The three young friends who devised the “happy ending” problem became some of the most influential mathematicians of the 20th century, but were never able to solve their own puzzle

Kevin Hartnett in Quanta:

One measure of a good math problem is that, in trying to solve it, you will make some unexpected discoveries. Such was Esther Klein’s experience in 1933.

At the time, Klein was 23 years old and living in her hometown of Budapest, Hungary. One day she brought a puzzle to two of her friends, Paul Erdős and George Szekeres: Given five points, and assuming no three fall exactly on a line, prove that it is always possible to form a convex quadrilateral — a four-sided shape that’s never indented (meaning that, as you travel around it, you make either all left turns or all right turns).

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Erdős and Szekeres eventually found a way to show that Klein’s statement was true (she had worked out the proof before bringing it to them), and it got them thinking: If five points are enough to guarantee that you can always connect four to form this kind of quadrilateral, how many points are needed to guarantee that you can form this same kind of shape with five sides, or 11 sides, or any number of sides?

By 1935 Erdős and Szekeres had solved this problem for shapes with three, four and five sides. They knew it took three points to guarantee you could construct a convex triangle, five points to guarantee a convex quadrilateral, and nine points to guarantee a convex pentagon.

More here.