How does the drawer principle work?

1. Know the number of drawers and at least the number (of the same kind), when finding objects: the number of objects = (at least number -1) the number of drawers + 1. When at least the number is 2, the number of objects = the number of drawers + 1.

2. Principle 1: If you put more than N+1 objects in N drawers, there will be no less than two things in at least one drawer.

3. Principle 2: Put more than mn (m times n) + 1 (n is not 0) objects into n drawers, then at least one drawer has no less than (m+1) objects.

4. Principle 3: If you put an infinite number of objects into n drawers, there will be at least one drawer with infinite objects.

Hail 1. Put three apples in two drawers, and there must be at least two apples in one drawer.

2. The common form of the drawer principle is to put all the objects of N+K(K 1) into N drawers in any way, and there must be at least two objects in a drawer.

3. Second, put all the mn+k(k 1) objects into n drawers in any way, and there must be at least m+1 objects in a drawer.

4. Three, put m1 + m2 + ....+mn+k(k 1) objects are all placed in n drawers in any way, then at least m1+1 objects are placed in one drawer, or at least m2+1 objects are placed in the second drawer,......Or put at least mn+1 object four in the nth drawer, and put all m objects in n drawers in any way, there are two cases: When n|m (n|m stands for n divisible m), there must be at least one object high band in a drawer; When n is not divisible by m, there must be a drawer with at least 1 object in it ([x] denotes the maximum integer not exceeding x).

The drawer principle can be explained as any natural number in which at least two numbers are multiples of the difference of yes. First of all, we need to understand this rule: if the remainder of the division of two natural numbers is the same, then the difference between the two natural numbers is a multiple.

And the remainder of any natural number divided, according to this situation, the natural number can be divided into classes, and this type is the "drawer" we want to make. We think of numbers as "apples", and according to the drawer principle, there must be at least one number in a drawer. In other words, natural numbers are divided into classes, and at least two of them are of the same class.

Since they are of the same class, the remainder of the two numbers being divided must be the same. Therefore, for any natural number, there must be at least one multiple of the difference between the natural numbers.

If you put m elements in n drawers, there will be at least [(m-1) n]+1 element in one of the drawers.

A more general formulation of the drawer principle is:

If you put more than kn+1 into n empty drawers (k is a positive integer), then there must be at least k+1 in one drawer. ”

Using the above principle, it is easy to prove: "In any 7 integers, at least 3 numbers are multiples of 3." "Because there are only three possible remainders when any integer is divided by 3, at least three of the seven integers divided by 3 give the same remainder, i.e., the difference between them is a multiple of 3.

Extended Information: How to construct a drawer:

The core of using the drawer principle is to analyze which is the object and which is the drawer in the problem. For example, if there are 12 genitals, then at least one of the 37 people is not less than 4 people.

At this time, the genus is regarded as 12 drawers, then there are 37 12 in a drawer, that is, 3 surplus 1, the remainder is not considered, but the whole number is considered upward, so here is 3+1=4 people, but it should be noted here that the previous remainder 1 is not the same as the 1 added here.

Therefore, in the problem, the more party is the object, and the less party is the drawer, for example, the 12 aspects in the above problem are the corresponding drawers, and the 37 people are the corresponding objects, because 37 is more than 12 <>