In combinatorics, the pigeonhole principle states that if or more pigeons are placed into holes, one hole must contain two or more pigeons. This seemingly trivial statement may be used with remarkable creativity to generate striking counting arguments; the principle bears particular fruit in Olympiad settings.
In older texts, the principle may be referred to as the Dirichlet box principle. A more useful phrasing of the problem through balls and boxes is that if balls are to be placed in boxes and , then at least one box must contain more than one ball.
The pigeonhole principle may be demonstrated intuitively by the following argument: assume that there exists a way to place balls into boxes, where such that all boxes contain at most one ball.
Let how many balls each box contains. Our condition that all boxes contain at most one ball implies that for all , so However, we know that there are a total of balls across all our boxes, so this sum must equal : Therefore, . This contradicts our condition that . Therefore, our assumption is incorrect; at least one box must contain two or more balls, as required.
In formal terms, the pigeonhole principle is a consequence of how we define one set to be larger than another set.
Let be a set of balls and be a set of boxes such that , or . The definition of this is that there exists a surjective mapping from to , but never a bijection. In other words, there exists a way to map every ball of to every box of , but not one that maps every ball to a distinct box. This is just a different way to phrase the fact that one box must contain more than one ball, which is the pigeonhole principle.
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