[MUSIC PLAYING] Super serious math question-- are there two people in the world with the exact same number of body hairs?

There are currently more than 7 billion people in the world.

That's a lot of people.

But how likely is it that two of them have the exact same number of body hairs?

Without going through the process of actually counting every hair on every person, you might think the answer is probabilistic, like it's 70% likely, or maybe even 99%.

But actually, it's 100% guaranteed.

The answer is yes.

Definitively, yes.

How can I be so confident about this?

The first step is to figure out how many body hairs a person could possibly have.

If we figure that somebody is covered head to toe with dense hair, what's the maximum number of hairs they could have?

Fortunately for us, this wonderful book called "The Heart of Mathematics" actually calculates that.

The author starts by examining his own head, and finds out that he has 1,600 hairs per square inch.

He figures nobody can have more than 10 times that much, 16,000 hairs per square inch.

This is definitely an overestimate.

You can try to draw 16,000 teeny tiny dots in a one inch by one inch square.

It's not really possible.

But that's OK. We're looking to overestimate the number of hairs on a human body.

Now, we need to know how much skin a human could have.

The book uses a cylinder as a model of a human body, easily calculating that the author has less than 2,500 square inches of skin.

Then, he concludes nobody could have more than 10 times this, 25,000 square inches of skin.

If we assume that all of the skin is covered in the maximum number of hairs, there would be 25,000 times 16,000 hairs, which equals 400 million.

All of these over estimates lead us to the conclusion that there's absolutely no way a person has more than 400 million hairs on their body.

Back to the question-- are there two people in the world with the exact same number of body hairs?

There must be, because there are more people on earth-- 7.3 billion-- then there are possible number of body hairs-- 400 million.

Here's a great way to think about it, also from the book "The Heart of Mathematics."

Imagine there are 400 million rooms, each number 1 through 400 million.

Every person on earth has to stand in the room that represents how many body hairs that person has.

It's not possible for everyone to stand in a different room.

There simply aren't enough rooms.

Some room has to have multiple people in it.

And those people have the exact same number of body hairs.

The method that we used to answer that big hairy question was surprisingly simple.

Mathematicians call it the pigeonhole principle, because if you want to put 12 pigeons in 10 holes, some of the pigeons have to share a hole.

That's it.

That's the whole idea.

Let's phrase it as a more general mathematical principle.

If I'm splitting things into categories, and I have more things than categories, then some of the things must be in the same category.

Sometimes people find the pigeonhole principle frustrating.

It doesn't tell you which pigeons share which hole.

But it's still capable of proving some amazing facts, despite its simplicity.

This idea of using the pigeonhole principle to prove silly things about human hair is pretty old.

Way back in 1692, it shows up in "The Athenian Mercury," an old British magazine that's sometimes called the world's first advice column.

In it, regular people could ask fancy people questions, like philosophers say there's no motion.

How can that be?

Or, is it possible my neighbor is a vampire?

Someone also wrote in to ask whether there are any two persons in the world that have an equal number of hairs on their head.

Then, the fancy person from the magazine responds, "The there are more men in the world than hairs upon any man's head is very certain.

And so, 'tis demonstratable there are an equal number of hairs on two people."

And then it goes on to give a convoluted explanation of the pigeonhole principle.

That's cute and all, but what's the point of this whole pigeonhole principle thing?

Can it help with anything besides a three century long discussion of body hair?

Yes, definitely.

This simple idea can be applied to so many things.

Check out these modern examples.

If you're in an elevator with three other people, and there are five buttons lit up, then the pigeonhole principle says that some prankster pressed the buttons for multiple floors.

Another example-- there are only 10,000 possible pins for your debit card.

So in a college of 10,001 students, at least two of them have the same pin.

We could make up these silly examples all day.

In fact, I encourage you to try just that.

But at this point, you might be wondering if the pigeonhole principle is actually used in serious mathematics.

It is.

This guy is credited as the first person to write down the pigeonhole principle in formal mathematics.

He used it to prove this theorem.

So while the pigeonhole principle seems totally obvious when you're talking about pigeons and holes, it can prove lots of mathematical statements that are completely not obvious.

Here's another example of a tricky math problem that requires the pigeonhole principle to solve.

Take a sphere in three-dimensional space and draw five dots on it anywhere you want.

Then, here's a weird fact-- we can always draw a closed hemisphere that contains at least four of the dots.

Let's give a proof of this fact that uses the pigeonhole principle.

First, pick your favorite two dots.

Then, we can draw a circle going through those two points that goes all the way around the sphere, like the equator does.

It's called a great circle, and it cuts the sphere into two hemispheres, or equal halves.

Two of the five points are on this circle, so we have three points left.

The pigeonhole principle says that one half of the sphere, as divided by our great circle, must contain at least two of the points.

So draw the hemisphere to fill that side.

That closed hemisphere contains four points.

The awesomely simple pigeonhole principle just helped us prove a really cool math fact.

But now I'm going to introduce you to a more complex version of the pigeonhole principle.

Using the basic version, we know that, if three goals have been scored in a soccer game, then some team has scored at least two points.

But if five goals have been scored, then some team has scored at least three points.

That's a more powerful version of the pigeonhole principle.

It says that, if we have K many categories and N things, there must be at least N divided by K things in some category.

If six teams are playing in a soccer tournament, and 28 goals have been scored in the tournament, then some team must have scored at least five goals.

28 divided by 6 is 4 and 3/4.

But if you've scored at least 4 and 3/4 goals, then you've scored five goals.

Here's another example-- there are only 366 possible birthdays and 8 and 1/2 million people in New York City.

That means that there is some day of the year when at least 23,225 New Yorkers are simultaneously celebrating their birthday.

Using this beefed up version of the pigeonhole principle, we can revise our answer to the original question-- are there two people on earth with the exact same number of body hairs?

Since we're fitting 7.3 billion people into 400 million categories, there must be some category with at least 7.3 billion divided by 400 million people in it.

That means that there are at least 19 people who all have the exact same number of body hairs.

They could form a club, until somebody loses or grows a hair.

Then, they get kicked out.

What's the wildest application of the pigeonhole principle you can come up with?

Shared it in the comments, and I'll see you next week on Infinite Series.

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