pocket change

You’re paying for a purchase with cash and the amount is not a whole number like $1 but a decimal number like $.01. How can you minimize the number of coins in your pocket after the transaction?

If you hand over $1 you’ll get back three quarters, two dimes, and four pennies, for a score of +9 coins. But if you hand over a penny ($.01) you won’t get any new coins back and you will dispose of one pre-existing coin, for a score of -1 coin. (*Assumptions listed below).

Low scores are better than high in this game.

The more change you start with, the more coins you have to choose from in forming your payment, the more optimal coin combination you can pick, and the lower your final score. And vice versa: the less change you start with, the higher your final score.

Let’s say you start off with a $infinity bill and no coins. On your first transaction your score will be high. As time goes on you will accumulate coins. Once you accumulate enough coins your scores will be low. As time goes on your count will reach a steady state and stop increasing.

Let’s say you have a change jar on your bedside table. When putting on your pajamas you put your pocket change in the jar. The next morning when you get dressed you do not return that change to your pocket, so that you always leave home with no coins. You will get the highest possible score over time.

This is because you will usually accumulate the maximum number of coins before hitting the steady state where you have the optimal number. Your score for each transaction will be high until you hit the point where you have so many coins that you can usually get a low score. The number of coins in your change jar will be the sum of those day scores.

Let’s say you have no change jar on your bedside table. Each day you leave home with the exact same change as the day before. You will end up with the lowest possible score over time. This is because there will only be one day (your very first day, the one in which you leave home with no change) when you have high scores. After that day you will have the optimal amount in your pocket and your scores will not increase.

*Assumptions: the clerk always gives the least amount of coins in return, for example a nickel instead of five pennies; ignore whole dollars for the sake of simplicity (but if you can’t do that, treat bills as just another form of coin, like a quarter, dime or nickel); American currency, because I’m American; the clerk insists on being paid the exact amount and won’t let you slide on a penny; a quarter = .25, a dime = $.10, a nickel = $.05, no other coins to be discussed.