Magic cube

Magic Cubes are a three-dimensional variant of magic squares. A large cube is divided by cuts in many smaller cubes. The aim of this Jazzy game is now to associate each of these cubes different numbers so that the sum of each edge parallels takes the same constant value X. In addition, the four space diagonals must be the same amount X ​​.

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

Example of a magic cube of order 3:

2 --------- 15-------- 25      / | / | / |     24 ------- 7 - ------- 11 |    / | | / | | / | |   16 - 20 ----- - ----- 6 | |   | | | | | | | | |   | | 18 ---- - -19 ---- - -5   | | / | | | / | | | / |   | 1 - ----- ----- -14 -27 |   | / | | | / | | | / | |   23 - ----- 9 - - ----- 10 | |   | | | | | | | | |   | | 22 ---- - ----- -8 - -12   | | / | | / | | /   | 17 ------ ------ -4 -21   | / | / | /   3 --------- 13-------- 26 magic sum X = 42

A magic cube of order 6 can be prepared by recursion from 27 magic cubes of order 2 by the smaller magic cube within one constant ( n-1) · 27 adds, where n represents in each sub-cubes a different number. The spatial distribution of these constants must be of the distribution of numbers in a magic cube of order 2 correspond to (ie as n = 16 for the front left cube up, 14 for the middle cube, ...).

From a perfect magic cube is when the face diagonals of these magic squares is the sum X. Such a magic cube of order 7 has shown A. Frost already in 1866, the English missionary. Perfect magic cubes of orders 2, 3 and 4 can not exist, which is already proven. But if there is this for orders 5 and 6, was unknown for 150 years. In September 2003, but was able to provide a perfect magic cube 6th order of the mathematician and teacher Walter Trump from Germany. Within two months now and with the help of Christian Boyer from France, a 5th-order cube was found. These can marvel at MathWorld.

With this discovery, the field of magic squares and cubes is fairly well known, the " Cubists " currently apply to multi magic cubes, where not only the numbers but also their squares magic squares and cubes arise.

The fee for a three -dimensional magic cube is calculated using the following formula:

Magic hypercube

There are magic hypercube having a dimension 3.

One can calculate the sum for a p -dimensional magic hypercube with the following formula: