A polyhedron has 20 triangular faces. Determine its number of vertices and edges.
Let's perform a double counting between the faces and the edges, using the notations $S, A, F$ from the previous exercise. Let $N$ be the number of pairs $(f, a)$ where $f$ is a face and $a$ is an edge that is a side of $f$. On one side, if we fix $f$, there are three edges that work because all the faces of the polyhedron are triangular, and thus $N=3 F$. On the other side, if we fix $a$, there are 2 faces that work because an edge is the side of exactly two triangles. We therefore get $N=2 A$, and thus

$$
A=\frac{3}{2} F=30
$$

Now, the number $S$ of vertices is obtained by Euler's formula

$$
S=A-F+2=12
$$