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((About|the mathematical concept|the flashbulb cartridges|Magicube|the puzzle|Rubik's Cube))
((more citations needed|date=September 2014))
[[File:Simple Magic Cube.svg|thumb|right|An example of a 3&thinsp;×&thinsp;3&thinsp;×&thinsp;3 magic cube. In this example, no slice is a magic square. In this case, the cube is classed as a [[simple magic cube]].]]
In [[mathematics]], a '''magic cube''' is the [[dimension|3-dimensional]] equivalent of a [[magic square]], that is, a collection of [[integers]] arranged in an ''n''&thinsp;×&thinsp;''n''&thinsp;×&thinsp;''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 diagonal]]s are equal, the so-called [[magic constant]] of the cube, denoted ''M''<sub>3</sub>(''n'').<ref name=":0">((Cite web|url=http://mathworld.wolfram.com/MagicCube.html|title=Magic Cube|last=W.|first=Weisstein, Eric|website=mathworld.wolfram.com|language=en|access-date=2016-12-04))</ref><ref>((Cite web|title=Magic Cube|url=https://archive.lib.msu.edu/crcmath/math/math/m/m022.htm|access-date=2021-04-20|website=archive.lib.msu.edu))</ref> It can be shown that if a magic cube consists of the numbers 1, 2, ..., ''n''<sup>3</sup>, then it has magic constant ((OEIS|id=A027441))

:<math>M_3(n) = \frac{n(n^3+1)}{2}.</math>

If, in addition, the numbers on every [[cross section (geometry)|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 diagonal]]s also equal the cube's magic number, the cube is called a [[pandiagonal magic cube]].

==Alternative definition==
In recent years, an alternative 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. That is not the case with the above definition for the cube.

==Multimagic cubes==
((Update|inaccurate=yes|reason = see the main article more information has been picked up from MathWorld and other sources about the known cubes|date=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.<ref name=":0" />  (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 [http://sites.google.com/site/aliskalligvaen/home-page/-magic-cube-with-duerer-s-square Magic cube with the magic square of Dürer], and [http://sites.google.com/site/aliskalligvaen/home-page/-magic-cube-with-gaudi-s-square Magic cube with the magic square of Gaudi]

==See also==
* [[Perfect magic cube]]
* [[Semiperfect magic cube]]
* [[Multimagic cube]]
* [[Magic hypercube]]
* [[Magic cube classes]]
* [[Magic series]]
* [[Nasik magic hypercube]]
* [[John R. Hendricks]]

== References ==
((Reflist))

((Citation |last=Andrews |first=William Symes |author-link=William Symes Andrews |title=Magic Squares and Cubes |chapter=Chapter II: Magic Cubes |url=https://djm.cc/library/Magic_Squares_Cubes_Andrews_edited.pdf |publisher=[[Dover Publications]] |edition=2nd |location=New York |year=1960 |pages=64–88 |doi=10.2307/3603128 |jstor=3603128 |isbn=9780486206585 |oclc=1136401 |mr=0114763 |zbl=1003.05500 |s2cid=121770908 ))

==External links==
* Harvey Heinz, [http://members.shaw.ca/hdhcubes/index.htm All about Magic Cubes]
* Marian Trenkler, [http://math.ku.sk/~trenkler/aa-cub-01.pdf Magic p-dimensional cubes]
* Marian Trenkler, [http://math.ku.sk/~trenkler/05-MagicCube.pdf An algorithm for making magic cubes]
* Marian Trenkler, [http://www.imi.ajd.czest.pl/zeszyty/zeszyt13/Trenkler.pdf On additive and multiplicative magic cubes]
* [http://sites.google.com/site/aliskalligvaen/home-page Ali Skalli's magic squares and magic cubes]
((Magic polygons))
((Authority control))

[[Category:Magic squares]]

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