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Unsolved Mathematical Mysteries: Happy End Problem Kevin Hartnett

In this section, we accept articles about fascinating mathematical problems that have not yet been solved. We ask the authors to describe the problem so that a middle school student could understand it. The piece should also reference the progress that has been made by the mathematics community in solving the problem.

The nickname of this problem "Happy End" comes from not mathematical reasons, but the fact that Esther Klien and George Szekeres worked collaboratively on the problem in the mid-1930's which resulted in their marriage. Their friend Paul Erdos also contributed to working on this problem. In 1933, a 23-year-old Esther, who lived in Budapest, Hungary, brought home a puzzle that she presented to her friends, Paul and George. Here's the puzzle: 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 (Figure 1). Esther derived a proof prior to presenting the puzzle to her friends and they were able to show that Esther's proof was true. As mathemati- cians always do, they decided to check if the proof is generalizable, meaning if this works for five points that guarantee forming a quadrilateral by connecting four of them, how many points are needed to guarantee a formation of a convex poly- gon with any number of sides (e.g. pentagon, hexa- gon, etc.)? Paul and George did solve this problem for triangles, quadrilaterals, and pentagons. They were

able to prove that it took three points to guarantee construction of a convex triangle, five points to construct a convex quadrilateral, and nine points to guarantee a construction of a convex pentagon. They also proposed a formula for counting the number of necessary points: 2 (n-2) + 1, where n is the number of sides of the convex polygon. How- ever, this was a conjecture and not a proof. The problem seems simple enough to un- derstand; many mathematicians worked on it for several decades. "Yet as the decades passed, math- ematicians made virtually no progress in proving the conjecture. (The only other shape whose result is known is a hexagon, which requires at least 17 points, as proved by Szekeres and Lindsay Peters in 2006.) Now, in work recently published in the Journal of the American Mathematical Socie- ty , Andrew Suk of the University of Illinois, Chica- go, provides nearly decisive evidence that the intui- tion that guided Erdos and Szekeres more than 80 years ago was correct" (Hartnett, 2017). There is still work to be done in solving this problem. We encourage you to introduce this con- jecture to your students, and maybe one of them will provide the solution someday and will get to publish their story in the Virginia Mathematics Teacher . This is a summary of the article titled "A Puzzle of Clever Connections Nears a Happy End" by Kevin Hartnett and is printed with permission from Quanta Magazine, May 2017 issue. The link to the original article can be found at https://www.quantamagazine.org/a-puzzle-of- clever-connections-nears-a-happy-end-20170530/.

Figure 1. The shape on the left is a convex polygon. The shape on the right is a concave (non-convex) polygon.

Virginia Mathematics Teacher vol. 44, no. 1

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