= Solution
A geodesic triangulation of a closed smooth surface decomposes it into finitely many triangular discs whose edges are geodesic arcs and whose intersections are common faces. Its Euler number is
$$
\chi=V-E+F.
$$
The <Gauss-Bonnet theorem> for a closed surface states
$$
\int_S K\,dA=2\pi\chi(S).
$$
For any triangulation of a closed surface, each triangular face has three edges and every edge belongs to two faces, so $3F=2E$. The sum of all vertex valencies is $2E$. Hence the average valency is
$$
\overline d=\frac{2E}{V}.
$$
The <torus> has <Euler characteristic> $\chi=V-E+F=0$. Using $F=2E/3$ gives $V-E/3=0$, so $E=3V$ and
$$
\boxed{\overline d=6}.
$$
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