At any party of six people, one group of three is unavoidable
At a party of six people, any two either know each other or are strangers. Prove that there must always be at least three people who all know each other, or three people who are all mutual strangers — no matter how the acquaintances are arranged.
Reveal the answer
It's always true, and six is the smallest number where it's guaranteed. Pick any one person: among the other five, they either know at least three or don't know at least three of them (pigeonhole). If they know three others, either two of those three know each other — completing a trio with the first person — or none of them do, making a mutual-stranger trio on their own. This is the smallest case of Ramsey's theorem, built on Frank P. Ramsey's 1930 logic paper, and it appeared as a Putnam Competition problem in 1953.