## Friday, September 30, 2011

### Handshaking at the UN meeting

A whole lot of people shook hands and a whole lot refused to shake hands with each other at the currently happening UN meeting in NYC.  As UN handshaking is not just a handshaking but a political statement, all these handshakes were carefully monitored and accounted for by the special math agent Ilya 011. Where 011 stands for his ranking in this puzzle marathon.

At the end of the meeting Ilya reported that something fishy is going on as the number of people that shook odd number of hands is an odd number and this is impossible. He was right. It turned out that President Obama extended a handshake to Prime Minister Netanyahu but the latter refused it. The rest of the recorded handshakes were mutual.
The question is: how did Ilya knew to suspect the handshaking statistics?

Disclosure: this puzzle celebrates our 10+ puzzle points winner Ilya. The rest of the characters and events in this puzzle are fictional. Any resemblance is purely accidental.

When two people extend hands for a handshake, this counts as one handshake for each of them. At that time two handshakes total are being recorded and added to the total number of handshakes.

Answer ideas accepted any time until midnight on Saturday October 1st (EST), on our Family Puzzle Marathon. They will be hidden till then and everyone who submitted a valid solution will get a puzzle point.

Bean said...

Since each handshake should be counted by two people, there should be an even number of handshakes reported. Yet we know there is an odd number of handshakes being reported. The group of people who report an even number of handshakes can be ignored, since we know that group comes out even. The number of people who shook an odd number of hands needs to come out even for the total to be even. Two people with an odd total will add up to an even number, and thus make our total remain even. But after we've paired up all those "odd" people, we have one guy left who shook an odd number of hands. So the total now must be odd. Which is odd.

Wang said...

If I understand the question correctly, it sounds like the total number of handshakes is always even (that is, either a handshake between two people is recorded or not and therefore, it's either +2 total handshakes or +0 total handshakes respectively)

If that's correct, then Ilya's statement about an odd number of people with an odd number of handshakes being impossible is right. If there was an odd number of people with an odd number of handshakes, that means that there would be an odd number of total handshakes (since the rest of the people would have an even number of handshakes).

In order for there to be an odd number of total handshakes, there must be +1 handshake somehow and I'm guessing you can't just shake hands with yourself and count that as 1 handshake.

In order for there to be an even number of total handshakes, that must mean that are either an even number of people with even handshakes or an even number of people with odd handshakes (ex. 2 people only shook hands with each other say) OR an odd number of people with even handshakes.

SteveGoodman18 said...

If you add up the number of hands that each person shook, the total will be double the actual number of handshakes. This is because each handshake is counted from two different points of view.

Since the total is double the actual number of handshakes, it must be an even number. But, if an odd number of people reported an odd number of handshakes, the sum of all handshakes would be odd, not even.

Thus, there was obviously something not reported correctly.

anne-marie said...

A.k.a the pigeonhole principle.
If a a party one person did not shake a hand, it means that the other people did at most n-2 handshakes. Then, if n is odd, n-2 is odd.
If nobody shaked hand zero time, then each person could do n-1 handshakes.
If n is odd, n-1 is even.

Jerome said...

This is a neat puzzle. It's a mind twister. I think the easiest way to solve it is to observe that every handshake involves TWO people. The total number of handshakes always increases by two.

If you begin with 0 handshakes the first two people to shake hands will create a total of 2 handshakes but both of them will have participated an odd number of handshakes ie one.

No matter how this continues, there is no way that an even number added to an even number will create an odd.

What all of this babble means is that starting from 0 every handshake will have to increase the total by 2 which will create another even number. The fact that one of the people in the room has an odd number of handshakes only means that there is another one somewhere in the same room who also has an odd number of handshakes to help create an even total.

Maria said...

You people are really really good.

The puzzle is tricky but the answer is simple.
Just repeating what Bean, Wang, SteveGoodman18 and Jerome explained so nicely:

- as each handshake is counted twice, the total number of handshakes is even

- this large even number could be split into two sums: handshakes reported by people who shook even number of hands and handshakes reported by people who shook odd number of hands

- the first sum is a sum of even numbers and is always even

- the second sum should also be even to produce an even total

- what is this second sum that should be an even number? Handshakes reported by people who shook odd number of hands. It is a sum of odd numbers. This sum could be even only if we have even number of additives.

Special agent Ilya 011 knew that number of people who reported shaking odd number of hands should be even.

I think Bean, Wang, SteveGoodman18 and Jerome deserve a big puzzle point. Anne-marie - you were on the right track but still a bit far from the full solution. Knowing how good you are, we won't give you a puzzle point for trying.