Counting argument
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.
The door that was not opened
Three doors, one prize, a host who opens a losing door and offers a swap. Switching wins two times in three, and the reason is not about doors — it is about what the host was allowed to do.
Named alongside it
The objects these essays reach for when they reach for this one.
Existence proofNonconstructivePigeonhole principleAreaBayes' theoremComplete graphConditional probabilityDirichletGraphIndependenceLikelihoodMonty hall