What is "magic cube"

Etymology 1 n. By analogy with magic square, an n-by-n-by-n arrangement of n³ numbers such that the numbers in each row, in all columns (horizontal and vertical) and in all main diagonals (of which there are 6*n+4) each have the same sum. Etymology 2

n. Rubik's cube

In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n × n × n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic constant of the cube, denoted M(n). It can be shown that if a magic cube consists of the numbers 1, 2, ..., n, then it has magic constant

$$M_3(n) = \frac{n(n^3+1)}{2}.$$

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 cube.