{\bf 4.}  (10 points altogether)

(a) (2 points) Define a {\bf Platonic soild}


(b) (2 points) Let $a$ be the number of edges meeting each vertex, and let $b$ be the number of edges surrounding each face.
Express $V$ (the number of vertices) and $F$ (the number of faces) in terms of $E$ (the number of edges), and $a$ and $b$.

(c) (2 points) Find an expressions for  $F$, in terms of $a$ and $b$.

(d) (4 points) Obviously both $a$ and $b$ must be at least $3$, and $F$ (and hence $V$ and $E$) must be positive. It is easy to see (you don't have to do it)
that $a,b$ must be both between $3$ and $5$, leaving $9$ potential scenarios. Find those values of $a$ and $b$ that make sense,
and thereby prove that there are exactly $5$ Platonic solids. For each of them, find $F$  (the number of faces) and give the name of the corresponding Platonic solid.