Magic cube

This article is about the mathematical concept. For the flashbulb cartridges, see Magicube. For the puzzle, see Rubik's Cube.

An example of a 3 × 3 × 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 3-dimensional equivalent of a magic square, that is, a collection of integers 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, the so-called magic constant of the cube, denoted M₃(n). If a magic cube consists of the numbers 1, 2, ..., n³, then it has magic constant (sequence A027441 in the OEIS)

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

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

Main article: Multimagic cube

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 faces Magic cube with the magic square of Dürer, and 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

Andrews, William Symes (1960), "Chapter II: Magic Cubes", Magic Squares and Cubes (PDF) (2nd ed.), New York: Dover Publications, pp. 64–88, doi:10.2307/3603128, ISBN 9780486206585, JSTOR 3603128, MR 0114763, OCLC 1136401, S2CID 121770908, Zbl 1003.05500

External links

• Harvey Heinz, All about Magic Cubes
• Marian Trenkler, Magic p-dimensional cubes
• Marian Trenkler, An algorithm for making magic cubes
• Marian Trenkler, On additive and multiplicative magic cubes
• Ali Skalli's magic squares and magic cubes