Let Me Tell You A Story

Probability Level 2

Let us take a journey to Handshake land.

Here, the residents have a custom of congregating at a predetermined place and time to shake hands and do so only on one day of the Handshake-ian calender.

The peculiar thing about this custom is that they can do only an odd number of handshakes with all the other residents present on the auspicious day, lest they incur the wrath of the Handshake lords.

Let the number of people who take part in this custom each year be N .

The question is, what is $N$ mod $2$ ?

The answer is 0.

2 solutions

Shreya R
Jan 2, 2015

To have an odd number of handshakes, there need to be an even number of people. Any even number mod 2= 0

B.S.Bharath Sai Guhan
Jul 13, 2014

The story, though a little amusing, does not give any important information other than the fact that a person takes part in an odd number of handshakes.

That is a person performs (N-1) handshakes.

This, we know to be odd.

Hence, $N-1 ≡ 1$ ( mod $2$ )

Therefore, $N ≡ 2 ≡0$ ( mod $2$ )

Another way of approaching this problem is by using Graph Theory.

Imagine a graph with N vertices (our residents).

Each has an odd degree (no. of handshakes each person performs) and the number of edges is given by $E = (N \times Degree)/2$

For the graph to exist, E must be an integer.

We know that $Degree ≡ 1$ ( mod $2$ )

Hence, N must be even i.e $N≡0$ ( mod $2$ )

Let Me Tell You A Story

The answer is 0.

2 solutions

0 pending reports