"Among any 367 people, there must be the same person in Amanome."

"Choose 6 pairs of gloves from any 5 pairs of gloves, of which at least 2 pairs are just a pair of gloves."

"From 1, 2, ..., 10, and there are at least two parity differences. "

......

Everyone will think that the above conclusion is correct. What principles are these conclusions based on? This principle is called pigeon hole principle. Its content can be expressed in vivid language as follows:

"put m things into n empty drawers at will (m >；; N), then there must be at least two things in a drawer. "

In the first conclusion above, because there are at most 366 days in a year, at least two of the 367 people were born on the same day in the same month. This is equivalent to putting 367 things into 366 drawers, and at least two things are in the same drawer. In the second conclusion, imagine that five gloves are numbered separately, that is, two gloves are numbered 1, 2, ..., 5, and these two gloves with the same number are just a pair. Take six gloves at random. They have at most five numbers, so at least two of them have the same number. This is equivalent to putting six things in five drawers, and at least two things are in the same drawer.

A more general expression of pigeon coop principle is:

"Put more than kn things into N empty drawers at will (k is a positive integer), then there must be at least k+ 1 things in one drawer."

Using the above principle, it is easy to prove: "In any seven integers, the difference between at least two numbers of three is a multiple of three." Because there are only three possible remainders when any integer is divisible by 3: 0, 1 and 2, at least three of the seven integers are divisible by 3 to get the same remainders, that is, the difference between them is a multiple of 3.

If there are infinitely many objects discussed in the problem, there is another way to express the pigeon hole principle:

"Put an infinite number of things into N empty drawers at will (N is a natural number), then there must be an infinite number of things in one drawer."

The pigeon hole principle is simple and easy to accept, and plays an important role in mathematical problems. Many proofs of existence can be solved with it.

In the June/July issue of 1958, the American Mathematical Monthly has such a topic:

"prove that at any party of six people, or three people have known each other before, or three people have not known each other before."

This problem can be proved simply and clearly in the following ways:

On the plane, six points, A, B, C, D, E and F, respectively represent any six people attending the meeting. If two people have known each other before, then connect a red line between the two points representing them; Otherwise, connect a blue line. Consider the AF between five connecting lines AB, AC, ..., A and other points, with no more than two colors. According to the pigeon hole principle, at least three lines have the same color, so let AB, AC and AD be red. If a line in BC, BD and CD3 is also red, then the triangle ABC is a red triangle, and the three people represented by A, B and C have known each other before; If the lines BC, BD and CD3 are all blue, then the triangle BCD is a blue triangle, and the three people represented by B, C and D have never known each other before. No matter what happens, it is consistent with the conclusion of the problem.

The six-person assembly problem is the simplest special case of the famous Ramsey theorem in combinatorial mathematics. The proof of this simple question can be used to draw other deeper conclusions. These conclusions constitute an important content in combinatorial mathematics-Ramsey theory. From the proof of the six-person meeting problem, we once again see the application of the pigeon hole principle.

Look at the commodity prices of various supermarkets, especially the original price, special price, buy X and get Y ... and compare (calculate) to get the answer. Buy the cheapest = =

Common, how much is X gram, how much is Y gram, how much is Z gram (big package) to send small products and so on.

There are many promotional activities on the roadside (on TV), and there are also "fake promotions" (seemingly low prices, but in fact high prices)