The Perfect Box as its known, is a box whose edges are all integers, whose face diagonals are integers and it's body diagonal is an integer also. No one has yet found a Perfect Box, but no one has yet proven that it can't exist.
The question of the box's existence has been around since about 1719. The smallest box with all integer face diagonals, but not an integer body diagonal was discovered around then, it has measurements 44 x 117 x 240. In all that time no one has come up with a proof as to why it can't exist. But some facts are known about it, should it exist:
- It must have 2 even sides and 1 odd side.
- The 2 even sides must also be divisible by 4 and at least one of them by 16.
- Exactly 2 of the sides must be divisible by 3 and at least one of those by 9.
- At least one of the sides must be divisible by 5 and at least one must be divisible by 11.