Counting Hairs

Oct 26, 2021

Let's say you have a town in which there are 200,000 people.

I want to know, what are the chances that two inhabitants of that town will have exactly the same number of hairs on their heads? Now, I will stipulate that no one in this town can have more than 100,000 hairs and nobody will have fewer than zero.

So you have a town of 200,000 people. And you have heads of hair that have between zero and 100,000 hairs. So I want to know what are the chances that there are two inhabitants of this town that have exactly the same number of hairs? Not necessarily in the same place. You could have like you want hairs on the front and somebody could have eight hairs on the back and that would be a match.



This is a branch of mathematics known as like common tutorial analysis. And this is a well known  pigeonhole principle. You take the first 100,000 people and you distribute them such that no two people have the same number of hairs.  You put each one of them on a little pigeonhole.

You say, "Okay, how many hairs you got? Zero, Okay, you go in the zero pigeonhole."

Next guy goes in "one" next person was in "two". So you've used up 100,000 residents, and you've used up all 100,000 pigeonholes.

So the very next person, the 100,001st person is back where? They must be a duplicate of one of the first 100,000.

So as a matter of fact, there are 100,000 people in this town that have the same numbers of hairs on their heads. They all have a mate (or many mates!)

I thought I'd share the reviews on this one. My brother said, "That is the most inane, great puzzler I've ever heard!"

And Doug left a little note for me on this one: "Raymond, I never had time to tell you last week, but this is one of the most inane puzzles you come up with in a long time. Love, your inestimable producer, Douggie."


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