Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
Is it possible to make an irregular polyhedron using only polygons of, say, six, seven and eight sides? The answer (rather surprisingly) is 'no', but how do we prove a statement like this?
How many different colours of paint would be needed to paint these
pictures by numbers?
Some simple ideas about graph theory with a discussion of a proof
of Euler's formula relating the numbers of vertces, edges and faces
of a graph.