MT1100 Problem Sheet 4 Due: Thursday 29th October 2020 at 10:00. Please write your name clearly at the top of each page. Scan your work as a single-file PDF. Do not use too high a resolution to ensure the file size remains below 20 MB (the upper limit is 50MB). Then upload your PDF file onto Moodle, as per the instructions available from the General Information box on Moodle. You are encouraged to leave your handwritten comments to the marker on the first or last page of your homework submission. This problem sheet is all about the 'counting twice' technique from the lectures. 1. Suppose we have a polyhedron with f faces, e edges, and v vertices. Suppose that each face of our polyhedron has n sides. Consider the collection of arrows from the middle of each edge to the centre of each face adjacent to that edge. By carefully counting these arrows in two ways (see the great rhombicuboctahedron example in the lectures), show that fn = 2e. [This fact was used in the proof of Theorem 4.2.] 2. In a truncated icosidodecahedron, every vertex is adjacent to one square, one regular hexagon and one regular decagon (10 sides). Suppose there are x4 square faces, x6 hexagonal faces and x10 decagonal faces. (i) Give a brief reason why f=x4+x6+ @10. (ii) Each of the following equations can be justified by counting a collection of arrows. For each equation, describe a suitable collection of arrows. (No need to count them.) 1 (4x4+ 6x6 + 10x10), e = 1 v = (4x4 +6x6 + 10x10), 3 v = 4x4, v= 6x6, v = 10x10. (iii) Using Euler's formula and the equations above, show that the truncated icosidodecahedron has 62 faces.