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Magic hypercube
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Magic hypercube
In mathematics, a magic hypercube is the k-dimensional generalization of magic squares and magic cubes, 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 diagonals are all the same. The common sum is called the magic constant of the hypercube, and is sometimes denoted Mk(n). If a magic hypercube consists of the numbers 1, 2, ..., nk, then it has magic number
For k = 4, a magic hypercube may be called a magic tesseract, with sequence of magic numbers given by OEIS: 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 includes a module, library(magic), that will create magic hypercubes of any dimension with n a multiple of 4.
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.
This definition of "perfect" assumes that one of the older definitions for perfect magic cubes is used. The Universal Classification System for Hypercubes (John R. Hendricks) requires that for any dimension hypercube, all possible lines sum correctly for the hypercube to be considered perfect magic. Because of the confusion with the term perfect, nasik is now the preferred term for any magic hypercube where all possible lines sum to S. Nasik was defined in this manner by C. Planck in 1905. A nasik magic hypercube has 1/2(3n − 1) lines of m numbers passing through each of the mn cells.
A Nasik magic hypercube is a magic hypercube with the added restriction that all possible lines through each cell sum correctly to S = m(mn+1)/2 where S is the magic constant, m the order and n the dimension of the hypercube.
Hub AI
Magic hypercube AI simulator
(@Magic hypercube_simulator)
Magic hypercube
In mathematics, a magic hypercube is the k-dimensional generalization of magic squares and magic cubes, 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 diagonals are all the same. The common sum is called the magic constant of the hypercube, and is sometimes denoted Mk(n). If a magic hypercube consists of the numbers 1, 2, ..., nk, then it has magic number
For k = 4, a magic hypercube may be called a magic tesseract, with sequence of magic numbers given by OEIS: 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 includes a module, library(magic), that will create magic hypercubes of any dimension with n a multiple of 4.
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.
This definition of "perfect" assumes that one of the older definitions for perfect magic cubes is used. The Universal Classification System for Hypercubes (John R. Hendricks) requires that for any dimension hypercube, all possible lines sum correctly for the hypercube to be considered perfect magic. Because of the confusion with the term perfect, nasik is now the preferred term for any magic hypercube where all possible lines sum to S. Nasik was defined in this manner by C. Planck in 1905. A nasik magic hypercube has 1/2(3n − 1) lines of m numbers passing through each of the mn cells.
A Nasik magic hypercube is a magic hypercube with the added restriction that all possible lines through each cell sum correctly to S = m(mn+1)/2 where S is the magic constant, m the order and n the dimension of the hypercube.