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The Magic Cafe Forum Index » » Puzzle me this... » » Courtesy of Martin Gardner (0 Likes) Printer Friendly Version

landmark
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A lovely little problem that doesn't seem to have enough info to solve it, but has a very elegant solution:

My wife and I go to a party attended by four other couples. Various handshakes occur. No one shakes their own hand, no one shakes their spouses' hand, no one shakes a given person's hand more than once.

At the end of the evening I ask each of the nine other people, how many hands they've shaken. Curious to say, each person (truthfully) gives me a different answer.

How many hands did my wife shake?


Jack Shalom
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9 people so someone was alone unles you asked your wife
landmark
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Yes, I asked all 9 other people, including my wife. Each of those nine shook a different number of hands. How many hands did my wife shake?

Jack
BradBrown
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I really like this one. At first, I thought it had to be a trick question, because there didn't seem to be anything to go on to logically arrive at an answer. Then it occurred that I could at least narrow it down somewhat, and finally I saw the key to approaching the solution.

I think your wife had to have shaken 4 hands.

-Brad
Wolflock
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16 Hands of they shook both Hands of each person?
Wolflock
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landmark
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Wolflock, you may only shake one hand of any given person.

Jack
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"Various handshakes occur. No one shakes their own hand, no one shakes their spouses' hand, no one shakes a given person's hand more than once."

It does not say that you may only shake one hand.
It says "no one shakes a given persons hand more than once" So you may not shake the same hand more than once. Some ladies will grab both hands one in each of their own hands and give it a little shake.
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leonard
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Here's my start.

0 handshakes minimum per individual.
8 handshakes maximum per individual.

Other peoples answers are 0, 1, 2, 3, 4, 5, 6, 7, and 8. Only way 9 individuals can answer uniquely.

Total number of handshakes must be even. "It takes two to shake hands."

0 + 1 + ... + 8 + "how many hands I shook" = 36 + "how many hands I shook",
implying I must have shook an even number of hands (0, 2, 4, 6, 8).

Correct so far?
leonard
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Continuing,

Using a picture with 10 dots, the layout satisfying 9 individuals answering with 0 through 9 is unique. In this layout the number of hands shaken by each individual is 0, 1, 2, 3, 4, 4, 5, 6, 7, and 8 (note the two fours). Therefore, I must have shaken four hands.
leonard
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Finally,

Pairing individuals into couples, based on the number of handshakes, yields the following: 8&0, 7&1, 6&2, 5&3, and 4&4. The individual who shook hands with 8 others must be coupled with the individual who shook no hands, etc.

Because I shook hands with four people, so did my wife.
BradBrown
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Here's how I approached it. It's logically equivilent to the solution leonard gave, but it may be easier to follow.

Each of the nine people gave different answers, and there are only nine possible answers, 0 through 8. I listed all the people at the party by the number of hands they shook. I used X for the person in the puzzle asking the questions, because I didn't know how many hands he shook. So, I started of with this list...

0
1
2
3
4
5
6
7
8
X

Then I want through the list and figured out who shook hands with who. I know person 8 had to shake hands with everybody except person 0, who shook no hands. This gave me the following. (The numbers after the colon are the people they shook hands with.)

0
1 : 8
2 : 8
3 : 8
4 : 8
5 : 8
6 : 8
7 : 8
8 : 7654321X
X : 8

Next I proceeded with person 7. I know he shook hands with person 8. He had to have shaken hands with six other people, and there are only six left who haven't maxed out their handshakes. After that, I went on to 6, 5 and so forth, until I identified all the handshakes...

0
1 : 8
2 : 87
3 : 876
4 : 8765
5 : 8764X
6 : 87543X
7 : 865432X
8 : 7654321X
X : 8765

Now I know Mr. X shook hands with 4 people. All that remains is to determine who is married to who. Person 8 shook hands with everyone but person 0, so 8 and 0 are a couple. 7 shook hands with everyone but person 0 and 1. We already know 8 and 0 are a couple, so 7 must be with 1. Continuning, we can identify 6 is with 2, and 5 is with 3. That leaves 4 to be Mr. X's wife.
landmark
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Nice work, all.

Jack
Wolflock
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Um... *Wolf still sits scratching his head and giving the blank stare* Is the answer "green"?

Regards
Wolflock
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landmark
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Only if "green" means four.


Jack
Wolflock
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Yes it does! *Wolf tries to look convincing*

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Wolflock
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