A logic problem by A A

Logic Level 2

At a party, everyone shook hands exactly once with everybody else.

If the number of handshakes was 2016 how many persons attended the party?

The answer is 64.

1 solution

Zee Ell
Aug 25, 2016

Let the number of partycipants (sic!) be n.

Each person shook hands with the (n-1) other people (but not with him/herself.) Any 2 person shook hands only once (so we have to eliminate double counting).

Hence, we can set up the following equation:

$\frac {n × (n - 1)}{2} = 2016$

$n^2 - n - 4032 = 0$

After solving the quadratic equation for n > 0, we get:

$\boxed { n = 64}$

Yes ,or for another reasoning.

Let m be the number of persons at the party. Then , we can characterize that everybody shakes hand with everybody else once by the fact that the first shakes with m-1 , the second with m-2 and so on leading to the equation that the total nubmer of hands shaken is m-1+m-2+m-3 etc that leads to , by Gauss , M(m-1)/2 = 2016. Solving smartly the equation you get the result m=64 . Note that this reasoning givesyou the same equation but the characterization is different Actually the 2 reasoning proposed for solving the problemby you and me are 2 ways of computing the series m-1+m-2 etc anyway. Therefore you can either say each person shakes hands with m-1 persons and we need to divide by 2 or say the first person shakes with m-1, the second with all the others minus the shake of hands with the first so m-2 and so on this 2 being therefore 2 ways of interpreting and determining the answer anyway.

A A - 4 years, 9 months ago

A logic problem by A A

The answer is 64.

1 solution

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