## Chapter 44 Happy Ending Problem: A Play

###### Project by: *Alvin Ho*, *Abhay Sahni*, and *Yifan Zuo*.

\(\textbf{Summary:}\) Our project was inspired by a well–known story about the Happy Ending Problem.

The story involves three friends, Paul Erdős, Esther Klein, and George Szekeres.

When the three friends were undergraduate students they would often discuss mathematical problems. The story goes that Esther proposed the following problem: Among any five points in general position in the Euclidean plane, it is always possible to select four points that form the vertices of a convex quadrilateral.

This question led to the famous result now known as the Erdős–Szekeres theorem. The plot of the story includes the fact that George was in love with Esther. This was the reason that George did everything he could to resolve the general statement of the Happy Ending Theorem: Given any positive integer \(n\text{,}\) there exists a number \(K(n)\) such that among any \(K(n)\) points in general position, it is possible to select \(n\) points that form the vertices of a convex \(n\)–gon.

Esther and George got married and, like in a fairy tale, lived happily together for many years.

Inspired by this story we have created our own story about three students who live and study in the present time. One of them is a math wiz, the second is in love with the third one. And there is a mathematical problem to be solved! As in the George and Esther's story, solving the problem will bring the happy ending to our story as well.

Here is our video: