How the ‘happy ending problem’ launched a new branch of math—and a romance

How the ‘happy ending problem’ launched a new branch of math—and a romance

A puzzle about dots on a page led to one of the most profound areas of modern math

In the early 1930s a group of bright university students held gatherings in Budapest to discuss math. Among them was Paul Erdős, an eccentric prodigy who would go on to become the most prolific mathematician in history . Other attendees included mathematicians George Szekeres and Esther Klein. One day Klein presented a puzzle to her friends: a deceptively simple question about dots scattered across a page. Unbeknownst to them, the puzzle would help launch a profound branch of modern math about order among chaos and spark a romance that would last the rest of Klein’s life.

Here’s what Klein presented to her pals: speckle five dots on a page wherever you like, as long as no three dots fall along the same straight line. Will four of those dots always form the corners of a four-sided shape with no dents or crossed sides? In mathematical terms, must some four of the five dots form a convex quadrilateral?

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With only four dots, it’s clearer to see that a convex quadrilateral might elude you, depending on where the dots fall. But the five-dot case changes things.

Grab a pencil and test it for yourself. Place five dots at random, with no three dots in the same line, and hunt for the four-cornered dentless shape. Then try to be clever by attempting to intentionally arrange the dots to deny yourself such a shape. You will fail every time. No matter how you place five dots, four of them will always form a convex quadrilateral. Klein proved this with an elegant argument.

Imagine that the dots protrude from the page to form pegs. If you stretch a rubber band around all of the pegs and let it snap taut against the outermost ones, it will trace a convex boundary because rubber bands don’t form spontaneous dents. There are three cases to consider: the band touches five pegs; the band touches four pegs with one trapped inside; or the band touches three pegs with two trapped inside. It can’t touch only two pegs unless all five lie in a straight line, which the rules forbid. If the band touches four pegs, then we’ve already found our convex quadrilateral (four sides) made of those four boundary points. If it touches five pegs, forming a convex pentagon, then you can lift the band off of any one of the pegs, and it will snap to the remaining four to form a convex quadrilateral. The tricky case occurs when the band touches only three points, forming a triangle with two interior points.

Does this three-point configuration always admit a convex quadrilateral? It does. Draw a straight line through the two interior points and extend it across the page. That line slices the plane into two halves, one of which will contain two corners of the outer triangle (the bottom half in the figure below).