Problem:
A convex polyhedron has vertices, edges, and faces, of which are triangular, and of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does have?
Solution:
The total number of diagonals and edges is , and there are 24 face diagonals, so has space diagonals. One such polyhedron can be obtained by gluing two dodecahedral pyramids onto the sided faces of a dodecahedral prism.
Note that one can determine that there are edges as follows. The triangles contribute edges, and the quadrilaterals contribute edges. Because each edge is in two faces, there are edges.
The problems on this page are the property of the MAA's American Mathematics Competitions