Let $S$ be a closed surface, equipped with a triangulation. Define the Euler characteristic $\chi(S)$ of $S$. How does $\chi(S)$ depend on the triangulation?

Let $V, E$ and $F$ denote the number of vertices, edges and faces of the triangulation. Show that $2 E=3 F$.

Suppose now the triangulation is tidy, meaning that it has the property that no two vertices are joined by more than one edge. Deduce that $V$ satisfies

$$V \geqslant \frac{7+\sqrt{49-24 \chi(S)}}{2} .$$

Hence compute the minimal number of vertices of a tidy triangulation of the real projective plane. [Hint: it may be helpful to consider the icosahedron as a triangulation of the sphere $\left.S^{2} .\right]$