At a party, everyone shook hands with everybody else. These were 66 handshakes. How many people were at the party?

In general, with n+1 people, the number of handshakes is the sum of the first n consecutive numbers: 1+2+3 + ... + n. Since this sum is n (n+1) / 2, we need to solve the equation n (n+1) / 2 = 66. This is the quadratic equation n2 + n - 132 = 0. Solving for n, we obtain 11 as the answer and deduce that there were 12 people at the party.