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### 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,

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/

Associated Topics:
High School Permutations and Combinations
High School Puzzles

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