Problem 24: Shake Shake Shake

There are lots of famous maths problems about handshakes. Here are three of increasing difficulty:

1. You attend a party with 9 other people (so there are 10 guests in total). During the party, each person shakes hands once with everyone else (and of course people don’t shake hands with themselves). How many handshakes happen in total?
2. You attend another party with 9 other guests. This time, people don’t necessarily shake hands with everyone else. Explain how you know there will always be two people who shake hands with the same number of people.
3. You attend a third party, but this time it’s for couples. You are there with your partner, and there are 4 other couples. Of course, people in a couple don’t shake hands with one another. You ask the other 9 guests how many hands they have shaken, and they each give you a different answer. How many hands did your partner shake?