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

Answer:

Handshakes are used traditionally as greetings, but they are also used to seal an agreement when
a business transaction has been mutually accepted. A handshake is sometimes used to
characterize the personality of an individual. A firm handshake is interpreted to indicate an assertive
person or an extroverted personality, whereas a less firm or limp handshake is viewed as a sign of weakness
and lack of confidence. Some diseases, such as influenza, can be spread by shaking hands with an
infected individual and then touching one's face. (See Hygiene)

With two people (A and B), there is one handshake
(A with B).

With three people (A, B, and C), there are three handshakes
(A with B and C; B with C).

With four people (A, B, C, and D), there are six handshakes
(A with B, C, and D; B with C and D; C with D).

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 n^{2}+ n -132 = 0. Solving for n, we
obtain 11 as the answer and deduce that there were 12 people at the party.

Since 66 is a relatively small number, you can also solve this problem with a hand
calculator. Add 1 + 2 = + 3 = +... etc. until the total is 66. The last number that
you entered (11) is n.

Carl Friedrich Gauss (1777-1855) is credited with finding the formula for computing the sum
of the first n consecutive numbers when he was an elementary school student, at age 8.
The teacher had asked the students to compute the sum (S) of the first 100 integers.
To the teacher's astonishment, Gauss was able to do it very quickly
by noticing that the sum of the sequence and the reverse sequence produced a series of constants.
S = 1 + 2 + 3 + ... + 100 S = 100 + 99 + 98 + ... + 1
2S = 101 +101 +101 + ... + 101 = 100×101

S = (100×101)/2 = 5,050

The symmetry of the solution can also be observed with a graphical representation: