Question 94
Planes through the points with integer coordinates in the three dimensional Euclidean space partition the space into unit cubes. Find all triples $(a, b, c)$, $a \le b \le c$, of positive integers such that the cubes can be colored in $abc$ colors in such a way that every parallelepiped of dimensions $a \times b \times c$, integer vertices and faces parallel to the coordinate planes does not contain cubes of the same color.