We consider a polyhedron with 20 triangular faces. How many edges and vertices does it have?
Let $S, A, F$ be the numbers of vertices, edges, faces respectively, so that $F=20$. We will proceed by double counting. We count the pairs $(a, f)$ where $a$ is an edge and $f$ is a face of which $a$ is a side. Each edge belongs to two faces, so this number is $2 A$. But since the faces are triangular, each face has three sides, so this number is also $3 F=60$. We deduce that $A=30$.

Now that we have $A$ and $F$, we think of Euler's formula (from the previous exercise) to find $S$. We have

$$
S=A+2-F=30+2-20
$$

so $S=12$.