Volume 41

2005 This issue of Parabola celebrates some of the mathematics of George and Esther Szekeres.

In the October 1964 issue of Parabola, the article on the Four Colour Problem called your attention to the existence of numerous unsolved mathematical problems which can be stated in quite simple non-technical terms.

The theory of combinatorial configurations abounds in unsolved problems, some of which can be stated in simple non-technical terms. One of the most famous among these is the following problem due to the French mathematician, J. Hadamard.

How many people must attend a party before you are sure that you can find either three people who all know each other, or three people who do not know each other? This is a question in an area called Ramsey Theory.

George Szekeres made many contributions to various areas of mathematics. In combinatorics, one of his most important contributions was to ask a question which we still don’t know how to answer.

We need counting in our daily life. Collecting cash from the supermarket, checking the bill at a restaurant, counting the number of place settings at a dinner party.... This sort of counting is pretty easy because we can count one by one.

In my last column, I showed the way in which the solution of cubic equations led to the introduction of complex numbers. Here I will concentrate on the solution of cubic equations themselves.

Q1191. In the triangle $ABC, M$ is the midpoint of $BC.$ Points $X$ and $AB$ and $Y$ on $AC$ are such that $XY\ \Vert\ BC.$  Show that $BY$ and $CX$ intersect at a point $P$ on $AM.$

Q1181. Consider the following set of linear equations \begin{eqnarray*}
x+2y+z&=&1\\
-2x+\lambda y-2z&=&-2\\
2x +6y+ 2\lambda z&=&3
\end{eqnarray*}