magic cube]]s, that is, an ''n'' × ''n'' × ''n'' × ... × ''n'' array of [[integers]] such that the sums of the numbers on each pillar (along any axis) as well as on the main [[space diagonal]]s are all the same.  The common sum is called the [[magic constant]] of the hypercube, and is sometimes denoted ''M''<sub>''k''</sub>(''n''). If a magic hypercube consists of the numbers 1, 2, ..., ''n''<sup>''k''</sup>, then it has magic number

:<math>M_k(n) = \frac{n(n^k+1)}{2}</math>.

For ''k'' = 4, a magic hypercube may be called a '''magic tesseract''', with sequence of magic numbers given by {{OEIS2C|id=A021003}}.

The side-length ''n'' of the magic hypercube is called its ''order''.  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)|R programming language]] includes a module, <code>library(magic)</code>, that will create magic hypercubes of any dimension with ''n'' a multiple of 4.