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Pigeonhole principle

The pigeonhole principle states that if n pigeons are put into m pigeonholes, and n > m, then at least one pigeonhole must contain more than one pigeon. Another way of stating this would be that m holes can hold at most m objects with one object to a hole, then adding another object will force you to reuse one of the holes. Although this may seem a trivial observation, it can be used to demonstrate unexpected results. The first statement of the principle is believed to have been made by Dirichlet in 1834 under the name Schubfachprinzip ("drawer principle").

For example, there must be at least two people in London with the same number of hairs on their heads. Demonstration: a typical head of hair has around 150,000 hairs. It is reasonable to assume that no-one has more than 1,000,000 hairs on their head. There are more than 1,000,000 people in London. If we assign a pigeonhole for each number of hairs on a head, and assign people to the pigeonhole with their number of hairs on it, there must be two people with the same number of hairs on their heads.

Another practical example of the pigeonhole principle involves the situation when there are five people who want to play softball, but only four teams. This wouldn't be a problem except that each of the five refuses to play on a team with any of the other 4. To prove that there is no way for all five people to play softball, the pigeonhole principle says that it is impossible to divide five people among four teams without putting two of the people on the same team. Since they refuse to play on the same team, at most four of the people will be able to play.

A generalized version of this principle states that, if n discrete objects are to be allocated to m containers, then at least one container must hold no fewer than objects, where

denotes the ceiling function.

The pigeonhole principle is an example of a counting argument which can be applied to many more serious problems, including ones involving infinite sets that cannot be put into one-to-one correspondence. In diophantine approximation the quantitative application of the principle to the existence of integer solutions of a system of linear equations goes under the name of Siegel's lemma.

See also

cardinal number
Ramsey's theorem