A Problem and a Conjecture

The dialogue takes place in an imaginary classroom. The class gets interested in a PROBLEM: is there a relation between the number of vertices V, the number of edges E and the number of faces F of polyhedra – particularly of regular polyhedra – analogous to the trivial relation between the number of vertices and edges of polygons, namely, that there are as many edges as vertices: V = E? This latter relation enables us to classify polygons according to the number of edges (or vertices): triangles, quadrangles, pentagons, etc. An analogous relation would help to classify polyhedra.

After much trial and error they notice that for all regular polyhedra V-E+F = 2. Somebody guesses that this may apply for any polyhedron whatsoever. Others try to falsify this conjecture, try to test it in many different ways – it holds good. The results corroborate the conjecture, and suggest that it could be proved. It is at this point – after the stages problem and conjecture – that we enter the classroom. The teacher is just going to offer a proof.

A Proof

Teacher: In our last lesson we arrived at a conjecture concerning polyhedra, namely, that for all polyhedra V-E+F = 2, where V is the number of vertices, E the number of edges and F the number of faces. We tested it by various methods.