The simplest form of the pigeonhole principle says
If n+1 letters are put into n pigeonholes, then some pigeonhole must contain more than one letter.
This result can be quantified, extends naturally to Ramsey’s Theorem, and lies at the base of diophantine approximation and other things.
But it was borne home to me last week, when I chaired a public lecture by Colva Roney-Dougal on the mathematics of communication, that there are in fact two kinds of mathematicians: those who put letters into pigeonholes, and those who put pigeons into pigeonholes.
Which kind are you?
It seems vitally important not to mix the two kinds!