Sometimes a piece of maths can be so neat and elegant, it makes you want to shout "eureka!" even if you haven't produced it yourself. One of our favourite examples is the art gallery problem.
The paths of billiard balls on a table can be long and complicated. To understand them mathematicians use a beautiful trick, turning tables into surfaces.
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.