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.
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).