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Making Change Efficiently

Suppose U.S. coinage were to be drastically reformed so that coins of only one denomination beyond the penny were minted. What value, k, of this coin would allow change to be made most efficiently? More generally consider a system of currency where the smallest paper note has value d¢. Making change then means being able to hand over, in 1¢ and k¢ coins, any value from 1¢ to (d − 1)¢. What value for k makes this system most efficient? Even this two-denomination problem is not so simple. The difficulty lies in the fact that the solution requires a minimum of a maximum of a minimum. In this paper, we describe how to lead a student to solve completely the two-denomination problem for any currency, while developing the skills needed to tackle the much harder three-denomination problem. While calculus is not needed for the two-denomination problem, it proves useful for the extension to three denominations. In any case, someone tackling this project needs the same level of comfort with functions and graphs as a calculus student. It is hoped such a student will find these exercises enjoyable as well as enlightening.