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# Getting to know you

## Question

bonanova    76

You and your spouse invite four other couples to a party. Ten people in all.

Prior to the party each person [other than you] knew a different number of people present.

How many people did your spouse know?

How many did you know?

You may assume the following:

If A knows B, then B knows A.

Every male knows his partner.

You might have invited people you don't know.

## 2 answers to this question

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Guest

if a knows b then b knows a

every male knows his partner

therefore every partner knows her male

everyone knows at least 1 person (their partner)

there are 9 people who must know at least 1 person

the most anyone can know is 9

no two can know the same number of people - other than one person who can (and must) know the same number of people as you

therefore one person knows only 1 other, one knows 2, etc through the last person who knows 9 other people

the partner of the person who knows 9 can only know one person (their partner)

you both know 5 people -

PS - great puzzle - keep 'em coming...

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It took me a while to follow Guest's reasoning, like that Far Side cartoon where a person is writing a solution on the blackboard, and step 2 is in a cloud, and the other person is saying, "We need more detail in step 2".

How did Guest get to "the partner of the person who knows 9 can only know one person (their partner)"?

Here's how: Let's say it is person A who only knows 1 person (B, his/her spouse).
Who can know 9 other people? Since we don't count a person knowing themself, that person knows everybody, including A. The only person who A knows is B, so the 9-knower must be B.

Now, Let's say it is person C who only knows 2 people (D, his/her spouse, and B, who knows everybody). Who can know 8 other people? That person knows everybody except A, including C. The only person who knows C (aside from B) is D. So the 8-knower must be D.

Similar argument shows that if a person knows N people, that person's spouse knows 10-N people. And that includes the two people who know 5.

But we know that YOU know the same number as somebody else. That must mean that YOU know 5. And we know that your spouse knows 10-5=5.

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