Posted by: thumpasery | December 4, 2008

The Pigeonhole Principle

The pigeonhole principle is a relatively simple counting argument that can be used to demonstrate some rather counterintuitive results. It is a simple principle:

If n>m, and n pigeons are placed in m pigeonholes, there must be at least one hole with more than one pigeon.  This principle relates in obvious ways to our study of isomorphisms and the use of bijectivity in various proofs. An equivalent mathematical statement that is more formal is that there is a 1-1 correspondence between two finite sets A and B if and only if the order of A equals the order of B.

More importantly, there are some interesting problems about/applications of the pigeonhole principle. Here is one good example:

1. Take a chessboard with the opposite corner squares missing. Is it possible to cover the entire board with domino pieces of length two squares?

The opposite corners are both either black or both white. Thus, either way, we have two more of one color remaining. But each domino covers one black and one white square. Thus there is a 1 to 1 correspondence between the white and black squares, established by the dominoes covering the chessboard. But removing the opposite corners will ensure that a 1-1 correspondence is impossible. So it is not possible to cover the entire board with such dominoes.

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