In mathematics, a magic cube is the 3-dimensional equivalent of a

magic square In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...

, that is, a collection of

integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

arranged in an ''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 The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is ...

of the cube, denoted ''M''₃(''n''). It can be shown that if a magic cube consists of the numbers 1, 2, ..., ''n''³, then it has magic constant :

M_3(n) = \frac.

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 Perfect commonly refers to: * Perfection, completeness, excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film * ''Perfect'' (1985 film), a romantic drama * ''Perfect'' (2018 film), a scien ...

; 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 In a magic cube, a broken space diagonal is a sequence of cells of the cube that follows a line parallel to a space diagonal of the cube, and continues on the corresponding point of an opposite face whenever it reaches a face of the cube. Broken s ...

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

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

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. (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 face
Magic cube with the magic square of Dürer
an
Magic cube with the magic square of Gaudi

Alternative definition

Multimagic cubes

Magic cubes based on Dürer's and Gaudi Magic squares

See also

References

External links