Magic hypercube

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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 x n x n x ... x 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

M_k(n) = \frac{n(n^k+1)}{2}

If, in addition, the numbers on every cross section diagonal also sum up to the hypercube's magic number, the hypercube is called a perfect magic hypercube; otherwise, it is called a semiperfect magic hypercube. The number n is called the order of the magic hypercube.

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).

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