How long has it been since we have had a good pop quiz???

I heard a great one today on Car Talk. Let's see if you can figure this out (no cheating please by looking on the Web! )

A woman and her husband attended a holiday office party with 53 people in attendance. Since there were some new employees, there were informal introductions and handshakes. When the woman and man were driving home, the woman said, "I know that at least 2 people shook hands with exactly the same number of people."

How did she know that?

Come on, people! This isn't that hard! Shufitz got it in minutes. Or, is it just too easy?

They were conjoined twins.


Actually, I think the lack of response might be due to us (me, anyway) not completely understanding the premise of the riddle. Are you sure you haven't left something out?

There are quite a few ways of approaching this.
Here's one that doesn't need any maths.

If it wasn't the case that at least two people shook the same number of hands then everybody must have shaken a different number.

The only way for that to happen would be for

person number one shook hands with no-one
person number two shook hands with exactly one other person
person number three shook hands with exactly two other people
.
.
.
.
person number fifty two shook hands with exactly fifty one other people
person number fifty three shook hands with exactly fifty two other people.

Look at person fifty three - he (or she) has shaken hands with every other person at the party, including person number 1 - but we already said that person number one shook hands with no-one so we have a contradiction. The premise that everyone shook hands with a different number of people is therefore impossible and at least two people MUST have shaken hands with the same number of people.

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quote:

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Are you sure you haven't left something out?

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Yes.

Well, on this wildly popular quiz , I will now say that Bob is right. Either shaking hands with no one or with everyone means that two people had the same number of handshakes. BTW, this works for any number of people being present in a room.

That's it for quizzes from me! For the record, it really was quite a fascinating quiz on CarTalk, and it was presented to the audience in precisely the same way as I presented it here.

PLUS they were conjoined twins!


Sorry, Kalleh. I really did think you had left something out. Had I given the matter some serious thought, I'm sure I would have gotten it (and invented Velcro, and written "Jabberwocky," and discovered a way to turn lead into gold, etc etc etc) but I didn't.

Humbled once again...