Pigeonhole principle — where it appears
More things than boxes
If there are more objects than containers, some container holds two. That is the entire principle, it is impossible to disagree with, and it settles questions that look nothing like it.
Six people at a party
Among any six people, three are mutual acquaintances or three are mutual strangers. Five is not enough, and the arrangement that saves five is a pentagon. Beyond that the numbers become unknowable.
How close a fraction can get
Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting argumentExistence proofNonconstructiveDirichletComplete graphContinued fraction convergentGolden ratioGraphLiouville numberParityProbabilistic methodRamsey number