A magic cube is a three-dimensional (or higher dimensional) analogue of a magic square. Specifically, an order n magic cube is an n x n x n array of numbers chosen from the set {1,2,...,n³} so that each number appears exactly once and the sum of the numbers in every row, column, file, and diagonal is the same, magic constant. The magic constant must therefore be the sum of all the numbers in the cube divided by the number of rows (columns or files). So, it is (1 + 2 + ... + n³)/n² = n(n³ +1)/2.

A perfect magic cube is such that each square parallel to a face of the cube is a magic square. Here is one:

If a magic cube is not perfect then it is called semi-perfect. A pandiagonal cube has the property that all of its broken space diagonals sum up to the magic constant. In...