The 'happy ending problem'

Thursday 7th May 2020

This problem was so-called by the mathematician Paul Erdos because two of his friends who worked on it, George Szekers and Esther Klein, fell in love and got married!

It involves investigating how many non-colinear points are needed so that convex polygons of different sizes can be drawn. A convex polygon of size 3 can be drawn from any 3 non-colinear points:

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We will always get a triangle, which could be said to be a convex polygon. We can say \(n(3)=3\).

However, what happens if we want a higher order polygon? What about \(n(4)\)?

If we have 4 points, we do not always get a convex polygon:

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We therefore know that \(n(4)\) does not equal \(4\). Below are some questions to try:

Question 1: Show that \(n(4)=5\) and investigate to find \(n(5)\) and \(n(6)\).

\(n(5)=9\) and \(n(6)=17\)

Question 2: Find a formula that will satisfy \(n(3)\) to \(n(6)\).

\(1+2^{n-2}\)

Erdos and Szekeres were unable to find the general result for \(n>6\). They proved that you needed at least the answer for \(n(3)\) to \(n(6)\), and that there will always be a convex \(n\)-sided polygon if there are enough points, i.e., the answer is finite, but they could not prove how much is enough.

This problem illustrates a phenomenon studied in an area of maths called Ramsey Theory which states that, if a system is large enough, i.e., has enough points, then it will contain some ‘order’ such as convex figures, even if the system as a whole is disordered.

Acknowledgements: +plus magazine, Wikipedia.

By Ann-Marie Newton

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