The maths of the World Cup

The FIFA World Cup starts this week and around the world millions of people are about to become football-obsessed.

But naturally, we are also interested in the maths underpinning the game. We have collected together some of our favourite articles from the last few years to help get the ball rolling (sorry) for the next few weeks of football-madness. We've got explainers for everything about the beautiful game from penalties to the stadiums to the ball itself.

Penalties, free kicks, and goals

The beautiful game (theory) of why penalty takers ignore statistics

Why do penalty takers not heed the advice of statisticians? We find out with a little game theory.

Hitting the spot

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

Making penalties fairer

Is the proposed ABBA rule for penalty shootouts really fairer than the existing rule? Maths shows that it is, and also suggests another, more subtle rule.

Blink and you'll miss it: The free kick in football (part I)

Free kicks deliver much drama. But how should strikers approach them and how does the design on the ball matter?

Blink and you'll miss it: The free kick in football (part II)

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.

When is a goal not a goal?

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?

About footballers

Where are England footballers born?

Which region of England produces the most England players?

Balls an stadiums

Stadium maths

In this podcast Paul Shepherd tells us about the maths of football stadiums and why his work required him to listen to Belgian techno.

A fly walks round a football

What makes a perfect football? 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.

If you can't bend it, model it!

Learn about the aerodynamics of footballs and perfect your free kick.