魔術正方體

此條目介紹的是the mathematical concept。关于the flashbulb cartridges，请见「Magicube」。关于the puzzle，请见「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 简易魔术正方体.

以数学方面論述，魔術正方體 在維度上相當於幻方，也就是以n × n 方式排列的方體，在每個線段交點填上任意不重複的整数，並使得每行、每列及每個柱上數字的和相同。而此立方體的幻方常數表示為M₃(n). It can be shown that if a magic cube consists of the numbers 1, 2, ..., n³, then it has magic constant （OEIS數列A027441）

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

If, in addition, the numbers on every 截面 (幾何) 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.

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.

Multimagic cubes

如同魔術方塊一般， 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.

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 class
• Magic series
• Asymptotic magic hyper-tesseract
• Nasik magic hypercube
• John R. Hendricks

External links

• 埃里克·韦斯坦因. Magic Cube. MathWorld. 
• 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