Homework problems:

Due Jan 23:

• Find all inscribed squares in a regular n-gon. (Hint: try three cases separately: n odd, n a multiple of 4, and the rest).
• Find all inscribed squares in the figure here.

For the curve r(s) given in class:

• describe (and draw) the curve
• verify that |r'(s)| = 1
• find κ(s) and τ(s)
• find the osculating plane at time s
• show that the line through r(s) in direction N(s) hits the z-axis, with a right angle

• Show that if there is a path of length n from 0 to k, then there exist non-negative integers a, b with n = a + b and k = a - b. (Use the picture!)
• In the situation above, show that N_n(0 --> k) is "n choose a".
• Use the previous problem to prove the second equality in the Ballot Theorem.

For these exercises, the random walk operates in such a way that a step of +1 has probability p, and a step of -1 has probability q (with p + q = 1).

• Show that the ruin probabilities r_i satisfy the difference equation x_i = q x_i-1 + p x_i+1, and the boundary conditions r_-a = 1, r_b = 0.
• Show that two solutions to the difference equation are x_i = 1 and x_i = (q/p)ⁱ, so that the general solution is given by x_i = C + D (q/p)ⁱ. Then use the boundary conditions to find C and D, and find r(a,b) = r₀.
• Suppose that the game is favorable to you (p > q), but your opponent has unlimited financial resources (b --> infinity). (Eg. you wish to open a casino.) Show that then r(a) = lim_{b --> inifnity} r(a,b) = (q/p)^a. How much capital "a" would you need if you give yourself a hopefully inconspicuous edge of just p = 0.51 and you want to run a risk of at most P = 0.01 of going broke?

• Consider the function A_n --> (S_n - A_n) taking a permutation s to (1 2)s. Show that this function is a bijection.
• Express (1 2 3)(4 5)(1 6 7 8 9)(1 5) as a product of disjoint cycles.
• Compute a^kba^-k where a = (1 2 ... n) and b = (1 2).
• Show that (1 2) and (1 2 ... n) generate S_n.

Consider the 3x3 "8"-puzzle, with 8 tiles in a 3x3 grid.

• Explain how to express a configuration as a permutation in S₈.
• Describe four moves which give non-trivial permutations.
• Find the permutations corresponding to these four moves.
• What subgroup of S₈ do the moves generate? Which configurations of tiles are solvable? All of them? The "even" ones? Something else?

• Compute the bracket polynomial of the figure eight knot.
• Compute the X-polynomial and the Jones polynomial of the figure eight knot.

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