Magic cube

Magic cubes are a three-dimensional variant of magic squares . A large cube is divided into many smaller cubes by cuts. The aim of this puzzle is now to assign different numbers to each of these cubes in such a way that the sum of each parallel line has the same constant value X. In addition, the four room diagonals must also form the same sum X.

Orders

Example of a 3rd order magic dice the magic total is 42

One speaks of a magic cube of the nth order if it has the edge length n . By making appropriate cuts parallel to the edges, you can create 3n different magic squares .

A magic cube of order 6 can be produced by recursion from 27 magic cubes of order 2 by adding a constant (n − 1) · 27 within each of the smaller magic cubes , where n represents a different number in each sub-cube. The spatial distribution of these constants must correspond to the distribution of the numbers in a magic cube of order 2 (i.e. about n = 16 for the cube in the front left upper corner, 14 for the middle cube, ...).

One speaks of a perfect magic cube if the diagonals of these magic squares also form the sum X. The English missionary A. Frost showed such a magic cube of the 7th order as early as 1866. There cannot be perfect magic cubes of orders 2, 3 and 4, this has already been proven. But whether these exist for orders 5 and 6 was unknown for 150 years. In September 2003, however, the mathematician and teacher Walter Trump from Germany was able to present a perfect 6th order magic cube. Within two months and with the help of Christian Boyer from France, a 5th order cube was found. These can be admired at MathWorld .

With this discovery, the field of magic squares and cubes is quite well explored, the Cubists are currently turning to multimagic cubes, where not only the numbers but also their square numbers result in magic squares and cubes.

The sum for a 3-dimensional magic cube is calculated using the following formula:

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

Magical hypercubes

There are also magical hypercubes that have a dimension greater than 3.

The sum for a p-dimensional magic hypercube can be calculated using the following formula:

${\displaystyle M_{p}(n)={\frac {n\,(n^{p}+1)}{2}}}$

Individual evidence

1. ↑ Magic Cube on MathWorld