A certain polyhedron has 6 vertices and 9 edges. Determine the number of faces on this polyhedron.

Descates (in 1639) has given us the relation between the 
vertices , faces and edges of any polyhedron  or any plane figure bounded by sraight
edges, plane polygonal faces and vertices.It is popularly known as Euler's formula(Euler
rediscovered it in 1751) or Euler's theorem and proved by Cahchy  also in
1811 :

The number of vertices+ number of faces = number of
edges +2. Or V+F = E+2. Substitute the given values , V=6 and E = 9  and we get: 6+F =
9+2 . So F = 9+2-6  = 5..