Pigeonhole principle

In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m > 0, then at least one container must contain more than one item. This theorem is exemplified in real life by truisms like "there must be at least two left gloves or two right gloves in a group of three gloves". It is an example of a counting argument, and despite seeming intuitive it can be used to demonstrate possibly unexpected results; for example, that two people in London have the same number of hairs on their heads.

The first formalization of the idea is believed to have been made by Peter Gustav Lejeune Dirichlet in 1834 under the name Schubfachprinzip ("drawer principle" or "shelf principle"). For this reason it is also commonly called Dirichlet's box principle or Dirichlet's drawer principle. This should not be confused with Dirichlet's principle, a term introduced by Riemann that refers to the minimum principle for harmonic functions.

The principle has several generalizations and can be stated in various ways. In a more quantified version: for natural numbers k and m, if n = km + 1 objects are distributed among m sets, then the pigeonhole principle asserts that at least one of the sets will contain at least k + 1 objects. For arbitrary n and m this generalizes to k + 1 = ⌊(n - 1)/m⌋ + 1, where ⌊...⌋ is the floor function.

Though the most straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence. To do so requires the formal statement of the pigeonhole principle, which is "there does not exist an injective function whose codomain is smaller than its domain". Advanced mathematical proofs like Siegel's lemma build upon this more general concept.

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