( Pigeonhole Principle )
This meme is taken from a scene in the Cohen brother’s 1998 comedy “The Big Lebowski”. During a game of bowling, Walter, in the picture, gets annoyed at the other characters constantly overstepping the line. Drawing a gun, he asks: “Am I the only around here who gives a shit about rules?”[ii]
Considering that there are roughly 7 billion people on earth, a positive answer seems highly unlikely. But it is possible to do better. We can know with certainty, i.e. prove, that the creator of the meme is not the only one. This is a simple and straightforward application of a fascinating, intuitive and yet powerful mathematical principle. It is usually called “pigeonhole principle” (for reasons to be explained below) or “Dirichlet’s principle”.
There exist many formulations of the Dirichlet’s principle, but a very simple one is the following: Suppose you have n holes (where n is a positive integer) and n+1 pigeons. Now, no matter how hard you try, it is impossible to fit all the pigeons into individual holes. There is at least one hole that contains two (or more). Similarly for hairs. The numbers vary of course, but an average blonde person is thought to have about 150’000 hairs on their head[v]. To be on the safe side, let us assume that the hairiest person on earth has 300’000 hairs. For ease of calculation, let us further assume that there are 7 billion people on earth. Then, at least two people will have the same number of hairs. Indeed, at least 23’333 people will do.
To demonstrate the truth of this rather obscure claim, suppose it is false; this means it is not the case that at least two people will have the same number of hairs. Say, we put the 7 billion people into a row, starting with the person of 0 hairs to our left and running up to the Guinness World Record Hairiest person of 300’000 hairs. So, person 1 has 0 hairs, person 2 has exactly one hair, etc., up to person 300’001, who holds the Guinness World Record. Now what about person 300’002? Remember she has to have 0,1,…,300’000 hairs (otherwise the World Record would be broken yet again!). But all those numbers of hairs are already taken by person 1 up to 300’001, so she necessarily has the same number of hairs as someone of them.
Of course, this is a rather silly application, but the principle can be generalised (in semi-mathematical terms): If you have two sets S and T, where S contains more elements than T, there is no way of assigning a single element in T to each element in S.
So far, we have only considered finite sets, but what happens if either S or T (or both) are infinite? The pigeonhole principle still applies, and has implicitly been put to use in some of G. Cantor’s (1845-1918) most beautiful proofs, notably in his diagonal argument.
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Posted By F. Sheikh