In the US, commonly used coins are worth 1, 5, 10 and 25 cents. Have you ever wondered whether the current system of coin denominations is efficient? Well, turns out it’s not. Actually, as Jeffrey Shallit (2003) shows in his short note, the US system could be improved by 17% just by changing the dime to an 18-cent coin!
The author’s aim is to show which denominations are optimal (in that they minimize the number of coins needed in an average change-making) for systems made of different numbers of coins. His results are shown in the figure below. For a 4-coin system, the most efficient combination would be any of the two sets (1,5,18,29) and (1,5,18,25), which would bring about a decrease in the cost of change-making by 17%. If we wanted to add one coin and not take away any of the beloved coins that are currently used, a 32-cent coin is the best option. Well, politicians – the ball’s in your court!
Reference: Shallit, J., 2003. What this country needs is an 18-cent piece, Math. Intelligencer 25 (2), 20-23. Available here.