Pigeonhole principle

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In mathematics and computer science, the pigeonhole principle states that if n items are put into m pigeonholes with n > m, then at least one pigeonhole 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.

The first formalization of the idea is believed to have been made by Johann 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. In Russian and some other languages, it is contracted to simply "Dirichlet principle"—a name that could also refer to the minimum principle for harmonic functions. The original name is still in use in Italian ("principio dei cassetti") and German ("Schubfachprinzip").

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 on finite sets whose codomain is smaller than its domain". Advanced mathematical proofs like Siegel's lemma build upon this more general concept.

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Examples

Softball team

Imagine five people who want to play softball (n = 5 items), with a limitation of only four softball teams (m = 4 holes) to choose from. A further limitation is imposed in the form of each of the five refusing to play on a team with any of the other four players. It is impossible to divide five people among four teams without putting two of the people on the same team, and since they refuse to play on the same team, by asserting the pigeonhole principle it is easily deducible that at most four of the five possible players will be able to play.

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