I have been getting into the puzzles section at the tail end of The Guardian newspaper which I get each week. Last week it asked you to prove that there are infinitely many different pairs of numbers (a,b) for which a+b=ab.
This is easy to prove but I admit I had to work at it while wandering around the golf course yesterday before the straightforward way of thinking about the issue presented itself. It is a few years since I have tried these sorts of mathematical recreational problems and I am rusty – as a school kid I went in maths competitions organised around the UNSW magazine Parabola which introduced advanced math thinking to school students. One problem that I agonised over for a day or so until I saw a clear solution was to prove that in any group of N people there are at least 2 people who have shaken hands with the same number of other people. I had a particular affection for such mathematical recreations that involved simple to state problems that involved real insight.
I remember reading in a biography of John Von Neumann that prior to WW2 the Hungarians held mathematics competitions in the parks of Budapest.
I worry a bit about is that I am so rarely forced to think things through carefully these days. University work can become routine because, as you age, you tend to draw on long-held intellectual capital rather than really taxing your brain. You might even be more efficient at solving problems by drawing on this capital but it does make you more machine-like.