1988 AIME Problems/Problem 10

Problem

A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?

Solution

The number of segments joining the vertices of the polyhedron is . We must now subtract out those segments that lie along an edge or a face.

Since every vertex of the polyhedron lies on exactly one vertex of a square/hexagon/octagon, we have that .

Each vertex is formed by the intersection of 3 edges. Since every edge is counted twice, once at each of its endpoints, the number of edges is .

Each of the segments lying on a face of the polyhedron must be a diagonal of that face. Each square contributes diagonals, each hexagon , and each octagon . The number of diagonals is thus .

Subtracting, we get that the number of space diagonals is .