Math Circle Sample Question

The Four Coins Problem
(adapted from Joshua Zucker)

You are the royal mathematician for the king of Florin. One day, the king comes to you and says:

I want you to create a new money system for our country. I want to have four different coin values. Carrying only coins of those four denominations, I want to pay for anything that costs between 1 and 10. Also, I want to pay quickly, so I never want to have to add more than two numbers to pay for something.

What four coin denominations would work?

The king tried 1, 2, 3 and 4. Can we make all numbers between 1 and 10 by adding at most two coins? Let’s try:
1 = 1
2 = 2
3 = 3
4 = 4
5 = 1+4
6 = 2+4
7 = 3+4
8 = 4+4
9 = Looks like there’s no way to make 9 because the largest coin we have is 4
10 = Again, we cannot make a 10 because our largest denomination is 4.

Looks like coin denominations of 1, 2, 3, and 4 cannot make 9 and 10.

As a royal mathematician, it’s your job to do that hard work of finding the coin values that work. As a reward, the king says you can name your coins anything you want (the king prefers the name Florins, after the country, but that would be boring).

Another Coin System

The queen of the neighboring country of Gildor heard about your coin system, and decided that she wants one too. It’s such a great idea to only ever have to add at most two coins to get number from 1 to 10! However, she doesn’t want to have an exactly the same coin system as Florin, so she asks you to come up with a different set of 4 coins for her. So, what set of four coin values, not all the same as the one you made for the king, will work for the queen.

How Many Can You Make?

By now, the rumors of your great inventions have spread far and wide. Other countries also want coin systems with four coins, such that at most two coins are needed to pay for anything valued between 1 and 10. How many different coin systems of four coins can you create? What are some patterns you see in your coin systems?

Where is the math?