## Background

When using a coin sorter like this, odds are you want to minimize the number of coins you’re dispensing for each customer, lest you have to press levers more times than are necessary. Fortunately, computer science has given cashiers everywhere ways to minimize numbers of coins due: greedy algorithms.

According to the National Institute of Standards and Technology (NIST), a greedy algorithm is one “that always takes the best immediate, or local, solution while finding an answer. Greedy algorithms find the overall, or globally, optimal solution for some optimization problems, but may find less-than-optimal solutions for some instances of other problems.”

What’s all that mean? Well, suppose that a cashier owes a customer some change and on that cashier’s belt are levers that dispense pound coins, 50p’s, 20p’s, 10p’s, 2p’s and 1p’s. Solving this “problem” requires one or more presses of one or more levers. Think of a “greedy” cashier as one who wants to take, with each press, the biggest bite out of this problem as possible. For instance, if some customer is owed 41p, the biggest first (i.e., best immediate, or local) bite that can be taken is 20p. (That bite is “best” inasmuch as it gets us closer to 0¢ faster than any other coin would.) Note that a bite of this size would whittle what was a 41p problem down to a 16p problem, since 41 – 25 = 16. That is, the remainder is a similar but smaller problem. Needless to say, another 20p bite would be too big (assuming the cashier prefers not to lose money), and so our greedy cashier would move on to a bite of size 10p, leaving him or her with a 6p problem. At that point, greed calls for one 5p bite followed by one 1p bite, at which point the problem is solved. The customer receives one 20p, one 10p, one 5p, and one 1p: four coins in total.

It turns out that this greedy approach (i.e., algorithm) is not only locally optimal but also globally so for America’s currency (and also the European Union’s). That is, so long as a cashier has enough of each coin, this largest-to-smallest approach will yield the fewest coins possible. How few? Well, you tell us!