RIOPLATENSE 1995 Problem 3

Given a regular tetrahedron with edge $a$ , its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered: $\bullet$ those that contain the faces of the tetrahedron, and $\bullet$ each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above. Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.