At a party 66 handshakes occurred. Each person shook hands exactly once with each of the other people present. How many people were present?
Let's start with small numbers of people and handshakes and move from there. I'll represent people with letters to show the handshakes:
If we have 2 people, there is 1 handshake
If we have 3 people, there are 3 handshakes
If we have 4 people, there are 6 handshakes
If we have 5 people, there are 10 handshakes
See that we can express the number of handshakes as the sum of consecutive positive integers, starting with 1, i.e.
Let's test this with 5 people. We have
So what we need to do is add up to 66 and we'll be able to find the number of people: