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The origin of pigeon hole principle
In the daily life of pigeon hole principle, it is not difficult to find something with regularity as long as people pay a little attention. For example, if you put 10 apples in nine drawers, there must be two or more apples in one drawer. This is a simple truth that everyone can understand, called the pigeon hole principle or the pigeon cage principle (compare pigeons to apples and cages to drawers). The general principle of pigeon hole is: put n+ 1 apples into n drawers, and at least one drawer will hold two or more apples. Don't underestimate this universal and simple principle. Many interesting problems can be solved by pigeon hole principle. For example, among any 13 people, there must be two people born in the same month. Only 13 people need to be regarded as apples and 12 months as drawers, so the pigeon hole principle is concluded. For another example, given any five points in a square with a side length of 1, there must be two points, and the distance between them will not be greater than 1/2. Then the side lengths of the four small squares are all 1/2, and the distance between any two points in each small square will not be greater than the diagonal length of the big square 1/2. If five points are regarded as apples and four small squares as drawers, then from the pigeon hole principle, there must be two points in a small square, so the distance between these two points is not more than 1/2.
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