I like Wechsler's solution better than my planned one: A long regular N-sided right prism topped with short pyramids; only the rectangle faces of the prism, not the triangle faces on the pyramids, are stable resting positions. Nice and elegant. [I was planning on an (N-1)-sided pyramid with the exact right value for the height/diam ratio to make the land-on-base probability come out equal to the land-on-each-other-face probability...] re Adam Goucher: rhombic hexecontahedron is a polyhedron such that the projection of the centroid to any face lies outside that face. that's a nonconvex polyhedron. For a convex one, any face whose plane has minimum distance to the center of mass, is always a stable resting face, and it is never stable to rest on an edge or corner not a face. This all also is true for polyhedra of nonuniform internal composition.