Puzzles
Any Five Points Hide a Convex Quadrilateral
In 1933, Esther Klein posed a puzzle to her friends in Budapest: take any five points on a plane, with no three of them in a straight line. She claimed you can always pick four of those five that form the corners of a convex quadrilateral, no matter how the points are scattered. Can you see why that has to be true?
Reveal the answer
Yes, it's always possible — provable by checking the few ways five points can be arranged (their convex hull is a pentagon, a quadrilateral, or a triangle with two points inside, and each case yields four points forming a convex quadrilateral). Paul Erdos and George Szekeres proved the general theorem in 1935; Erdos dubbed it the 'Happy Ending problem' because working on it led Klein and Szekeres to fall in love and marry.