OME 2006 Problem 2

The dimensions of a wooden octahedron are natural numbers. We painted all its surface (the six faces), cut it by planes parallel to the cubed faces of an edge unit and observed that exactly half of the cubes did not have any painted faces. Prove that the number of octahedra with such property is finite.

(It may be useful to keep in mind that $\sqrt[3]{\frac{1}{2}}=1,79 ... <1,8$ ).