## Saturday, December 20, 2014

### Change for a Dollar

So it turns out that there are 293 ways to make change for a dollar using pennies, nickels, dimes, quarters, half-dollars, or dollar coins. If you limit yourself to the more common coins (pennies, nickels, dimes, or quarters), then there are 242 ways to make change for a dollar. I copied all of the combinations from Frank Morgan's post on this topic and transferred them to this online spreadsheet, so you can play with the data if you like.

Having all of the combinations handy on a computer makes possible some wacky puzzle ideas, like this one:

A customer went to the bank and gave the teller \$242 in dollar bills. The customer said, "Give me change for each one of these dollar bills, please—pennies, nickels, dimes, or quarters—and furthermore, I want no two of these dollar bills to be changed the same way." The teller obliged, and soon the customer had a large pile of coins in front of him. "On second thought," said the customer, concerned about the weight of the coins, "let's change as many of these pennies as we can for dollar bills." The teller did so. "And you know what?" the customer said. "Let's also make dollar bills out of as many of these nickels as we can." This was done. "OK," the customer said, "let's do the same for the dimes." When that was done, the customer said, "What the heck, let's change as many of these quarters as we can for dollar bills too." After this was done, the customer had \$241 in dollar bills, plus change for a dollar, and he walked happily out of the bank. How many pennies, nickels, dimes, and quarters was the customer finally left with?

I don't know if there is any realistic way to find the answer without using a computer. Anybody who's sufficiently spreadsheet-savvy should feel free to post it in the comments!

Exactly $1 in change. JasonZimba said... You are correct! From now on, any time I see that combination of coins, I'm going to have a shaggy-dog story to tell about it. Unknown said... I came up with the same result. Just add the columns of pennies, nickels,dimes, quarters and its easy to see the answer. You have 184 quarters for$46 dollars, you have 616 dimes for 61.6 dollar (leaving 60c in dimes or 6 of 'em). Repeat for nickels and pennies and you get 4 left over nickels for 20c, and 20 leftover pennies for 20c.