Problem:
Euler's formula states that for a convex polyhedron with vertices, edges, and faces, . A particular convex polyhedron has faces, each of which is either a triangle or a pentagon. At each of its vertices, triangular faces and pentagonal faces meet. What is the value of
Solution:
Since , and it follows that
Since faces meet at each vertex, there are edges that meet at each vertex. Hence , from which and
Each triangular face has three vertices, so the product counts each triangular face times. Thus the total number of triangular faces is . Similarly, the total number of pentagonal faces is . Because every face is a triangle or a pentagon,
Combining and we have
from which
The only non-negative integer solution of this equation is . From we find , so .
The problems on this page are the property of the MAA's American Mathematics Competitions