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
- <math>M_k(n) = \frac{1}{2}n(n^k+1)<math>
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. Marián 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. From the proof follows a construction of magic hypercube.
The R programming language includes a module, library(magic), that will create magic hypercubes of any dimension (with n a multiple of 4).
External links
- page about J. R. Hendricks (http://perso.club-internet.fr/cboyer/multimagie/English/Hendricks.htm)
- papers on magic cubes and hypercubes (http://kosice.upjs.sk/~trenkler/Cube-Ref.html)
- Magic Cubes - Introduction by Harvey D. Heinz (http://members.shaw.ca/hdhcubes/)