smc

Question

A convex polyhedron Q has vertices V_1,V_2,,V_n, and 100 edges. The polyhedron is cut by planes P_1,P_2,,P_n in such a way that plane P_k cuts only those edges that meet at vertex V_k. In addition, no two planes intersect inside or on Q. The cuts produce n pyramids and a new polyhedron R. How many edges does R have?

Accepted Answer

Each edge of Q is cut by two planes, so R has 200 vertices. Three edges of R meet at each vertex, so R has 12 3 200 = 300 edges.