Maths can be many things depending on how we use it. Sometimes it can just be our servant, our slave – providing simple calculations to answer such routine questions as ‘Have I got enough money in my bank account to last to the next pay day?’ or ‘How long will it take us to drive to the coast?’ This is useful Maths, but, it has to be said, rather dull. At other times, Maths can be cleverer than we are - more like a magician’s hat, producing amazing rabbits that take us by complete surprise. This is Maths at its most magical and mysterious. Wizard Maths, you might call it – and altogether rather more exciting!

Consider, for example, the accidental discovery by the Scottish scientist James Clerk Maxwell of the speed of light - a classic case of serendipity. In the 1860’s, he was trying to devise a set of equations that would explain the laboratory results obtained by Michael Faraday and others during experiments linking electricity and magnetism. This was the new field of “electromagnetism”. As he wrestled with his equations, trying to make them work, Maxwell found that he was forced to add an extra term to make everything hang together. It was not so much that this special extra term was essential to explaining Faraday’s results, but more that it was needed to make Maxwell’s equations consistent, to keep them mathematically tidy, you might say – and the consequences of including it were truly profound. It led to the derivation of Maxwell’s wave equations that described the behaviour of electromagnetic waves, including the remarkable fact that these waves had a constant universal speed of 299,792,458 meters per second. And, of course, as you may well know, light is an electromagnetic wave. Are you getting that tingling feeling down your spine yet?

So here is our rabbit from the mathematical hat, a little piece of magic – that a physical phenomenon, the speed of light, could have been discovered entirely by the manipulation of a set of equations. Maxwell wasn’t looking for it – it leapt out at him, demanding his attention. I say 'little' piece of magic with my tongue firmly in my cheek. It wasn’t so much the number assigned to the speed of light that was so remarkable, but the fact that it held constant under all circumstances. It was ‘absolute’. This was one of the curiosities that set Einstein thinking in the early 1900s, leading to his formulation of the theory of special relativity. And, under the scrutiny of Einstein’s rigorous logic, other assumed absolutes, like space and time, came tumbling down like dominoes. So – some rabbit, you might say!

Another famous mathematician and physicist, Isaac Newton, once reputedly said ‘I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.’ Newton’s evocative quote gives us this sense that there is much more out there to find, more wonders to emerge from the magician’s hat – and that Maths is one of the keys to unlocking further mind-bending discoveries.

So, while we’re on the trail of the next piece of truly wizard Maths, here’s a simple little teaser for you to get the mental cogs turning. Why, when we differentiate the area of a circle (πr^{2}, where r is the radius) with respect to the radius do we get the circumference, 2πr? That's my mini-rabbit. If you can’t crack it, don’t let it drive you too crazy!