Does there exist a rectangular parallelepiped (box) all of whose edges and face diagonals are of integral length, and the length of whose main diagonal is also an integer?Put another way, can you construct a rectangular solid where all three edge lengths, all three face diagonals, and the long space diagonal have integer values (or prove that such a box cannot be constructed)?
This deceptively simple-sounding problem is known as the Perfect Cuboid Problem (among other names). It is closely related to the problem of finding an Euler Brick, which has many solutions, the smallest of which is a box with edges 125, 244, and 267.
No perfect cuboids were found during an exhaustive search of the integers up to 100 billion (1010). This lends evidence (but is far from a proof) to the suspicion that no perfect cuboids exist. One near miss is the cuboid that has edges with lengths 44, 117, and 240, and diagonals with lengths 125, 244, and 267.