A "Perfect cuboid" (also known as a Perfect box) is a cuboid where the lengths of the edges, face diagonals and body diagonals are all integers. The following are some interesting facts about the side lengths of a Perfect cuboid:

Two sides must be even and one side must be odd.
One side must be divisible by 4 and one other must be divisible by 16.
One side must be divisible by 3 and one other must be divisible by 9.
One side must be divisible by 5.
One side must be divisible by 11.

As of March 14, 2005, no example of a Perfect cuboid has been found and no one has proved that it does not exist. Exhaustive computer searches have shown that the smallest side of a Perfect box is at least 4.3 billion.