Question:

Picture the successful candidate for this position. He or she is hired, and there’s a big welcome party with 153 fellow employees wanting to shake the candidate’s hand. Since a number of people in the room don’t know each other, there’s a lot of random handshaking among the staff, too.

No one can know exactly how many handshakes there were in total. The question is, can we say with certainty that there were at least two people present who shook the exact same number of hands? How can we be sure?

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Answer: Let’s start with what we do know. We know that each person could have shaken the hands of any number of people from zero to 152. There were 153 people present but no one shakes hands with himself. One way to conceptualize this puzzle is to distribute the 153 people by putting each one of them into a pigeonhole. The individual who shook hands with 1 person goes into hole 1, the individual who shook hands with 2 people, goes into hole 2, and so on. With 153 people, you can get 153 different pigeon holes filled before anyone has to “share” a pigeonhole.

But what about pigeonhole zero? Is it occupied by someone who didn’t shake hands with anyone? If so, then no one can occupy pigeonhole 152. Here’s the crux of the puzzle stated another way. If anyone had the number zero, meaning he or she shook no hands, then no one could have the num-ber 152. For that to happen, that person would have had to shake hands with himself. As a result, when you get to the end, when you get to the 153rd per-son, you say how many hands did you shake? He’s got no choice but to repeat a number that’s already been used and then has to share that pigeonhole.

Answer: Yes, it is certain that at least 2 people shook the same number of hands.