Who’s who
Did each problem have a unique solution?
Yes each problem has a unique solution. I can see this because the majority of letters in each problem have to be in a fixed position. They cannot move. For example A has to be in the middle because that is the one connected with the most friends.
Were there any clues you didn't need to use?
I do not think there were any clues that I did not need to use. This is because they are all useful when it comes to checking over your work. Although I think each problem has clues which are more useful than the others. For instance in the first problem the first and third clue was more useful than the second.
If you label each node with the number of friends the person has, and add together all the numbers, what can you say about the answer? Can you explain why?
When you add together the number of friends each person has in the second question you find that there are more friends than people. This is because lots of people have more than one friend and they are all are the same.