There's no doubt that information is power, but could it be converted into physical energy you could heat a room with or run a machine on? In the 19th century James Clerk Maxwell invented a hypothetical being — a "demon" — that seemed to be able to do just that. The problem was that the little devil blatantly contravened the laws of physics. What is Maxwell's demon and how was it resolved?

A Gömböc is a strange thing. It wriggles and rolls around with an apparent will of its own. Until quite recently, no-one knew whether Gömböcs even existed. Even now, Gábor Domokos, one of their discoverers, reckons that in some sense they barely exists at all.

Geometric hallucinations are very common: people get them after taking drugs, following sensory deprivation, or even after rubbing their eyes. What can they tell us about how our brain works?

In 2004 three physicists decided to dabble in a field they knew little about. Within weeks they had developed a new technique that transforms weeks' worth of computer calculations into something that could be done on a single page in an hour. It's used in particle accelerators such as the LHC at CERN.

The natural logarithm is intimately related to the number e and that's how we learn about it at school. When it was first invented, though, people hadn't even heard of the number e and they weren't thinking about exponentiation either. How is that possible?

A 1 in 14 million chance to win the lottery, a 5% risk of cancer, a 50:50 chance of heads on a coin — we deal with probabilities all the time, but do they actually mean anything? We explore the philosophy of probability and ask whether the probabilities that come up in physics differ from those in every day life.

Are there objective chances in the world?

Why does time only ever move in one direction? We talk to philosophers of physics Jeremy Butterfield and David Wallace, as well as the eminent Roger Penrose about the puzzle time poses to physicists and what it has to do with the Big Bang and the second law of thermodynamics.

Agreeing to pay £50,000 for something worth £2 wouldn't win you any haggling competitions. In mathematics, however, a similar result can bring you international acclaim. This is the case with recent progress towards the famous twin prime conjecture.

Space is the stage on which physics happens. It's unaffected by what happens in it and it would still be there if everything in it disappeared. This is how we learn to think about space at school. But the idea is as novel as it is out-dated.