In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an 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, the so-called magic constant 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 (sequence A027441 in the OEIS)

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

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 magic cube.