geometry
How to derive the famous quadratic formula from pictures, just like the Babylonians did. 
Euclid's fourth axiom says that all right angles are equal. But isn't that obvious? 
This week would have been the 100th birthday of Martin Gardner, who is deservedly credited with turning on several generations of people worldwide to the pleasures of maths! To mark the occasion here are some favourite puzzles that, apart from being fun, also lead to some serious maths. 

The Mayan civilisation brought forth many great things — including this clever way of making a right angle. 
Maryam Mirzakhani is being honoured for her "rare combination of superb technical ability, bold ambition, farreaching vision, and deep curiosity". 
From a smile to a line drawing by Picasso, curves bring great beauty to our world. But how curvy is a curve? 
Free kicks will deliver much of the drama in the football world cup this summer. But how should strikers approach them and how does the design on the ball impact on its behaviour in flight? Maths can give us answers... 