This is the first in a series of recreational math problems that I will be asking and then answering (to the best of my ability) in this space. Hello? Is anyone still reading this?
I have been doing recreational math for thirty-some years, but not so much lately. I have had several articles published in Journal of Recreational Mathematics (which seems to have suspended publication several years ago). But I still enjoy posing and then solving math problems of the type I present below.
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I hate to carry pocket change. [This is not a complaint — this is the setup to the problem.] I hate it so much that I want to minimize the number of coins I carry when I have to make a cash transaction. If I want to carry the fewest possible number of coins, defined as the average number of coins before and after the transaction, what coins should I start with?
Here are some assumptions that will help you solve the problem. [Recreational math always involves assumptions.] Assume that I have an unlimited supply of dollar bills and the only thing I care about here are the coins. Assume that the only coins in circulation are pennies, nickels, dimes and quarters. Assume that I can’t ask the cashier to convert one denomination to another as a side transaction (e.g., give her five pennies in exchange for a nickel). Assume that the cashier makes change in a way that returns the fewest possible number of coins. Finally, assume that it is equally likely for the transaction price to end in $.01, $.02, $.03, etc. In other words, we want to minimize the number of coins we carry, on average, given some arbitrary number of independent, randomly-priced transactions.
Take as long as you would like to work on this. The answer is given below. Good luck.
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Wow, that was fast. Congratulations on your mental agility. Now let’s see if you are right.
I initially guessed that I should carry at least one coin of each denomination. The first case I studied was 2Q – 1D – 1N – 3P. If you want to get rid of change, this is not a bad choice. On average, you will unload 2.30 coins per transaction. But your goal here is to minimize the average number of coins carried in your pocket or purse, and in this case the result is 5.85 coins (you carry 7 coins before the transaction and 4.70 coins afterward).
I suspected that this result could be improved upon. Out of curiosity, I looked at the case where one carries no change at all and pays for every transaction with dollar bills. This is in fact the best answer. The average number of coins in your pocket in this case is 2.35. While you are gaining 4.70 coins per transaction, you are starting out with zero coins.
If you wonder why you accumulate change, this is the reason. If you are like me, you offer only currency for 90% of your cash purchases. For every ten currency-only transactions, you will come home with 47 coins (15 quarters, 8 dimes, 4 nickels and 20 pennies). I had thought about verifying this result by counting the coins in my own change dish, but since I fish out the quarters and dimes to use in parking meters, my ratios would be off.
To close, let’s consider the effect of the “take-a-penny” dish next to the cashier. If I carry no coins with me and use one penny from the dish whenever the price ends in 1 or 6, then the number of coins I accumulate per transaction drops from 4.70 to 3.90. If I were also to “balance the books” by putting a penny in the dish whenever the price ends in 4 or 9, then I would accumulate only 3.70 coins per transaction.
So, here’s the bottom line — if you hate change, only carry currency, use the penny dish, and vote for conservatives. Asked and answered.