Classic Hand shake question

Question

I am asked the following question:

My wife and I were invited to a party attended by four other husband-wife couples, making a total of ten people. As people arrived, there was some hand shaking. No one shook their own hand, and there were no husband-wife hand shakes. When it was over, I asked each person "How many people did you shake hands with?" I asked nine people (not including myself) and got nine different answers. How many people did my wife shake hands with?

Since there are 10 people in the room should I use the formula n(n-1)2, which results in 10(10-1)/2=45 which would be the sum of all the handshakes. If I divide that by ten I would get 4.5 people shook his wife's hand.

This doesn't seem correct at all. What is the right direction to head in order to solve this question?

EDIT:

I ended up trying to solve by pairing everyone together and drawing a graph showing the relations between everyone: enter image description here

But this wasn't complete apparently.

score 5 · Accepted Answer · answered Oct 03 '12 at 16:50

5

Hint: No person shook more than eight hands, so you must have gotten answers from 0 to 8 inclusive. Who is married to the person who shook 8?

answered Oct 03 '12 at 16:50

Ross Millikan

374,822

Did she shake 4 hands? – Nick Oct 03 '12 at 17:06
@Nick: yes, she did. – Ross Millikan Oct 03 '12 at 17:16
I do not see how she shook 4 hands. Why could no one have shaken 0 hands? He got answers from 0 to 8 inclusive right? – Moderat Nov 21 '12 at 18:10
1

@jmi4: yes, somebody shook 0 hands. The person who shook 8 had to shake with everybody except his/her spouse, who must be the person who shook 0. The the ones that shook 7 and 1 are married. It keeps going, with each couple summing to 8. That requires two 4's. To avoid getting two like answers, he must be one of the two and his wife the other. – Ross Millikan Nov 25 '12 at 23:11
@RossMillikan can you take a look at my edits. I didn't end up getting the answer the proper way could you expand on your answer so I can see the proper way. My full hw is here – Nick Dec 12 '12 at 21:52
@Nick: your graph seems to reflect the point of my answer to jmi4. You correctly pair H5 with W5 as the person who shook 8 must be married to the person who shook 0. Then you pair H4 with W4 as the 7 and 1, going on to your wife with 4. What do you mean not the proper way? – Ross Millikan Dec 12 '12 at 22:42
I'm not sure, my professor gave me a 6 out of 10 on the homework. So I assumed I did something incorrect. I thought maybe there was a better math representation of the answer versus my graphical one – Nick Dec 13 '12 at 00:59
@Nick: I think it is difficult to follow. I knew what to look for based on our discussion. If I were doing it, I would write up the logic of my last two comments, rather than the diagram. – Ross Millikan Dec 13 '12 at 01:44
Okay, I will do that. Thanks! – Nick Dec 13 '12 at 01:51

score -1 · Answer 2 · answered Oct 03 '12 at 17:35

-1

The answer lies in the question itself. It says, there were no hand shakes between husband-wife and the man asking everyone has not included himself in the count, so he and his wife would not be counted. Therefore the answer is 8...His wife shook hands with 8 people(excluding her husband).

answered Oct 03 '12 at 17:35

laique mehdi

1

Who shook 0 then? – Karolis Juodelė Oct 03 '12 at 17:37
2

No.if his wife shook 8 hands, she shook hands with everybody except her husband. Nobody shook 0 hands so there must be two people who shook the same number. – Ross Millikan Oct 03 '12 at 17:41
How? any explanation? – laique mehdi Oct 03 '12 at 17:44
Yes, you're right Ross – laique mehdi Oct 03 '12 at 17:46
1

It sounds like you are reading it that everybody shook all hands except their spouse's. Then each person would shake 8. But then when he asked how many hands people shook, all answers would be the same. – Ross Millikan Oct 03 '12 at 17:50

Classic Hand shake question

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