Puzzles
Six strangers at a party
At any party of six people, prove that there must be either three people who all know each other, or three who are all strangers to one another. Why does a party of only five people not guarantee this?
Reveal the answer
It's always true for six — a result known as R(3,3)=6 — but false for five, where a five-person 'friendship cycle' avoids both a mutual trio and a mutual-stranger trio. This is the founding case of Ramsey theory, from Frank Ramsey's 1930 paper 'On a Problem of Formal Logic,' later popularised as a 1953 Putnam Competition question.