Professor Halfbrain has spent his entire weekend by placing colored dots on the surface of a huge wooden cube. His objective was to find large groups of dots that form the vertices of a regular polygon (= a convex polygon whose angles all are the same and whose edges all have the same length).
The professor managed to prove the following two deep theorems.
Professor Halfbrain's first theorem: There exist three points on the surface of a cube that form the vertices of a regular polygon and that do not all lie on the same face.
Professor Halfbrain's second theorem: There is no group of 37 points on the surface of a cube that form the vertices of a regular polygon and that do not all lie on the same face.
This puzzle asks you to improve the two theorems of professor Halfbrain and to make them even deeper. Find an integer $x$, so that "three points" in the first theorem may be replaced by "$x$ points", and so that "37 points" in the second theorem may be replaced by "$x+1$ points" (again yielding true statements, of course).