This problem involves the question of computing change for a given coin system. A coin system is defined to be a sequence of coin values v1 < v2 < . . . < vn, such that v1 = 1. For example, in the U.S. coin system we have six coins with values h1, 5, 10, 25, 50, 100i. The question is what is the best way to make change for a given integer amount A.

(a) Let c ≥ 2 be an integer constant. Suppose that you have a coin system where there are n types of coins of integer values v1 < v2 < . . . < vn, such that v1 = 1 and, for 1 < i ≤ n, vi = c · vi-1. (For example, for c = 3 and n = 4, an example would be h1, 3, 9, 27i.) Describe an algorithm which given n, c, and an initial amount A, outputs an n-element vector that indicates the minimum number of coins in this system that sums up to this amount. (Hint: Use a greedy approach.)

(b) Given an initial amount A ≥ 0, let hm1, . . . ,mni be the number of coins output by your algorithm. Prove that the algorithm is correct. In particular, prove the following:

(i) For 1 ≤ i ≤ n, mi ≥ 0

(ii) Pn

i=1mi · vi = A

(iii) The number of coins used is as small as possible

Prove that your algorithm is optimal (in the sense that of generating the minimum number of coins) for any such currency system.

(c) Give an example of a coin system (either occurring in history, or one of your own invention) for which the greedy algorithm may fail to produce the minimum number of coins for some amount.

Your coin system must have a 1-cent coin.

Hint on problem 3: Suppose that the input is n=4, c=2, A=53 (meaning four coins, growing exponentially in 2, that is, {1, 2, 4, 8}, and 53 cents as the amount). The output would be (1,0,1,6), meaning:

The input consists of n, c, and the amount A. The output is a list of the numbers of coins of each denomination that sums to A.