Marcus du Sautoy: Behind Every Great Game…

A lot of people love games. A lot of people hate maths. But what if they realised that actually the two are intimately related. Bubbling underneath every great game there is often a fascinating bit of mathematics, and if you understand that maths then it can give you an edge in winning the game.

Many games involve dice so an understanding of probability will allow you to make the best choice of moves in the game. Take Monopoly. The most visited square on the board is the jail square because there are so many ways to go to jail. But that’s not much help because you can’t buy the jail. But from jail the most common throw of two dice is a score of 7 because there are so many ways to make 7 from two dice. (6,1), (5,2), (4,3), (3,4), (2,5) and (1,6). Similarly scores of 6 and 8 are very common. These throws are going to put you in the orange region of properties. So my mathematical tip for winning monopoly: buy up the orange properties, stack them with hotels and bankrupt the other players as they leave jail and end up on your land.

The game of Risk also involves dice but the mathematics involved turned out to be much trickier to analyse. This is a game about world domination and you use dice to attack your opponent’s countries. The defending county uses up to two dice and the attacker three dice. You compare the dice to see whose armies are defeated. The first analysis of the game seemed to show that the advantage lay with the defending country. But that ignored an interesting dependence in the comparison that had been missed. When this was taken into account the mathematics revealed that the advantage was actually with the attacker.

As well as helping you to play games more successfully, mathematics is also important to those creating new games. Getting the maths right in the way points are scored for example in a game like Ticket to Ride is key to creating a game that has the perfect balance between risk and reward.

There is one game I’ve always enjoyed playing with my kids that I discovered has a beautiful piece of mathematical geometry that makes the game work. Called Dobble, the game consists of 55 cards and on each card are eight icons. If you choose two cards at random there will always be one icon in common on both cards. The game is to spot which icon it is and claim the cards.

But how is it possible to make a game that ensures that any two cards will have an icon in common? It feels like magic. But actually mathematics is the secret ingredient. In particular a beautiful piece of geometry. The cards are actually points in this geometry. And in any geometry if you take two points you can draw a line between those points. The line corresponds to the icon. The geometry hiding behind Dobble is called the projective plane of order seven. It is the mathematics that makes this game work.

Games are really just a way of playing maths. If more people realised this then maybe maths could be as popular as Monopoly, although hopefully it won’t be the cause of so many family arguments.

Marcus du Sautoy is Professor of Mathematics at the University of Oxford and author of Around the World in 80 Games (Fourth Estate).