A riddle question number 19

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.

$\displaystyle \frac { x ( x - 1 ) } { 2 } = 66 \rightarrow x ( x - 1 ) = 132 \rightarrow x ^ { 2 } - x - 132 = 0 \rightarrow ( x - 12 ) ( x + 11 ) = 0 \rightarrow x = 12, x = -11$ . But the number of people cannot be negative, so the answer is $\boxed { 12 }$ .