KöMaL 2009 A490

The two base faces of a prism are equilateral triangles and the other three faces are squares. At the beginning it stands on its triangle face. Then it is rolled around one of its edges that lays on the table. After some rollings, the prism will stand in the original position. Prove that then all vertices will be in the same position as at the beginning. (Suggested by L. Csirmaz, Budapest)