XXVI OBM

Consider the planar graph which vertices are the vertices of the triangles and edges are the sides of the triangles. Let $V$, $E$ and $F$ be the number of vertices, edges and faces of such graph and $d$ be the degree of each vertex. Note that $n=F-1$. Then $E=\frac{3F}{2}=\frac{dV}{2}⟺V=\frac{2E}{d}$ and $F=\frac{2E}{3}$. By Euler's theorem, $V-E+F=2⟺\frac{1}{d}+\frac{1}{3}=\frac{1}{2E}+\frac{1}{2}>\frac{1}{2}⟹\frac{1}{d}>\frac{1}{6}⟺d<6$. It's not hard to see that $d\ge 3$ and that each value for $d$ determines $V$, $E$ and $F$. Indeed $d=3⟹F=4$; $d=4⟹F=8$; $d=5⟹F=20$. The following examples show that the possible values for $n$ are indeed $3$, $7$ and $19$.