For every Positive Integer , there exists a Square in the plane with exactly Lattice Points in its interior. This was extended by Schinzel and Kulikowski to all plane figures of a given shape. The generalization of the Square in 2-D to the Cube in 3-D was also proved by Browkin.
See also Cube, Schinzel's Theorem, Square
Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 121-125, 1973.