In this paper we deal with trivalent planar maps in which the boundary of each country (or “face“) is a simple closed curve. One vertex is distinguished as the root and its three incident edges are distinguished as the first, second, and third major edges. We determine the average number of Hamiltonian polygons, passing through the first and second major edges, in such a “rooted map” of 2n vertices. Next we consider the corresponding problem for 3-connected rooted maps. In this case we obtain a functional equation from which the average can be computed for small values of n.