Magic cube

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For the flashbulb cartridges, see the "Magicube" section within Flash (photography).

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

$M_3(n) = \frac{n(n^3+1)}{2}$

An example of a 3 × 3 × 3 magic cube follows:

Top slice:

   8   24   10
  12    7   23
  22   11    9

Middle slice:

  15    1   26
  25   14    3
   2   27   13

Bottom slice:

  19   17    6
   5   21   16
  18    4   20

Note that 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.

1 Alternate definition
2 Bimagic cubes and beyond
3 See also
4 External links

[edit] 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.

[edit] Bimagic cubes and beyond

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.