My younger brother is often vocal about his dislike of maths. He never paid much interest in it at school and when he did he would complain constantly about how hard it was. However, come April and May each year, his mathematical ability would suddenly become exceptional, and unfortunately not because he was studying for his GCSEs… Any football fan will know that this is the time of year when the leagues will finish and hence a very tense time if your team is close to the top (or relegation...) On the final day there are several games happening simultaneously with 3 points on offer for the winners, 1 for a draw and also goal-difference to account for when points are tied. The mental arithmetic required to calculate your teams finish in real time is not easy and yet for many football fans, such as my brother, it’s easy and obvious.

The number crunching required to follow points in a league table is a clear example of mathematics in the game – however, there are many other places it can be found! Here are 3 of the most interesting:

1. The shape of a perfect football

A standard football is based on the shape of a truncated icosahedron, a shape with 12 pentagonal and 20 hexagonal faces, 60 vertices and 90 edges. The shape was first defined by ancient Greek mathematician Archimedes; however, it wasn’t used as a football until more than 2000 years later!

2. Winning penalty shoot-outs with statistics!

There are several models that can be used during a penalty shoot-out to predict the results in real time. One example is using the Markov chain, named after its creator Russian mathematician Andrey Markov. It is a series of events where each step depends solely on the previous one. It might not be massively useful for a goalkeeper or penalty taker in the moment, but statistics like this are the tools that make betting companies a lot of money!

3. Passing patterns and graph theory

The way the game is played has adapted over the years and recently most teams focus on a possession- and passing-based game plan. Teams will do anything to improve by that 1% including considering mathematical modelling techniques provided by graph theory. Using mathematical theorems and considering the players as nodes of a graph, the number of paths and which ones are optimal can be determined and then translated back to practical solutions on the pitch!