A magic cube is a three-dimensional array of whole numbers, in which each row, column, and body diagonal (corner to corner, through the center) adds up to the same total. A *perfect* magic cube is one in which the diagonals of each vertical or horizontal slice through the cube also sum to the same value.

Suppose each row, column, and diagonal contains *N* numbers. The magic cube would then consist of *N*^{3} numbers, usually the integers from 1 to *N*^{3}. In this case, the sum (or magic constant) is 1/2(*N*^{4} + *N*).

So, for *N* = 5, there would be 125 numbers in the array and the sum of each row, column, and diagonal would be 315. The number *N* is called the *order* of a magic cube.

There are no perfect magic cubes of order 2, order 3, or order 4.

For a long time, no one was sure whether there existed a perfect magic cube of order 5.

In 1972, Richard Schroeppel proved that, if a perfect magic square of order 5 exists, its center number must be 63.

Last November, Walter Trump and Christian Boyer finally found a perfect magic cube of order 5.

The first known perfect magic cube of order 5. |

As expected, the cube’s magic constant is 315, and its central value is 63. Moreover, there are 109 ways to get the magic sum: 25 rows, 25 columns, 25 pillars, 4 body diagonals, and 30 face diagonals.

Boyer and Trump ran five computers for several weeks to come up with their solution, checking a large number of “auxiliary” cubes of order 3 to find the right combination.

Interestingly, just 2 months earlier, Trump had found the first perfect cube of order 6.

These recent discoveries leave the existence of perfect magic cubes of order 10 as the next unresolved question.