1st Irish 1988 Problem 1 One face of a pyramid with square base and all edges 2 is glued to a face of a regular tetrahedron with edge length 2 to form a polyhedron. What is the total edge length of the polyhedron?
18 (not 22). Solution The polyhedron has two less edges than you might expect. We start with 8 + 6 = 14 edges. We lose 3 when the two faces are glued. But we lose another 2 because in two cases adjacent faces are parallel and merge to form a rhombus. (That is easily seen if you note that the midpoints of the edges of a tetrahedron form the vertices of a regular octahedron, which can be regarded as two square pyramids joined at the base. ) Thus the total edge length is 9 x 2 = 18. The polyhedron is a triangular prism with one square side face and two rhombic side faces (a total of 5 faces). It has 6 vertices, and 9 edges (note V + F = E + 2). 1st Irish 1988 © John Scholes jscholes@kalva.demon.co.uk 11 June 2003 Last updated/corrected 11 Jun 03