mathematics of sport

My first – and only – argument with the fascinating How to Take a Penalty – the Hidden Mathematics of Sport came in the introduction (which I had saved until last). ‘This is a dip-in book without any obvious beginning, middle or end,’ write authors Rob Eastaway and John Haigh. Maybe they intended it to be that way, but they undersell their skills, for this is one of those rare books which I can genuinely claim to have read in one sitting, writes Andy Etchells.

And that’s an impressive claim for a book written by two university lecturers, one of whom (Eastaway – he doesn’t have a maths degree, but does set puzzles for national newspapers) has previously written books with such arcane titles as How Long is a Piece of String?

It’s partly a question of a light, readable style and partly the unexpected links the authors make between sports as diverse as ice skating and boxing, cricket and American football. But something is also owed to the fact that they heed the advice of Professor Steven Hawking, who once said that every equation halves the readership of a book (and he should know!). Consequently, the text zips along with just enough technical stuff – and a few fun quizzes – to start making you scratch your head, but with the comfort of knowing that there’s a detailed appendix at the back if you really want to burrow further into the science.

Let’s start with the chapter on those eponymous penalties – entitled ‘Why didn’t you belt it, son?’ in honour of Stuart Pearce’s mum’s classic post-Euro 96 penalty shoot-out bit of hindsight. It was to be expected that the authors would refer to one study about the angle of the penalty-taker’s body, but they take the maths much further by showing that unpredictability is the key to penalty success. And that takes us into ‘game theory’ and what it means to be random. They call their notional penalty-taker Beckham and, with no hint of disrespect towards the England football captain, offer him a penalty-taking guide involving the position of the second hand of the stadium clock. (Come to think of it, maybe they are poking a bit of fun here: when did you last see a real clock with two hands in a stadium?)

Then there’s the mathematics of record breaking, and at last someone has definitively nailed the nonsense put out shortly after the last Olympics about how women will be sprinting faster than men next century. This was based on an item in Nature magazine which the authors say was written tongue-in-cheek, although this didn’t stop the claims being reported with a straight face in places where they really should know better. Use the same straight-line extrapolations and you could get some gullible sap to predict that, somewhere down the line, Olympic titles will be won in negative time. Whereas, in reality, we are possibly already seeing the reverse of the trend Nature reported on: take an average of the top 10 performances per year – not the extreme of the top-ranked time – and this is the result:

‘The graph of the men’s sprint times still follows what looks like a linear decline… though it appears to flatten out very slightly in the later years. For the women, though, the shape of the graph takes a much more dramatic turn. Sometime around 1984, the line suddenly appears to level off. If anything, the gap between men’s and women’s performances increases after that year.’

So there! And for record junkies, there’s worse to come as the ‘number of athletes outside China has now reached a plateau, partly because the population in the West is no longer growing. And in the next century, the forecasts suggest that the number of people in the athletics age range of 18-30 will dramatically decline worldwide’. That means a smaller pool from which record breakers will emerge and the authors show this attrition may already be at work in an event like the high jump. With a final poke at the Nature approach to statistics, they surmise that by AD 3339 ‘men won’t be able to get off the ground at all’.

However, there’s good news for deposed world 100m record holder Maurice Greene: adjusting for wind, his 9.79 with a minimal following wind of 0.1m/sec is worth, after calculating the value of each m/sec as 0.06sec, a ‘pure’ 9.80; while the Tim Montgomery performance that eclipsed Greene’s record (9.78 run with the aid of the only just legal 2m/sec) equates to a pedestrian 9.90 in the dead calm of 0m/sec.

Staying with track and field, there are fascinating insights into why national coaches in search of the next decathlon champion should be analysing the junior long jump ranking lists; why the Fosbury flop may be at the end of its 40-year dominance of the high jump; why shot putters – not just runners – do better in Mexico than Helsinki and why (wherever they practise their art) they should release the iron ball at an angle of 42º to the horizontal.

Forty-two? Douglas Adams of Hitchhiker’s Guide fame would have something to say about that particular figure, but did even he know about the way cricket influenced American football? Or the nine-day cricket test that still didn’t produce a result? Or why in football a 1-1 draw is the most likely result, even though 46% of matches result in a home win? Or why golf balls have dimples and what shape they are?

Andy Etchells

How to Take a Penalty – the Hidden Mathematics of Sport by Rob Eastaway and John Haigh (Robson Books £12.99)