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A selection of equilateral triangles, squares and regular pentagons, all with edges of the same length, is available to construct a semi-regular polyhedron with four faces meeting at each vertex. (i) Prove that, irrespective of the number of square faces, the number of triangular faces in any such polyhedron must exceed the number of pentagonal faces by 8. (ii) Name the one such polyhedron that can be made using all three types of polygonal faces. Given that it has 12 pentagonal faces, deduce the numbers of triangular and square faces that it has. (iii) Explain why it is not possible to have a semi-regular polyhedron in which exactly three faces, namely a triangle, a square and a pentagon, meet at each vertex. (It is not sufficient merely to observe that such a polyhedron is not in the list on page 35 of the unit.) A selection of equilateral triangles, squares and regular pentagons, all with edges of the same length, is available to construct a semi-regular polyhedron with four faces meeting at each vertex. (i) Prove that, irrespective of the number of square faces, the number of triangular faces in any such polyhedron must exceed the number of pentagonal faces by 8. (ii) Name the one such polyhedron that can be made using all three types of polygonal faces. Given that it has 12 pentagonal faces, deduce the numbers of triangular and square faces that it has. (iii) Explain why it is not possible to have a semi-regular polyhedron in which exactly three faces, namely a triangle, a square and a pentagon, meet at each vertex. (It is not sufficient merely to observe that such a polyhedron is not in the list on page 35 of the unit.)