Magic cube
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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 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 Template:OEIS
- <math>M_3(n) = \frac{1}{2}n(n^3+1)<math>
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.
An 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.
See also
- Magic square
- Perfect magic cube
- Semiperfect magic cube
- Bimagic cube
- Trimagic cube
- Multimagic cube
- Magic tesseract
- Magic hypercube
- Magic cube classes
External link
- MathWorld: Magic Cube (http://mathworld.wolfram.com/MagicCube.html)
- Harvey Heinz: All about Magic Cubes (http://members.shaw.ca/hdhcubes/index.htm)
- Marián Trenkler: Magic p-dimensional cubes (http://kosice.upjs.sk/~trenkler/aa-cub-01.pdf)de:Magischer Würfel