
Here are three regular polyhedra, the tetrahedron, cube, and
octahedron:
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\begin{center}
\includegraphics[height=1.25in]{polyhedra}
\end{center}
\end{problem}
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Three variables characterize a polyhedron:
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\begin{itemize}
\item The number of edges, $e$.  Each straight line in the drawing of
a polyhedron is a edge.  For example, the tetrahedron has 6 edges and
the cube has 12.
\item The number of sides on each face, $n$.  The tetrahedron and
octahedron have triangular faces, so $n = 3$.  The cube has square
faces, so $n = 4$.
\item The number of edges that meet at each corner, $m$.  For the
tetrahedron and cube, $m = 3$.  For the octahedron, $m = 4$.
\end{itemize}

For every polyhedron, $e > 0$, $n \geq 3$, and $m \geq 3$.  Later,
we'll prove that these variables must also satisfy two equations:
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\begin{align*}
\frac{2e}{m} - e + \frac{2e}{n} & = 2 &
\frac{1}{m} + \frac{1}{n} & = \frac{1}{e} + \frac{1}{2} \\
\end{align*}
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Use case analysis to prove that the only solutions to these equations
are those given in the table below; in other words, there are only
five regular polyhedra.
