Here are the US denominations less than $5 (not including $1):
We're looking for ways to get a total value using some number of each of these coins. We can write this as an equation, for example
Since it slows down exponentially as the total value gets larger, I was only able to run it for values up to 4 dollars/euros. The results are still interesting though. Here's the minimum number of coins needed to make a total:
As each larger denomination becomes available, there's a sudden drop in the number of coins needed. Since euros include many more denominations, they're able to use fewer.
We won't always get the minimum in change though, so it's useful to look at the different ways to make a value, and find out what the average number of coins is:
There are many things I'll need to unlearn when I return to the US next month (like saying "merci" instead of "thanks"), but since I never really got the hang of euros I suppose making change won't be one of them.