graph theory
Can we always find order in systems that are disordered? If so, just how large does a system have to be to contain a certain amount of order? 
How a cute 18th century puzzle laid the foundations for one of the most modern areas of maths: network theory. 
We've read the book. We've bought the Tshirt. And now, finally, here it is: the movie of one of our favourite maths problems 
How to make a hard problem easy by changing the way you look at it. 
The London Underground turns 150 today! It's probably the most famous rail network in the world and much of that fame is due to the iconic London Underground map. But what makes this map so special? 
Mathematicians and psychologists don't cross paths that often and when they do you wouldn't expect it to involve an (apparently) unassuming puzzle like the Tower of Hanoi. Yet, the puzzle holds fascination in both fields. 
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? 
The human brain faces a
difficult tradeoff. On the one hand it needs to be complex to ensure high performance, and on the other it needs to minimise "wiring cost" — the sum of the length of all the connections —
because communication over distance takes a lot of energy. It's a problem wellknown to computer scientists. And it seems that market driven human invention and natural selection have come up with similar solutions.

It is thought that the next great advances in biology and medicine will be discovered with mathematics. As biology stands on the brink of becoming a theoretical science, Thomas Fink asks if there is more to this collaboration than maths acting as biology's newest microscope. Will theoretical biology lead to new and exciting maths, just as theoretical physics did in the last two centuries? And is there a mathematically elegant story behind life? 
A new mathematical analysis of team tactics predicts a Spanish win in Sunday's FIFA World Cup final and also sheds some light on why England were trashed by Germany. 