Magic Cube

A magic cube is an version of a magic square in which the rows, columns, pillars, and four space diagonals each sum to a single number known as the cube's magic constant. Magic cubes are most commonly assumed to be "normal," i.e., to have elements that are the consecutive integers 1, 2, ..., . However, this requirement is dropped (as it must be) in the consideration of so-called multimagic cubes.

If it exists, a normal magic cube has magic constant

For , 2, ..., the magic constants are given by 1, 9, 42, 130, 315, 651, ... (OEIS A027441).

If only rows, columns, pillars, and space diagonals sum to , a magic cube is called a semiperfect magic cube, or sometimes an Andrews cube (Gardner 1988, p. 219). If, in addition, the diagonals of each orthogonal slice sum to , then the magic cube is called a perfect magic cube. If a perfect or semiperfect magic cube is magic not only along the main space diagonals, but also on the broken space diagonals, it is known as a pandiagonal magic cube.

There is a trivial perfect magic cube of order one, but no perfect cubes exist for orders 2-4. While normal perfect magic cubes of orders 7 and 9 have been known since the late 1800s, it was long not known if perfect magic cubes of orders 5 or 6 could exist. A perfect magic cube was subsequently discovered by C. Boyer and W. Trump on Nov. 14 2003.

A perfect or semiperfect magic cube that yields another magic cube of the same type when its elements are squared is known as a bimagic cube. Similarly, a magic cube that remains magic when its elements are both squared and cubed is known as a trimagic cube.

The smallest known multiplication magic cube is with largest term 416 and magic product 8648640, or (Boyer 2006).