One would be hard put to ﬁnd a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery — and more totally useless — than the perfect numbers.
— Martin Gardner
There are countless ways to classify integers. Happy, perfect, friendly, sociable, abundant, extravagant, cute, interesting, frugal, deficient, hungry, undulating, weird, vampire… the list goes on. But how useful are such classifications, beyond their inherent interestingness, and as a hook to get people into number theory?
A classic maths puzzle involves a line of one hundred prisoners, who have each been given a black or white hat by their nefarious captor, and must each correctly shout out the colour of their hat to win freedom. The twist is that the prisoners don’t know the colour of their own hat, and though they can see the colours of the hats in front of them, they don’t know many of each colour there are overall. They can confer on a strategy beforehand, and the aim is to get as many of them to correctly identify their hat colour as possible. You can find a full explanation here (and in many other places!)
There are several ‘sequels’ to this puzzle, some involving an infinite number of prisoners and requiring the axiom of choice to solve. This post is about a nice variation on the theme that I heard about at a recent MathsJam. It can (just about) be solved without knowledge of higher mathematics, and though it seems impossible at first glance, the prisoners in this situation can in fact save themselves with 100% certainty.
Science Showoff is a monthly night which takes place in a pub in London, and features acts from all areas of science, who each have 9 minutes to perform an act – a science demo, a routine, songs, experiments – anything entertaining or fun. Having tried a little bit of the short-set, trying-to-be-funny type of science communication involved in Bright Club (a similar venture, giving researchers the chance to try stand-up comedy, which started in London and has now spread all over the country), I thought it would be good to give it another go – in fact, Science Showoff was recommended to me by someone who saw my Bright Club set in Manchester. I had prepared an 8-minute piece about Fibonacci numbers to perform in Manchester, inspired by my artist friend’s admission that she didn’t see how maths could be interesting in the same way as art; she wasn’t there to watch, but I went down well (and ran horribly over time). So I decided to reprise my set at Science Showoff in February 2012 – and this time it would be the right length, and would be new and improved with all the best jokes left in and the duds taken out.