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**Example text**

G acts on X in the obvious way. A pair of colourings is indistinguishable precisely if they are in the same orbit. By the Burnside formula, the number of distinguishable colourings is given by 1 | FixG (σ)|. number of orbits = 6 σ∈G The fixed sets of elements of the various cycle types in G are as follows. Identity element ι: FixG (ι) = X, | FixG (ι)| = 64. , σ = (A B C), (A C B)): these give rotations and can only fix a colouring that has all sides the same colour, hence | FixG (σ)| = 4. , σ = (A B), (A C), (B C)): each of these gives a reflection in a line through a vertex and the midpoint of the opposite edge.

1 1-19. Investigate the continued fraction expansions of 6 and √ . Determine as many con6 vergents as you can. 1-20. [Challenge question] Try to determine the first 10 terms in the continued fraction expansion of e using the series expansion 1 1 1 e = 1 + + + + ··· . 1! 2! 3! 1-21. Find all the solutions of each of the following Diophantine equations: (a) 64x + 108y = 4, (b) 64x + 108y = 2, (c) 64x + 108y = 12. 1-22. Let n be a positive integer. a) Prove the identities n+ n2 + 1 = 2n + ( n2 + 1 − n) = 2n + 1 √ .

A) Show that τ is an arithmetic function. b) Suppose that n = pr11 pr22 · · · prt t is the prime power factorization of n, where 2 p1 < p2 < · · · < pt and rj > 0. Show that τ (pr11 pr22 · · · prt t ) = (r1 + 1)(r2 + 1) · · · (rt + 1). c) Is τ multiplicative? d) Show that η ∗ η = τ . 3-2. 1 are multiplicative. 3-3. For each r ∈ N0 define the arithmetic function [r] : Z+ −→ R by [r](n) = nr . In particular, [0] = η and [1] = id. a) Show that [r] is multiplicative. b) If r > 0, show that σr = [r] ∗ η.