Thursday, February 23, 2006

Maths problems

As a kid at school I solved maths problems in the UNSW school mathematics magazine Parabola which is still published today. Years later I tried to get my own school-age kids interested in this magazine without success.

When I first went to Macquarie University, as an undergraduate, I didn't enjoy the way mathematics was taught. But, to digress, I did have a fascinating Hungarian tutor in first-year algebra and calculus, a Mrs Esther Szekeres. She told me I had talent but I was really better-read than able in maths. Ester, it turns out, was married to Professor George Szekeres, the foundation professor of pure mathematics at UNSW and a leading Australian mathematician of his day. Esther was mainly interested in geometry and was a very good mathematician. Both were active in promoting the Parabola magazine and, a few years ago, I was told Esther, in her nineties, was still active promoting mathematics in schools and Parabola.

The reason I am writing this is that I just found out today, that after nearly 70 years of marriage, George and Esther Szekeres, both died within an hour of each other at the end of 2005. George was 94 and Esther 95. A tribute is here.

The Szekeres were Jews who fled Hungary in the 1930s with people like John Von Neumann who they knew. George was part of a Budapest group of brilliant students, including Paul Erdos and Paul Turan. One of the problems the group considered was proposed by Esther and solved by George to declare his suit. Erdos called it the "Happy Ending Problem", as it led to the pair marrying in 1937.

I am fascinated by the history of Hungary and of the Jews who fled Hungary with the rise of Nazism. I have read biographies of von Neumann and Erdos showing that in Hungary, during this time, they held mathematics competitions in the parks! Intelligence and thinking ability were things to be celebrated. What a difference from the mass cultures of today! I am also interested in the fact that, while some American universities were anti-Semitic at this time, the smart ones like Princeton University were not. They were very appreciative of ultra-intelligent Jews like Einstein and Von Newmann (and of ultra-intelligent gentiles like John Nash). The lack of prejudice was moral and at the same time advanced their self-interest.

A math problem that I remember solving in Parabola years ago as a school student was the following:

'Prove that in any group of n>1 people at least 2 have shaken hands with the same number of people'.

It is an ingenious problem because it is so simple to state. I had to think about it for days until I saw the solution was so simple. Can you prove it? I am positive that Esther could.

1 comment:

hc said...

Proof is by contradiction. Suppose everone shakes hands with a different number of other people.

The maximum range of numbers of people one person can shake hands with ranges from 0 to (n-1) assuming no-one shakes hands with anyone else twice. Order them from least to highest numbers of other people they have shaken hands with.

One person (#1) shakes hands with nobody else.
One person (#2) shakes hands with one other person.

The nth person (#n) shakes hands with (n-1) other people.

But this contradicts the hypothesis that person #1 shook hands with nobody else.

Hence it cannot be true that everyone shakes hands with a different number of other people.