4 coin problem

The Four Coins Problem

You’re creating a new coin system for your country. You must use only four coin values and you must be able to create the values 1 through 10 using one coin at a minimum and two coins maximum.

Example:

What happens if your four coins have values 1, 2, 3 and 4? You’re able to make the values 1-4 with a single coin, 5 (4+1 or 3+2), 6 (3+3 or 4+2), 7 (2+3), 8 (4+4), but it’s impossible to make 9 or 10 with two coins, so this coin system fails!

After you create a successful coin system for your country, the neighboring countries ask you to do the same for them, but to please not repeat the same coin system for any of them.

What are the other unique coin systems that you can create?

For how many countries can you create a unique coin system with four coins?

Source: Jo Boaler - Mathematical Mindsets