mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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

magic square In mathematics, especially History of mathematics, historical and 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 diago ...

, that is, a collection of

integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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 diagonal In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with '' face diagonals'', which connect vertices on the same face (b ...

s are equal, 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''). 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; 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 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

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. John R. Hendricks of Canada (1929–2007) has listed four bimagic cubes, two trimagic cubes, and two tetramagic cubes. Two more bimagic cubes (of the same order as those of Hendricks, but differently arranged) were found by Zhong Ming, a mathematics teacher in China. Several of these are perfect magic cubes, and remain perfect after taking powers.

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