How to change coins, M&M's, or chicken nuggets: The linear Diophantine problem of Frobenius

Abstract

Summary Let's imagine that we introduce a new coin system. Instead of using pennies, nickels, dimes, and quarters, let's say we agree on using 4-cent, 7-cent, 9-cent, and 34-cent coins. The reader might point out the following flaw of this new system: certain amounts cannot be exchanged, for example, 1, 2, or 5 cents. On the other hand, this deficiency makes our new coin system more interesting than the old one, because we can ask the question: “which amounts can be changed?” In the next section, we will prove that there are only finitely many integer amounts that cannot be exchanged using our new coin system. A natural question, first tackled by Ferdinand Georg Frobenius and James Joseph Sylvester in the 19 th century, is: “what is the largest amount that cannot be exchanged?” As mathematicians, we like to keep questions as general as possible, and so we ask: given coins of denominations a 1 , a 2 , …, a d , which are positive integers without any common factor, can you give a formula for the largest amount that cannot be exchanged using the coins a 1 , a 2 , …, a d ? This problem is known as the Frobenius coin-exchange problem . One of the appeals of this famous problem is that it can be stated in every-day language and in many disguises, as the title of these notes suggests.