## Task 11/87

How many adhesive flaps are (at least) required if a simply connected planar net of a convex polyhedron with $f$ faces and 9 vertices is to be folded into the solid?
The network consists of $f$ convex polygons connected by $(f-1)$ edges. (Proof of this claim: The first two polygons are connected by exactly one edge; each additional polygon shares exactly one edge with one of the already existing polygons.) According to Euler's polyhedral formula, for the number $k$ of edges, $k=e+f-2$. For the number $n$ of necessary folds, it holds that

$$
n=k-(f-1)=k-f+1=e+f-2-f+1=e-1
$$

It is surprising that $n$ depends only on the number of vertices $e$, not on the number of faces $f$ or the number of edges $k$. Specifically, for the tetrahedron $n=3$, for the octahedron $n=5$, and for the hexahedron $n=7$.