3.I.4E

State Euler's formula for a graph $\mathcal{G}$ with $F$ faces, $E$ edges and $V$ vertices on the surface of a sphere.

Suppose that every face in $\mathcal{G}$ has at least three edges, and that at least three edges meet at every vertex of $\mathcal{G}$. Let $F_{n}$ be the number of faces of $\mathcal{G}$ that have exactly $n$ edges $(n \geqslant 3)$, and let $V_{m}$ be the number of vertices at which exactly $m$ edges meet $(m \geqslant 3)$. By expressing $6 F-\sum_{n} n F_{n}$ in terms of the $V_{j}$, or otherwise, show that every convex polyhedron has at least four faces each of which is a triangle, a quadrilateral or a pentagon.

*Typos? Please submit corrections to this page on GitHub.*