= Solution
The <Euler formula for a sphere>, applied to the surface of a convex polyhedron, is \b[$V-E+F=2$]. In a regular polyhedron let $q\ge3$ be the common number of edges incident at a vertex. Counting vertex-edge and face-edge incidences, pentagonal faces give $qV=2E$ and $5F=2E$. Hence
$$
2=E\left(\frac2q-1+\frac25\right),
$$
whose right-hand side can be positive only if $q<10/3$. Thus $q=3$. Substitution then yields $2=E/15$, proving
$$
\boxed{E=30,\qquad V=20,\qquad F=12.}
$$
These are the counts of the <dodecahedron>. Here regular polyhedron is understood in the convex, ordinary-face sense used by the stated Euler formula; self-intersecting star constructions do not share these hypotheses.
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