A regular dodecahedron is a convex polyhedra that its faces are regular pentagons. The regular dodecahedron has twenty vertices and there are three edges connected to each vertex. Suppose that we have marked ten vertices of the regular dodecahedron. a) prove that we can rotate the dodecahedron in such a way that at most four marked vertices go to a place that there was a marked vertex before. b) prove that the number four in previous part can't be replaced with three. proposed by Kasra Alishahi