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
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).
Change to more modern conventions here-after (basically k ==> n and n ==> m)
Read more about Magic Hypercube: Conventions, Notations, Aspects, Basic Manipulations, Pathfinders, Qualifications
Famous quotes containing the word magic:
“Magic is akin to science in that it always has a definite aim intimately associated with human instincts, needs, and pursuits. The magic art is directed towards the attainment of practical aims. Like other arts and crafts, it is also governed by a theory, by a system of principles which dictate the manner in which the act has to be performed in order to be effective.”
—Bronislaw Malinowski (1984–1942)