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In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal to the same number, the so-called magic constant of the cube, denoted M3(n).[1] It can be shown that if a magic cube consists of the numbers 1, 2, ..., n3, then it has magic constant (sequence A027441 in the OEIS)

\( M_{3}(n)={\frac {n(n^{3}+1)}{2}}. \)

Simple Magic Cube

An example of a 3 × 3 × 3 magic cube. In this example, no slice is a magic square. In this case, the cube is classed as a simple magic cube.

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

Alternate definition

In recent years, an alternate definition for the perfect magic cube has gradually come into use. It is based on the fact that a pandiagonal magic square has traditionally been called perfect, because all possible lines sum correctly. This is not the case with the above definition for the cube.
Multimagic cubes
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This article's factual accuracy may be compromised due to out-of-date information. The reason given is: see the main article more information has been picked up from MathWorld and other sources about the known cubes. Please update this article to reflect recent events or newly available information. (October 2011)

As in the case of magic squares, a bimagic cube has the additional property of remaining a magic cube when all of the entries are squared, a trimagic cube remains a magic cube under both the operations of squaring the entries and of cubing the entries.[1] (Only two of these are known, as of 2005.) A tetramagic cube remains a magic cube when the entries are squared, cubed, or raised to the fourth power.
Magic cubes based on Dürer's and Gaudi Magic squares

A magic cube can be built with the constraint of a given magic square appearing on one of its faces Magic cube with the magic square of Dürer, and Magic cube with the magic square of Gaudi
See also

Perfect magic cube
Semiperfect magic cube
Multimagic cube
Magic hypercube
Magic cube class
Magic series
Asymptotic magic hyper-tesseract
Nasik magic hypercube
John R. Hendricks


W., Weisstein, Eric. "Magic Cube". Retrieved 2016-12-04.

Magic polygons

Magic circle Magic hexagon Magic hexagram Magic square Magic star Magic triangle

Related shapes

Alphamagic square Antimagic square Geomagic square Heterosquare Pandiagonal magic square Most-perfect magic square

Higher dimensional shapes

Magic cube
classes Magic hypercube Magic hyperbeam


Associative magic square Pandiagonal magic square Multimagic square

Related concepts

Latin square Word square Number Scrabble Eight queens puzzle Magic constant Magic graph Magic series

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia



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