The Handshake Puzzle !!

A nice way to represent this is by using graphs (not the ${xy}$ plane, rather a series of nodes and edges). It soon becomes clear that the number of handshakes between n people is equal to $\frac{n(n-1)}{2}$ (as explained neatly by Nayan Kundu). Equating this to 66 and factosising gives ( ${x}$ +36)( ${x}$ -37). Hence, n=37

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 = 666. This is the quadratic equation n2+ n -1332 = 0. Solving for n, we obtain 36 as the answer and deduce that there were 37 people at the party.

For a party having 'n' people each of a pair shakes their hands.So there will be 'nC2' i.e. 'n(n-1)/2' number of handshakes.Putting the given value & taking the positive value of 'n' you will get the required answer.

THAT WILL RESULT TO QUADRATIC EQUATION: a = 1, b = -1 and c = -1332 the positive root = 37

Although I'm skipping a few steps, The solution to the problem is ½n(n+1) = 666 meaning n(n+1) = 1332. By using approximation, n(n+1) is approximately n² and 1332 is approxomately 1300 which is 13 x 100. This means n² ≈ 13x100; n ≈ root 13 x 10. As 3 = root 9 < root 13 < root 16 = 4 so 30 < 10xroot13 < 40. As there is only one answer in this range, it must be that which is 37.

This is similar to finding the total number of diagonals + number of sides for a polygon : n + n(n-3)/2 = 666