# It's a game of two equations, my son

9th July 2004 at 01:00
As pupils around the country put their pens down at the end of this summer's GCSE and A-level maths exams, many will breathe a huge sigh of relief. Time to celebrate calculating their last sines and cosines and solving their final quadratic equations. At last they can forget maths and immerse themselves in a summer of sporting drama, from the thrills of Euro 2004 to the delights of Wimbledon, from England's one-day cricket series to the Olympics.

It may then come as a shock to them to learn that maths is an important player in the world of sport. The trajectory of a football or a tennis ball depends on equations that mathematicians calculated centuries ago. Although it was ballistics rather than ball skills that scientists were trying to understand, the same equations apply to both a cannonball and a football.

It is probably stretching the imagination to believe that Beckham is calculating the necessary angles as he stands, hands on hips, ready to take a trademark free-kick. But Sven might be advised to add some mathematicians to his backroom team in future tournaments to avoid the embarrassment of seeing another penalty fly over the bar.

In the US, people are obsessed by the statistics of their favourite sports stars. The famous baseball player Babe Ruth had a career slugging average of .690; Sid Luckman of the Chicago Bears had a touchdown percentage of 13.9. Statistics may not be quite so popular on this side of the Atlantic but they play a crucial role in dealing with another significant player in British sport - the rain.

Every time a one-day cricket match is rained off before the end of 40 overs, it is the equations of mathematics that determine who has won the game. In the past, before mathematicians got involved, if your cricket team scored 150 runs after 30 overs, it was assumed that your total score at the end of 40 overs would have been 200 runs. But this calculation did not take into account how many wickets you might still have left, or the fact that teams tend to hit out at the end of matches.

So two mathematicians, Duckworth and Lewis, did a statistical analysis of one-day cricket matches and produced an equation which modelled much more accurately the story of a cricket match if the rain hadn't intervened.

Not understanding the maths can have unfortunate consequences: South Africa went out of the last world cup cricket tournament because they miscalculated the score required to win their match when the rain came down.

Maths is also important in determining how a game should be scored. Prior to 1981 the football league awarded only two points for a win and shared the two points if there was a draw. This meant that with 20 teams playing 38 games, the total number of points at the end of the season would always be 760.

But in order to spice up the league and encourage teams to play less defensively, it was decided it should be three points for a win and one each for a draw. Suddenly it was no longer clear how many points would be awarded by the end of the season. It could be anywhere between 760 points and 1,140 points.

This uncertainty makes it hard to predict mid-season whether a team still has a theoretical chance of winning the Premiership. If Bolton, say, are mid-table after 19 games, is it possible to arrange the results of the remaining matches for all the teams in the Premiership to leave them champions?

In the old system of two points for a win, mathematicians knew a quick way to tell if Bolton could win the league. But with the complexity of three points for a win and a total of two for a draw, assessing Bolton's chances becomes very difficult.

Indeed the Premiership problem is so deep that one businessman has offered a million dollars to anyone who can solve it.

Maybe fewer pupils would be so happy at the prospect of dropping maths this summer if they realised how powerful equations can be not only to win a football match or avoid defeat at cricket but also to become a mathematical millionaire.

Marcus du Sautoy is professor of mathematics at Oxford university, and author of The Music of the Primes (Fourth Estate). He plays in the number 17 shirt for Recreativo FC who finished bottom of the Super Sunday Super League Division 2 last season

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