There's a popular game at fetes where you have to guess how many sweets there are in a jar. Making an estimate like this can be an enjoyable intellectual exercise. In a similar, but more complex, vein, the Cambridge entrance exams and the general exam sat by science students there used to feature fun little challenges. Two still stick in my head. One was A violin plays the A above middle C. Estimate the tension in the string. And the other: Estimate the distance from the North Pole to the Equator around a great circle. Under exam conditions you just had to use your brain to achieve a result - challenging but fun. (Incidentally, although most people fiddled around with geometric calculations, the second one has a very quick way of coming to a surprisingly accurate answer. Solution at the bottom of the post.) Now there's a blog dedicated to this kind of mental exercise - and it has a competition at the moment too. Run by Aaron Santos, the blog, A Diary of Numbers features c...

As someone who has written about infinity (and good fun it is too), I quite often pick up on big numbers. Regular commenter on this blog Ian Campbell kindly pointed out to me a piece on the BBC news site about the number pi being determined to 2.7 trillion digits . Many people will be wondering, 'what's the point?' The news article gives an excuse, but really this is justification after the fact. You might as well say that you can justify trainspotting because it provides useful statistics on train movements. Calculating pi to n digits is the trainspotting of the mathematical world. It's not particularly clever - anyone can do it given a simple algorithm and enough time - and it has no great value. It's just something to tick off. However, I don't want to put down Fabrice Bellard. There is no doubt it's an achievement. Why should everything in science and maths have a use? What's the use of a poem or a painting? Let him get on with it, I say. And g...