The authors describe their method for counting planar Feynman diagrams of quantum field theory. Mathematicians who are not familiar with the current jargon of quantum field theory are likely to find this paper rather difficult to follow. In order to demonstrate the usefulness of their method, the authors calculate the ground state energy of a system of coupled anharmonic oscillators with a \(\phi^4\) interaction in the planar approximation. They obtain the value \(0.58993g^{1/3}\) in the limit of large \(g\), where \(g\) is the coupling constant. By comparing this with the value \(0.66799g^{1/3}\) which is described as exact, the authors conclude that the planar approximation is at most 12\% wrong!