Handshake Problem Variant

Date: 07/08/99 at 20:25:32
From: Stephanie Bixler
Subject: Handshakes at a party

Taking the handshake question further:

Ten people (five couples) go to a party and start shaking hands. You 
don't shake your spouse's hand or (of course) your own. One of the 
men, Jake, shouts, "Stop! How many hands did you shake?" Every person 
gives a different answer. Jake's wife is the last to answer. How many 
handshakes does she have? Is it 0?

Date: 07/11/99 at 04:03:20
From: Doctor Floor
Subject: Re: handshakes at a party

Dear Stephanie,

Thanks for your question.

Nine people have answered, and there are nine possible answers.

Nine possible answers? Yes, people can shake hands with 8 people or 
fewer, and thus the possible answers are 0, 1, 2, 3, 4, 5, 6, 7 or 8.

The person who answers 8 has shaken hands at least with the persons 
that have answered 1, 2, ..., 7. That's only 7 persons, so she or 
he has shaken hands with Jake's wife too. This also means that the 
person who answers 0 is the only one who has not shaken hands with 
the person who answered 8, so they must be a couple.

We can now leave out this couple. As a thought experiment, we could 
re-ask the question restricting ourselves to the 8 people left. 
Originally, the 7 persons in this group answered 1, 2, 3, 4, 5, 6, 7. 
All these people have shaken hands with the person who answered 8 and 
none of them has shaken hands with the person who answered 0. So when 
restricting themselves to the new group, they would have answered: 
0, 1, 2, 3, 4, 5 and 6.

We can now repeat our argument:
The person who now answers 6 has shaken hands with Jake's wife. The 
persons who answer 0 and 6 form a couple. We can leave out this 
couple.

This leaves 6 people. When restricting themselves to the new group, 
they would have answered: 0, 1, 2, 3 and 4. The person who now 
answers 4 has shaken hands with Jake's wife. The persons who answer 0 
and 4 form a couple. We can leave out this couple.

This leaves 4 people. When restricting themselves to the new group, 
they would have answered: 0, 1 and 2. The person who answers 2 has 
shaken hands with Jake's wife (and with Jake). The persons who answer 
0 and 2 form a couple. The other couple is Jake and his wife.

We can conclude that Jake's wife has shaken hands with 4 people.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/