Assuming I correctly understand the problem.

Each side takes the form of #1 + #1 + unknown = 100. You simply fill in the numbers, and solve for unknown.

ie first side is

31 + 10 + unknown = 100

41 + unknown = 100

unknown = 100 - 41

unknown = 59

The 59 plugs in as one of the two numbers on the next side, allowing you to follow the same process to solve for that side, which allows you to solve the next side and so on. There is only one solution because 59 is the only answer that will make the side with 10 on it be equal to 100. And since that gives two values for the next side, there is only one solution for that side (20) and this gives you two numbers on the next side, meaning only a single solution for that one (33), etc.