What's the best spot to aim a penalty at?

Marcus du Sautoy talks about football, cryptography, and numbers.

Three steps for ensuring penalty success.

In the first part of this article we let maths set the scene for a free kick. Now we continue the drama, tracing the trajectory of the ball throughout the milliseconds it takes it to reach the goal line.

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...

Remember Frank Lampard's disallowed goal in the 2010 World Cup match against Germany? The ball hit the crossbar, landed well behind the line but then bounced out again. And it all happened too quickly for the ref to spot it was a goal. How these kind of (non)-goals happen and what can we do about them?

What makes a perfect football? Anyone who plays or simply watches the game could quickly list the qualities. The ball must be round, retain its shape, be bouncy but not too lively and, most importantly, be capable of impressive speeds. We find out that this last point is all down to the ball's surface, the most prized research goal in ball design.

This is the second part of our new column on risk and uncertainty. David Spiegelhalter, Winton Professor for the Public Understanding of Risk at the University of Cambridge, continues examining league tables using the Premier League as an example. Find out just how much — or how little — these simple rankings can tell you.

• Plus new writers award - Maths is the language of the universe, so what have you got to say?
• Reader's corner - 6174 and other mysterious numbers.
• World Cup maths - How Plus can help you with your football.