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.

In 数学, a magic cube is the 維度 equivalent of a 幻方, that is, a number of 整数 arranged in a 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 to the same number, the so-called 幻方常數 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 （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