Are there two people in the world who are equally hairy? Contrary to expectations, even without statistical analysis, we can answer this statement with a resounding ‘yes’. This requires the “Pigeonhole Principle”, also known as the “Dirichlet Principle”.
It sounds ridiculously simple: n object k There are drawers, and more objects than drawers (n > k), then several objects are placed in the same drawer. This simple statement, which sounds more like common sense than a mathematical theorem, was first mentioned by the French scholar Jean Lelechon in his 1622 book. The principle of the pigeonhole is usually attributed to Peter Gustave Lejeune Dirichlet, who lived about 200 years after Rouréchon. Despite its simplicity, the pigeonhole principle makes it possible to prove very complex relationships.
Coming back to hair, how can we find out if two people in the world have exactly the same number of hairs? You have to find it. The average person has between 90,000 and 150,000 hairs on their head, depending on the color of their hair. It’s safe to say that no one has more than a million hairs, but he has eight billion people on our planet. This means that there will always be two people with exactly the same number of hairs on their head. However, after a few strokes with the comb, it is more likely that another group with the same number of hairs as that person will emerge. .
Much more can be said about human hair volume. For example, the smallest number of people in the world who have the same amount of hair. To calculate this, it helps to consider two extreme cases. One, if each person has exactly the same number of hairs on their head (perhaps if they all shaved themselves), and another for him, human hair.
For this purpose, imagine a million rooms numbered in ascending order. Each person enters a numbered room corresponding to the number of hairs on their head. If everyone on the planet was equally hairy, we would all be in the same room. The remaining 999,999 rooms are empty, but in one room he has 8 billion humans.
But at the opposite extreme, people divide themselves so that they have as few people in the same room as possible. If so, what is the minimum number of people to share the room with? To calculate this, you can fill the room piecemeal. At first he was one per room, then two, then three, and so on. If he divides the 8 billion people evenly among the 1 million rooms, then in each room he will have 8,000 people. A little redeployment and you should soon have room for over 8,000 people. This means that no matter how people are divided, in any scenario, the largest room will contain at least 8,000 he. That means there are at least 8,000 people on the planet with the same amount of hair.
Thus, we presented an even stronger version of the pigeonhole principle. n the object is split k category, and n > kthen at least n ⁄ k Objects belong to the same category. If the objects are evenly distributed among the drawers, on average, n ⁄ k Objects are put in the same drawer. As soon as the objects are even slightly redistributed, one of the drawers will inevitably contain more. n ⁄ k object.for the quotient n ⁄ k is not an integer. The minimum value you are looking for corresponds to the rounded up value. This is because one drawer necessarily contains this number of objects.
For example, if a soccer match scored 7 goals, one team scored at least 4 goals (7 ⁄ 2 rounded up). It is also possible that the same team he scored 5 goals, 6 goals or all 7 goals. Or just think of a bigger number. At least 23,000 New York City residents celebrate their birthdays on the same day. The city has a population of about 8.5 million, and in 366 calendar days a person can be born (year of birth doesn’t matter here). So at least 8,500,000 / 366 = 23,000 people share the same birthday.
An interesting (and certainly less important) statement can be derived from the pigeonhole principle. For mathematicians, one related implication has to do with the distribution of points on the sphere. Choose any 5 locations on the sphere, at least 4 of which are on the same hemisphere. To demonstrate this, the hemisphere must be skillfully chosen. First, select 2 of the 5 marked points (it doesn’t matter which one) and mark the equator where the 2 points are. This splits the sphere in two, with three more points on top of it. According to the pigeonhole principle, two of them must be on the same hemisphere. When adding points on the equator, there will always be at least four points in the same half of the sphere, no matter how they are distributed.
The pigeonhole principle shows that even seemingly obvious statements are of great value in mathematics. However, this shouldn’t be too surprising. Ultimately, research in this area rests on some basic assumptions that are as simple as possible, such as the empty set, which can infer results as complex as Gödel’s incompleteness theorem. Simple systems can lead to complex results.
This article originally appeared on science spectrum Reproduced with permission.
