## Handshaking

Posted by Chris on February 13, 2011 – 8:47 am

My wife and I invited four other couples to our New Year’s Eve bash. After yelling “Happy New Year!”, everyone either shook hands with or hugged each other. The couples only hugged each other.

When I asked everyone how many people they shook hands with, each gave a different answer.

How many people did my wife shake hands with?

How many people did I shake hands with?

February 14th, 2011 at 2:29 am

your wife shoke hands with 8 people. you also did the same

February 14th, 2011 at 2:53 am

Since the truth is always somewhere in the middle and due to the equality between sexes I would say 4 handshakes for both me and my wife

February 14th, 2011 at 8:00 am

Well. I believe that you as a man would have shaken hands with all the other man in the room, so 4 handshakes for you, and for your wife I would say none. I believe that it is weird for women to shake hands for new years eve and it would be more likely to hug everybody else, as for men it would be weird to shake hands with women and weird to hug other men.

My answer is based on social acceptability, and how well you know everybody else…

February 14th, 2011 at 8:18 am

Hi chege. Sorry, that’s not right.

Hi slavy. I see you used the “new math” to do it successfully

Hi IMGthis. It seems that social maths almost works.

February 14th, 2011 at 1:38 pm

well…there’s just so many questions left unanswered.

you both are the hosts so you better have shook hands w/ everyone. Also, were they two handed shaking…or handed shaking? what about hi fives? when did that go out of style? not to mention, this is a new year’s eve party..so has anyone passed out from over-boozing yet? straight up, i wouldn’t shake hands with anyone who had new years eve chili all over them.

not to mention, jim’s been eye balling greg’s wife carol all year, so would they have hugged?

using my analytical brain of amazingness,

i’m gonna have to say 2.5 for you, and -3 for your wife.

February 14th, 2011 at 4:28 pm

This is a logic problem. It has a unique and unequivocal solution. The New Year’s Eve bash is just mind candy.

Slavy gave the correct number of handshakes, but deliberately withheld his reasoning so that others can have a shot at it.

It’s a fairly simple problem to solve, but you’ll need to get your head straight first.

In case there’s any doubt: if two people shake hands with one another, each would be able to say that s/he shook hands once.

I had even asked my wife how many people she had shaken hands with.

February 15th, 2011 at 8:22 pm

Does everyone give up? (I know that slavy has cracked it).

It is quite a nice puzzle. You will even be able to fully identify the couples (in terms of the number of handshakes each person gave).

I’m embarrassed to say that it took me about 15 minutes to solve. In mitigation the source version was confusing to read.

February 16th, 2011 at 10:01 am

I’ll post my pre-written solution tonight.

February 16th, 2011 at 5:50 pm

I can’t bring myself to just post the answer. So instead:

What is the maximum possible number of handshakes that any person could have made?

What is the only possible set of distinct numbers of handshakes that must have been made?

For the person who made the most handshakes, who must s/he have shaked/shaken/shook hands with, and who can’t s/he have shaked hands with?

The first two questions are very easy to answer. The third is a bit harder, so it may take a few minutes to convince yourself that you’ve got it right.

After that, the complete solution is a doddle.

February 17th, 2011 at 9:20 am

There were 10 people altogether. So the most handshakes that any of them could have made is 8 (you don’t shake hand with yourself or your partner). As each of the 9 people gave a different answer, the answers must have been 0,1,2,3,4,5,6,7,8.

I’ll identify the people by the number of hands that they shook, and I’ll use M to identify myself. The obvious bit is that 8 can’t have shook with 0, as 0 shook nobody’s hand. 8 must have shook with everyone else, so none of them is 8’s partner, so 0 must be 8’s partner. Now 7 can’t have shook hands with 0 (as nobody shook 0’s hand), neither can 7 shake hands with 1 as 1’s handshake was used up on 8. 7 must have shook hands with everyone except 0 and 1, so 0 or 1 must be 7’s partner. 0 can’t be 7’s partner as 0 is 8’s partner, so 1 is 7’s partner. I won’t bother to spell out the detailed reasoning for the other cases.

8 must have shook hands with 7,6,5,4,3,2,1,M and must be coupled with 0.

7 must have shook hands with 8,6,5,4,3,2,M and must be coupled with 1.

6 must have shook hands with 8,7,5,4,3,M and must be coupled with 2.

5 must have shook hands with 8,7,6,4,M and must be coupled with 3.

4 shook hands with 8,7,6,5 (see above) and must be coupled with M(e) as I’m the last turkey in the shop.

Just for completeness:

3 shook hands with 8,7,6 and is coupled with 5.

2 shook hands with 8,7 and is coupled with 6.

1 shook hands with 8 and is coupled with 7.

0 shook hands with nobody and is coupled with 8.

M shook hands with 8,7,6,5 and my wife is 4

So my both my wife and I shook hands 4 times. We both shook hands with the same people.

Karl, you’re the grammar king; should it be shaked, shaken or shook?

February 17th, 2011 at 10:00 am

Nice puzzle Chris.

I almost had it but couldn’t quite get the logic clear enough in my head to see it through to solution.

P.S. I vote for “shaken” when preceded by “must have” as in the first five examples, but “shook” when preceded by a number or letter as in the second five examples.

February 17th, 2011 at 10:08 am

Hi Dual. LOL, I’ve changed shaken/shaked/shook several times, none of them seem quite right. I’ve now adjusted my answer as I agree that your forms of shookended seem to be the nicest.

Later: LOL, I’ve gone back to shaked.

It’s a shame that (other than slavy) nobody did it – it is a nice problem. I especially like that it seemed impossible to solve at first. I lifted it from the old ToM site.

February 17th, 2011 at 1:44 pm

I’ve been out of pocket and want to answer before I read above.

I considered 5 pairs of people, each hugs their spouse.

AaBbCcDdEe (A hugs a, et cetera)

now for the handshakes…

A = 0

a = 8 (all others) = BbCcDdEe

B = 1 = a (already considered)

b = 7 = a + CcDdEe = 1 already considered + each remaining

C = 2 = ab (already considered)

c = 6 = ab + DdEe (already considered + each remaining)

D = 3 = abc

d = 5 = abc + Ee

E = 4 = abcd

e = 4 = abcd

If all have different answers, then I must have the same answer as one of them.

Therefore, I must be E or e, which makes my spouse the other.

So my wife and I each shook 4 hands.

For X guest couples, my wife and I each shake X peoples’ hands.

February 17th, 2011 at 1:48 pm

Thanks Chris. Enjoyed it.

February 17th, 2011 at 1:52 pm

Hi cazayoux. Well done. It is obvious that you didn’t cheat. Did it make your head hurt?

February 18th, 2011 at 8:49 am

The headache wasn’t too bad.

First thought was to create a 2-dimensional drawing of a 3-dimensional idea. Something with a couple of pentagons, each couple having representation on one of them. The idea was to draw out all possibilities …. back up … too complicated. Need to simplify.

Then I drew out 10 ‘people’(dots) in a cirle (like a clock with only 10 hours).

Blue line along the perimeter to show the couple pairings (and a hug).

Red lines for the handshakes through the ‘clockface’.

First attempt was to start by showing one person with 8 handshakes, the next with 7 …. failure

Second was to start with 0 and increase … failure

Then went back to starting with 8, recognized that that persons spouse would HAVE to be 0 and that the ‘1′ person was already accounted for, whichever person that was.

Tried alternating around the ‘clock’ 0,8,1,7, …

Definitely enjoyable to get the eureka moment!

February 18th, 2011 at 9:20 am

It took me several goes too. I just found that I seemed to need to remember too many things at the same time. I can’t see more than at most two moves ahead when playing chess, or any game come to that