Keats complained that a mathematical explanation of rainbows robs them of their magic, conquering "all mysteries by rule and line". But rainbow geometry is just as elegant as the rainbows themselves.

It's 21st of October and for puzzle lovers this can only mean one thing: the G4G Celebration of mind. This annual party celebrates the legacy of Martin Gardner, magician, writer and father of recreational maths, with mathemagical events in his honour happening all over the world.

This year's Nobel Prize in Physics was awarded for a discovery that proved Einstein wrong and right at the same time.

Convex or concave? It's a question we usually answer just by looking at something. It's convex if it bulges outwards, and concave if it bulges inwards. But when it comes to mathematical functions, things aren't that simple. A team of computer scientists from the Massachusetts Institute of Technology have recently shown that deciding whether a mathematical function is convex can be very hard indeed.

In the 1930s the logician Kurt Gödel showed that if you set out proper rules for mathematics, you lose the ability to decide whether certain statements are true or false. This is rather shocking and you may wonder why Gödel's result hasn't wiped out mathematics once and for all. The answer is that, initially at least, the unprovable statements logicians came up with were quite contrived. But are they about to enter mainstream mathematics?

Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way. But it's not quite like that. You can actually build different versions of maths in which statements are true or false depending on your preference. So is maths just a game in which we choose the rules to suit our purpose? Or is there a "correct" set of rules to use? We find out with the mathematician Hugh Woodin.

The human genome is represented by a sequence of 3 billion As, Cs, Gs, and Ts. With such large numbers, sequencing the entire genome of a complex organism isn't just a challenge in biochemistry. It's a logistical nightmare, which can only be solved with clever algorithms.