Are there two people in the world who are equally hairy? Contrary to what you might expect, this statement can be answered with a resounding yes, even without statistical analysis. For this, you need nothing more than the “pigeonhole principle,” also called “Dirichlet’s principle.”
It sounds almost ridiculously simple: if you want to divide n objects among k drawers, and there are more objects than drawers (n > k), then several objects end up in the same drawer. This simple statement, which sounds more like common sense than a mathematical theorem, was first mentioned by French scholar Jean Leurechon in a book in 1622. As usual, Stigler’s law—according to which no scientific discovery is named after its true discoverer—applies here. The pigeonhole principle is usually attributed to Peter Gustav Lejeune Dirichlet, who lived about 200 years after Leurechon. Despite its simplicity, the pigeonhole principle makes it possible to prove quite complex relationships—for example, that out of five randomly arranged points on a spherical surface, at least four are on the same hemisphere.
But back to hair: How do you find out whether two people in the world have exactly the same number of hairs on their head? To do this, you first have to find out the maximum amount of hair people can have. Depending on hair color, the average person has between 90,000 and 150,000 hairs on their head. It is safe to say that no one has more than a million hairs. There are eight billion people living on our planet, however. This means that there are bound to be people who have exactly the same number of hairs on their head—at least until the moment when one of these individuals combs their hair and loses a few. But then, after a few more strokes of the comb, there will most likely be another group of people who have the same number of hairs as that person does. In fact, Leurechon also chose the example of hairiness to introduce the pigeonhole principle.
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Even more can be said about humanity's hairiness—for example, the minimum number of people in the world who have the same amount of hair. To calculate this, it helps to consider two extreme cases: one in which the number of hairs on each person’s head is exactly the same (that would probably be the case if everyone shaved themselves bald) and one in which people’s hair varies as much as possible.
For this purpose, imagine a million rooms numbered in ascending order. Each person enters a room with the number corresponding to the number of hairs on their head. If everyone on the planet is equally hairy, all end up in the same room. Then there are eight billion humans in one room while the remaining 999,999 rooms are empty.
At the other extreme, however, people divide themselves in such a way that as few as possible end up in the same room. What is the minimum number of people sharing a room then? To calculate this, you can fill up the rooms little by little: first one person per room, then two, then three, and so on. If you divide eight billion people evenly among a million rooms, you end up with 8,000 people in each room. As soon as you redistribute people a bit, there is bound to be a room that houses more than 8,000 people. This means that no matter how people are divided, in any scenario, the fullest room contains at least 8,000 people. So there are at least 8,000 people on the planet with the same amount of hair.
Thus, we have shown an even stronger version of the pigeonhole principle: if n objects are divided among k categories, and n > k, then at least n ⁄ k objects belong to the same category. If the objects are distributed evenly among the drawers, then, on average, n ⁄ k objects end up in the same drawer. As soon as the objects get redistributed even slightly, one of the drawers will inevitably contain more than n ⁄ k objects. If the quotient n ⁄ k is not an integer, the minimum we are looking for corresponds to the rounded-up value because one of the drawers then inevitably contains this number of objects.
For example, if seven goals were scored in a soccer game, one team scored at least four of them (7 ⁄ 2 rounded up). That same team could also have made five, six or all seven goals. Or consider something with larger numbers: At least 23,000 residents of New York City have their birthday on the same day. The city’s population is about 8.5 million, and there are 366 different calendar days on which someone can be born (not worrying here about their year of birth). Accordingly, at least 8,500,000 / 366 = 23,000 people have the same birthday.
Entertaining—and admittedly not too significant—statements can be derived from the pigeonhole principle. For mathematicians, one of the relevant implications has to do with the distribution of points on a spherical surface. If you pick five arbitrary locations on a sphere, then at least four of them are on the same hemisphere. To show this, you have to choose the hemisphere cleverly: First, you pick two of the five marked points—it doesn't matter which—and mark an equator on which the two points lie. This divides the sphere into two halves, on which there are three more points. According to the pigeonhole principle, two of them must necessarily be on the same hemisphere. If you add in the points on the equator, there are always at least four points on the same half of the sphere's surface, regardless of how they are distributed.
The pigeonhole principle illustrates that even seemingly obvious statements have great value in mathematics. This should not be too surprising, however. After all, work in this field is based on a few basic assumptions that are as simple as possible—such as that there is an empty set—from which results as complicated as Gödel's incompleteness theorems can be inferred. Simple systems can have complex consequences.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.