Journey to distant islands to discover if topology can overcome topography and bring peace to rival towns.

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?

One of the most surprising things about mathematics is its many unsolved mysteries. Mathematics is far from "done and dusted", and Steve Humble shows us how we can come up with some mathematical mysteries of our own.

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.

Insurance companies offer protection against rare but catastrophic events like hurricanes or earthquakes. But how do they work out the financial risks associated to these disasters? Shane Latchman investigates.

Florence Nightingale died a hundred years ago, in August 1910. She survives in our imaginations as an inspired nurse, who cared passionately for injured and dying soldiers during the Crimean war, and then radically reformed professional nursing as a result of the horrors she witnessed. But the "lady with the lamp" was also a pioneering and passionate statistician. She understood the influential role of statistics and used them to support her convictions. So to commemorate her on the centenary of her death, we'll have a look at her life and work as a statistician.