Lovász and Plummer conjectured, in the mid 1970's, that every bridgeless cubic graph has exponentially many perfect matchings. In this work we show that every cubic planar graph G whose geometric dual graph is a stack triangulation (planar 3-tree) has at least 3φ|V(G)|/72 distinct perfect matchings, where φ is the golden ratio. Our work builds on a novel approach relating Lovász and Plummer's claim and the number of so called groundstates of the widely studied Ising model. © 2011 Elsevier B.V.