Euler's formula states that for a map on the sphere,

, where

is the number of vertices,

is the number of faces, and

is the number of edges. This Demonstration shows a map in the plane (so the exterior face counts as a face). The formula is proved by deleting edges lying in a cycle (which causes

and

to each decrease by one) until there are no cycles left. Then one has a tree, and one can delete vertices of degree one and the edges connected to them until only a point is left. Each such move decreases

and

by one. So all the moves leave

unchanged, but at the end

and

are each 1 and

is 0, so

* *must have been 2 at the start.