A perfect cuboid is a cuboid having integer side lengths, integer face diagonals
and an integer space diagonal
The problem of finding such a cuboid is also called the brick problem, diagonals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem.
No perfect cuboids are known despite an exhaustive search for all "odd sides" up to 100 billion (10^11) (Weisstein, Eric W., "Perfect Cuboid" from MathWorld).
Solving the perfect cuboid problem is equivalent to solving the Diophantine equations
A solution with integer space diagonal
and two out of three face diagonals is
, which was known to Euler. A solution giving integer space and face diagonals with only a single nonintegral polyhedron edge