I was recently interviewed by Lucia Taboada for La Redada Podcast about my love of maths and how it is used in today’s world to model everything from penalty kicks to the next TV series you watch on Netflix. The interview was translated into Spanish for the actual podcast so I’ve also included the original recording of my answers in English – enjoy!

On your YouTube channel, you present science in an entertaining way. Why is maths so unpopular sometimes, maybe students are afraid of maths?

How would you define the importance of mathematics in our life?

Tom, I’m a huge supporter of a Spanish team called Celta de Vigo. You explain the possibilities using maths to improve the performance of football players. How can Celta de Vigo use this to improve? (unfortunately, we are now in the last positions)

Penalty kicks are a science? Can you predict them?

Have you been hired by any football team?

Do you think football teams should hire math workers?

You are a tutor in St John’s College at the University of Oxford where you teach maths to the first and second year undergraduates. Oxford is a traditional university – how are your methods received there?

You have some maths tattoos on your body, thats right? Explain them to us?

Tom Crawford talks about how mathematics can help win a football league or the real ability of algorithms to manipulate people’s behaviour.

Tom Crawford (Warrington, United Kingdom, 1989) is presented as an atypical math teacher. He teaches mathematics to first and second year students at the University of Oxford (United Kingdom) and carries out an intense dissemination work in which he tries to approach a discipline that is not usually found among the favourites of young students.

In his attempt to popularise science, he does not hesitate to stay in his underpants , using the striptease as a metaphor for his work deepening the meaning of equations such as Navier-Stokes, unveiling them layer by layer, to make something affordable that can result in principle esoteric.

This week, Crawford visited the Student Residence, in Madrid, where, within the Mathematics in Residence cycle organised by the ICMAT, he offered the conference Mathematics of sport . In it, he uses sport as an example of a daily activity that can be better understood and practiced using mathematical equations.

Question. You undress or use sports to make mathematics impose less. Why is it necessary to show that mathematics is fun? I don’t see lawyers or judges, who also deal with very complex issues, trying to present the law as something fun.

Answer. I think it’s because people, for whatever reason, happily admit that they don’t like math, it’s socially acceptable. If you tell someone that you are a lawyer, their default answer is not going to be “I don’t like the law,” and that does happen with math. And it shouldn’t be like that. Everyone should have a basic understanding of math, but many people don’t have it. For me, that is why I want to emphasize that mathematics is fun and accessible. It doesn’t have to be something very hard or something that was taught badly in school.

Q. Do you think mathematics is taught especially badly in school, worse than other subjects?

A. Mathematics has a hard time competing with other subjects in the sense of teaching them through stories. When you learn something, if they can teach you through stories, it is something very powerful, which serves to catch people. And that is easier with literature or history.

A very simple example of how to add stories to mathematics would be trigonometry. The properties of the triangles you learn in high school. If you think about how these functions were discovered or invented, why we invented the sine, the cosine and the tangent, it was the ancient architects who tried to build buildings, churches, pyramids and created those intellectual tools. This is how trigonometry should be taught to me. Imagine they are in ancient Rome and you have to build a concrete building. How would you do it with the technologies available at that time? This prompts you to think about angles and distances and that is where trigonometry is useful and what it was invented for.

Q. A little more than a century ago, in a country like Spain, more than half of the population was illiterate. Do you think it would be possible and desirable to get a large majority of people to be able to handle basic mathematical tools?

A. It is completely possible and I would say that we are already doing it. It depends on what you consider a basic level of mathematics. Most people can, for example, looking at a clock know that the needles return to the same place every 12 hours, it is modular arithmetic, something you don’t study until you get to college. Even being able to calculate changes when they give you a ticket is to do mental arithmetic. Or calculate when you have to leave home if it takes 35 minutes to the station and the train leaves at 12.45. There are many things you do without thinking, but that involve mathematical calculations. So it depends on what you consider a desirable level of mathematics, but a large part of the population already has some capacity to use them.

“You can question whether trying to influence voters is good or bad”

Q. He also talks about the possibilities of mathematics to improve the performance of athletes. There is a movie like Money Ball , which talks about the experience of a baseball coach who uses mathematical analysis to lead a small team to compete against the big ones in the league with much less budget. Do you use math a lot in elite sport?

A. As far as I know, it is an important part of the scout systems of large teams. Today, these scouts, in addition to the classic analysis of a player’s performance, strengths and weaknesses, include teams of mathematicians and data scientists. As in Moneyball , your job is to analyse large amounts of data and detect marginal gains to take advantage of. That works well in baseball, because you have many controllable factors: The pitching of the pitcher, the batter, the race to the base. It is very formulable and they are repetitive behaviours. In football it is more difficult to find those marginal gains because it is less controllable.

The best example I can think of in football is Leicester City, which won the Premiere League in 2016. A big surprise. They had climbed to the first few years before and suddenly they win. In that victory, N’Golo Kanté was very important. He was the star of the season and won the player of the year award. He had been signed by a French second division team because the scout network had identified him among all the defensive midfielders in Europe at any level. As a defensive camper centre, one of your jobs is to stop the attacks of opponents. You can measure this in tickets, but one of the best ways to do this is through interceptions, which has to do with the player’s ability to read a game. It is something very difficult to assess with a number, quite subjective. But interceptions suggest that you are very often in the right place. And from that point of view, their number of interceptions was much higher statistically than the rest of midfielders. If the average of all midfielders in Europe is two, but most of the players are between 1.9 and 2.1 and Kanté is at 3, we see that it is an atypical case. It was not just a statistical analysis, because the human element is valued, but it was a factor to hire him.

Q. Can mathematics tell us what is the limit of human performance in sport? There have already been examples in the past, such as Roger Bannister’s, which went down four minutes on the mile when almost everyone said it was impossible, in which the predictions were completely wrong. Can these limits be accurately identified using mathematics?

R.If you look at the men’s marathon record during the last century, the marks descend, but not at a constant pace. You can estimate, for example, that every 10 years, 10 minutes are trimmed at the beginning, but then, in the 1940s and 1950s, the curve begins to flatten out and already in the 1990s it seems completely flat. So if we had sat here 30 years ago, when the record was around two hours and five minutes, we could have thought we would never run below two hours, because even if it keeps going down, the pace is getting slower. But in recent years, there has been much progress in long-distance races, such as new shoes that can provide 4% more energy. In addition, there is a professionalisation that allows you to train all day and not have a job besides running.

“I could predict with some confidence that the human limit for the marathon would be about an hour and 55 minutes”

So these are new factors that modify our calculations. In the future, in 30 years, new improvements may appear, but it is certain that we will not run a marathon in less than an hour. Given what has happened in the past, I think I could predict with some confidence that the human limit for the marathon would be about an hour and 55 minutes.

Q. Some people, when talking about the possibilities of mathematics to bring humans to the limit of perfection, may think that sports will become more boring, because there will be less and less space for the unexpected.

A. I think that also has to do with the human psychological trait that is nostalgia. But sport evolves and there is always a human factor. If the study allows you to perfect the place where it is better to throw a penalty, the goalkeepers can also work with that information. And then, there are some players who do not shoot at that supposedly perfect space, such as Eden Hazard, of Real Madrid, who when he threw the penalties for Chelsea waited until the last moment to decide where he threw it, a method that goes against what he says The mathematical model. In the end there are many variables in sports.

Q. Can mathematics help us better understand human groups? Does that technology have the potential to improve living together or to make it worse?

A. With all the data available, there are huge technology companies that can make profiles of people. Knowing that you are white, American, that you earn so much money and live in such a state, they can try to predict what you like or what you do and influence your vote in one direction. But this technology could also be used for good and you can also question whether trying to influence voters is good or bad. I think that ultimately we depend on the big companies that have control over these data so that they assume their moral responsibility and use the data well.

In any case, I think that most of the mathematicians working in this field would say that the idea of using mathematical data, algorithms and models to try to predict people’s behaviour is incredibly new and we don’t know exactly what we are doing. Algorithms may be a part of the decision making process, but not the only criteria for making a decision.

You can read the original article on El Pais here.

During my recent trip to Madrid to speak at the Residencia de Estudiantes, I was interviewed by national newspaper ‘El Mundo’ about my talk on the ‘Maths of Sport’ and my mission to popularise maths. The original interview can be found here.

Known as Tom Rocks Maths, the Oxford University scientist transforms boring formulas into fascinating models that he applies to sport to improve records and reduce errors.

MAR DE MIGUEL | Madrid

Football World Cup, 2018. 1-1 on the scoreboard. Spain plays its pass to the quarterfinals in the penalty round against Russia. Koke has failed one. Cheryshev is ready to throw. He scores. It’s up to Aspas. Expectation. Whistle. Launch and … Akinfeev stops it. We miss the game.

Could it have been avoided? The answer is Tom Crawford, l’ enfant terrible of numbers, a punkrocker in the court of mathematicians at the University of Oxford. And he explains it with a worn shirt, leather jacket, curled hair, piercing and tattoos. Because Crawford is not a common scientist. He is Tom Rocks Maths, an alternative researcher and communicator who transforms boring formulas into fascinating models that he applies to sports science, his second passion as a marathoner and a follower of Manchester United.

But, since all science is not exact, nor is Crawford a fortune teller, his predictions are based on data, taking into account all possible variables and, above all, on the highest probability of hitting. It is about getting ahead of the facts, of having all the necessary information to reduce errors and improve the records.

The mathematics of sport consist of “building models using data from the past to predict the future. When you don’t have them, you have to go to the field, contact the athletes and gather new information, ”explains Crawford in an interview with EL MUNDO after the talk he gave Tuesday in Madrid during the cycle of conferences ‘Mathematics in the Residence’, organized by ICMAT, the Student Residence and the Deputy Vice Presidency of Scientific Culture of the CSIC.

Win or lose penalties

The countries that best know how to throw penalties are Uruguay, Germany, Argentina and Brazil. Spain is not good, not bad. We are 50% among this list of experts and 50% of the worst, Mexico. We share media with France and Ireland. But how a team is better than another is not a matter of tradition or genetics, but numbers.

The first thing is to know how the players have responded before to the penalties, their statistics of failures and successes. According to Crawford, in the case of the 2018 World Cup, while Iniesta had four hits of five shots, Koke had zero of one and Aspas 16 of 17. The great surprise could have been given by Thiago, a substitute with a full in hits, four of four.

There is also a way to measure their stress responses with glasses that observe the movements of the eye. Footballers who are not immuted by pressure keep their eyes fixed and the most distracted move them. Knowing this in advance could decide the choice of a coach to choose the most focused players on those decisive penalties of a World Cup. “Football clubs now have entire teams of mathematicians and scientists who analyze all this data,” says Crawford.

But math doesn’t end there. In a goal you can make as many measures as your imagination, as Archimedes, to find the radius that indicates the exact area where you should place the ball without being stopped by the goalkeeper. It is called an insurmountable area and it depends, among other things, on the distance the goalkeeper moves in the shortest possible time from his position in the center. It looks like this: r^{2}-2r(a+b+R)+a^{2}+b^{2}-R^{2}. “These calculations are going to help you but they don’t guarantee that the penalty is perfect. In fact, the goalkeeper may also have trained against these formulas, ” Crawford alerts.

Roberto Carlos, the king of the Magnus effect

If penalties are a science, free kicks are not far behind. Defining its trajectory is one of Crawford’s favorite equations when the ball is given effect, as we have allways called it, which also has its scientific name: Magnus effect. “The ball that spins does not go in a straight line, because the rotation moves it to the side,” he said.

In this modality, for Crawford there is a master: Roberto Carlos, king of the Magnus effect in a match against France in 1997. It happened like this: he carefully placed the ball with his hands on the ground. He kicked. The ball passed over the barrier of players, turned in the air to the right, then to the left, hit a stick and entered as a stroke.

“I saw it when i was eight years old and I thought that it was impossible, that it was magic. But years later, using this equation to model Roberto Carlos’s shot, by entering the correct data (the speed of the ball, the distance to the door and the spin that applies to the ball) the formula accurately predicted that movement. It is still amazing. Although now I have an explanation that tells me that it did not break the barriers of physics.”

Marathon in less than two hours

In Tom Crawford’s mind there are not only favorite sport formulas but also graphics. And if there is one that drives him crazy it is the one that calculates, with a curve, when you will be able to run, with conventional methods, a marathon in less than 2 hours, something that it could happen between 2027 and 2035.

The record is owned by Kenyan Eliud Kipchoge, 2:01:39. He obtained it in Berlin in September 2018. In October of this year, the same athlete beat it at 1:59:40, but his feat was not accepted by the International Athletics Federation. “It has not been taken into account because they have broken the rules. As a new official mark, this record below two hours does not count, ”says Crawford.

How they did it? “Creating the perfect race,” says Crawford. And something else: a flat route in a straight line to go through the center of the track; a pair of shoes with carbon fiber that balances and saves 4% of energy; a tape on the leg with lumps (like golf balls) that create streams; a squad of escorts in V to cut the wind (called hares); a car that laser marks the ground so that these satellite corridors maintain the perfect position; a scanner that controls the muscle accumulation of carbohydrates and an enriched diet.

“Where you draw the line between what is due to the human element or an incredible shoe. What is the next? Putting rockets in our soles? ”Crawford wonders. We saw it in swimming a few years ago with high-tech swimsuits that reduce friction with water. They were even questioned for increasing buoyancy. With them, in some competitions 130 records were broken in just two years.

It is clear that mathematics helps to overcome tests and marks. However, in sports there are uncontrollable factors, such as the mental control of athletes, to disarm algorithms. “You can never add that factor to your models. You can never really predict a sport with total certainty. There are many unknown variables, ”reflects the English mathematician.

And, returning to the football game that we lost in 2018, would we have win if we knew the data in depth and having other players thrown the penalties that took us out of the World Cup? According to Crawford, we could have reduced the risk of losing, but this is something we will never know. What is highly certain is that their talks not only reinforce the devotion to sports, but also awake the mathematical vocation of the youngest students.

Following Manchester City’s penalty shootout victory over Liverpool in the Community Shield, I was asked by BBC Oxford to explain how scientists are trying to find the formula for the perfect penalty…

Cast your mind back to the summer of 2018… we saw the warmest ever weather in the UK, Brexit was not yet a complete and utter disaster, and seemingly against all the odds the England football team reached the semi-finals of the World Cup for the first time since 1990. No doubt the team had a huge celebration together afterwards – but it wouldn’t be the first time that two of them have celebrated an occasion at the same time. As well as playing together at the heart of England’s defence, Manchester City duo Kyle Walker and John Stones also share the same birthday! Stones was born on 28^{th} May 1994, making him 24 years old; Walker was born on the same day in 1990, meaning that he is exactly four years older than his teammate. How strange! Or is it…?

On the face of it, it seems quite surprising that in an England squad of just 23 players, two of them happen to share a birthday. However, as we’re about to see, this isn’t a freakish coincidence – maths says that it’s quite likely! What we’re talking about here is commonly known as the birthday problem: if there are a group of people of a certain size, what is the likelihood that at least two of them have the same birthday?

Let’s start by saying that we have a group of N people, and assume that birthdays are equally likely on every day of the year. (There is some evidence to suggest that this isn’t the case for top athletes; some say that they tend to be born early in the school year, such as around September in England. This is because they are slightly older than the other children in the year, and so they have a slight head-start in their physical development. However, we don’t want to make things too complicated, so we’ll ignore that for now.)

The easiest way to think about the problem is to first try to work out what the probability is that none of the N people share a birthday. Suppose our N people walk into a room, that is empty at first, one at a time. When the first person walks in, it’s obvious that they don’t share a birthday with anyone else in the room, because there isn’t anyone else. Therefore, they have the maximum probability of not sharing a birthday with anyone else in the room, which is 1.

Now think of the second person who walks in. The only way that they could share a birthday with someone in the room is if it happens to be exactly the same day as the first person. That means there is a 1 in 365 chance that they do share a birthday, so there is a 364 in 365 chance that they don’t.

Suppose that the first two birthdays don’t match, and then the third person walks in. They now have 2 days that they can’t share a birthday with, so there are 363 possible choices out of 365. Because we assumed that the first two didn’t match, we multiply the probabilities, so now the chance that none of them share a birthday is (364/365) * (363/365).

We can repeat this process until we get to our final person, number N. For example, the fourth person has 3 birthdays that they cannot share, so we multiply by a chance of 362/365; the fifth person has 4 days to avoid, so we include a probability of 361/365… By the time the Nth person walks in, there are N-1 people already in the room, so there are N-1 days that their birthday cannot fall on. This leaves them with 365-(N-1) possibilities out of 365.

To work out the total probability, we multiply all of these terms together which gives the likelihood that none of the N people share a birthday as

You might be thinking that this still looks like quite a big probability that none of them share a birthday, because all of the terms are very close to 1. But, if we try some values of N in a calculator, then it tells a very different story. (The percentages are calculated by finding the probability from the equation above and multiplying by 100.)

When N = 10, we get an 88% chance that none of them share a birthday. However, this drops down to 59% when there are N = 20 people. When we get to N = 23, the number of players in the England squad, the probability reaches just under 50%. That means that, incredibly, the likelihood that at least two of the 23 people share a birthday is just bigger than 50%!

So, in a random group of 23 people, it’s more likely than not that two of them share a birthday! This seems very strange at first; surely you’d need more than 23 people for a shared birthday to be more likely than not?! This is why the problem is commonly known as the birthday paradox – it might be very hard to get your head around, but the maths doesn’t lie!

Perhaps, in order to convince ourselves, we should look at some real-life examples. This is where the World Cup squads come into play: each team is restricted to bringing 23 players to the tournament. (We’ve seen that number before…) If our calculations above are correct, then if we picked any one of the World Cup squads, there would be roughly a 50:50 chance that at least two of the squad members share a birthday, which means that out of all of the squads that went to Russia, we would expect about half of them to have a birthday match. Well, let’s take a look…

Of the 32 teams, which were divided into 8 groups of 4, the following teams have at least one pair of players who share a birthday:

Group A

Russia

Group B

Iran, Morocco, Portugal, Spain

Group C

Australia, France, Peru

Group D

Croatia, Nigeria

Group E

Brazil, Costa Rica

Group F

Germany, South Korea

Group G

England

Group H

Poland

So, not only is there at least one team in every group with a birthday match, but if we count the total, there are 16 squads with a shared birthday pair – exactly half of the teams! The experimental results have matched up with the mathematical theory to perfection. Hopefully that’s enough to convince you that our calculations were indeed sound!

A slightly different question that you might ask is as follows: if I am in a group with a certain number of people, what are the chances that at least one of them shares my birthday? Is it the same idea? What we have worked out above is the probability that any two people in the room share a birthday (or rather, we worked out the opposite, but we can find the right answer from our working). Note that the pair doesn’t necessarily include you; it’s a lot more likely that it’s some other pair in the group.

In order to work out the answer to this similar sounding question, we work the other way around again, by calculating the probability that none of the N people share my birthday. For each of the N people, there is only one birthday that they cannot have, and that is mine (14^{th} November, in case you were wondering), which means there are 364 out of 365 possibilities for each person. We no longer care whether their birthdays match up; we only care if they match with mine. So each person has a 364/365 chance of not sharing my birthday; and the overall probability is just 364/365 * 364/365 * … * 364/365, N times, which we write as (364/365)^{N}.

Once again, we can plug some values of N into a calculator: N = 10 gives a 97% chance that no-one else has my birthday. For N = 50 the probability is still very high: there is an 87% chance that none of these 50 people have the same birthday as me. N = 100 gives 76%; N = 200 gives 58%; you have to go all the way to N = 253 before the probability dips below 50%, and it becomes more likely than not that at least one person will celebrate their birthday with me.

Applying this idea to all 736 players (32 squads of 23 players) involved in the World Cup, we should expect around 3 of them to have been born on the same day as me – 14^{th} November. And I am very happy to confirm that France’s Samuel Umtiti, Switzerland’s Roman Burki, and Belgium’s Thomas Vermaelen all have what is undoubtedly the best birthday of the year… Two similar problems with two very different solutions!

I was asked by the Daily Mirror to analyse the England football team’s penalty kicks against Colombia in the World Cup second round. You can find the key insights below and the full article online here.

Image: Dr Ken Bray, University of Bath

Harry Kane – Kane’s very calm and confident in his walk up to the penalty spot showing that he has prepared well mentally. He carefully places the ball and adjusts his socks before firing low and hard into the bottom left-hand corner of the net. The keeper goes the right way but it’s too accurate and right in the corner of the ‘unsaveable zone’.

Marcus Rashford – A different approach on the walk up as he keeps his head down to make sure he doesn’t give anything away to the Colombia keeper. He curves his run-up to add extra disguise to the shot and puts it in almost exactly the same place as Harry Kane. Again, the Colombia keeper goes the right way but it’s too fast, too accurate and right in the bottom corner of the ‘unsaveable zone’.

Jordan Henderson – The ‘kick-ups’ on the walk to the penalty area show he’s nervous and the look on his face also hints at a lack of confidence. The placement of the shot is actually very good as he hits the ‘unsaveable zone’ to the left of the keeper, but his shot is a little higher than the previous two making it a more comfortable height for the goalie, and his wide run-up gives the game away as he opens his body to go to the right. If you look closely you’ll see that Ospina moves before Henderson kicks the ball which is why he’s able to reach beyond the ‘diving envelope’ and make the save.

Kieran Trippier – He has his head down and a look of complete focus on his face as he approaches the penalty spot. After a little glance up to make sure he knows where he’s going, he buries it in the top left corner in the perfect spot. Comparing Trippier’s penalty to the fourth Colombian taker, Uribe, who missed, it’s the use of the inside of his foot that makes all of the difference. Despite them both aiming for the top corner of the ‘unsaveable zone’, Uribe leant back and went with his laces making it less controlled than Trippier’s side foot. It’s also interesting that England’s nominated set piece taker went fourth in the line-up. No doubt, because Gareth Southgate knew that the fourth penalty would be key to victory as one that goalkeepers are likely to save.

Eric Dier – Positionally, probably the worst of the five England penalties as it was the closest to the centre of the goal and the edge of the ‘diving envelope’ which is within reach of Ospina. The key aspect of Dier’s penalty that allowed him to score was the fact that it was along the ground. Ospina dives the correct way, but can’t reach close enough to his body to make the save. Compare this to Jordan Henderson’s penalty, which was much closer to the corner, but at a more comfortable height for the save.

Summary:

4 of the 5 penalties went to the left of the goalkeeper and were all scored, whereas the one that went to the right of the keeper was saved.

All of England’s penalty takers were right-footed.

2 of the 5 penalty takers were substitutes, likely brought on to take a penalty in the shootout.

All of England’s penalties hit the ‘unsaveable zone’, maximising the chances of scoring. For Colombia only 2 of the 5 penalties hit the ‘unsaveable zone’.

Jordan Pickford saved the fifth and final penalty, demonstrating how it is more likely for a goalkeeper to make a save later in the shootout.

England benefitted from good preparation from the manager in selecting his line-up months in advance, aiming consistently for the ‘unsaveable zone’ which is the most difficult area for the goalkeeper to reach, and by preparing well mentally and taking their time with each shot. Ultimately, these 3 things were key to the victory.

The 2018 World Cup in Russia kicks off today and so I bring you a special double-edition of Throwback Thursday looking at the science behind the perfect penalty kick… Fingers crossed the England players listen/read my website and we don’t lose to Germany in a penalty shootout (though let’s be honest we probably will).

Live interview with BBC Radio Cambridgeshire looking at the ‘unsaveable zone’ and the best way to mentally prepare for a penalty.

And if that wasn’t enough, here’s a full description of the ‘Penalty Kick Equation’…

For all of the footballers out there who have missed penalties recently, I thought I would explain the idea of the science behind the perfect penalty a little further, and in particular the maths equation that describes the movement of the ball. On the radio of course I couldn’t really describe the equation, so here it is:

If you’re not a mathematician it might look a little scary, but it’s really not too bad. The term on the left-hand side, D, gives the movement of the ball in the direction perpendicular to the direction in which the ball is kicked. In other words, how much the ball curves either left or right. This is what we want to know when a player is lining up to take a penalty, because knowing how much the ball will curl will tell us where it will end up. To work this out we need to input the variables of the system – basically use the information that we have about the kick and input it into the equation to get the result. It’s like one of those ‘function machines’ that teachers used to talk about at school: I input 4 into the ‘machine’ and it gives me 8, then I put in 5 and I get 10, what will happen if I input 6? The equation above works on the same idea, except we input a few different things and the result tells us how much the ball will curl.

So, what are the inputs on the right-hand side? The symbol p just represents the number 3.141… and it appears in the equation because footballs are round. Anytime we are using circles or spheres in maths, you can bet that p will pop up in the equations – it’s sort of its job. The ball itself is represented by R which gives the ball’s radius, i.e. how big it is, and the ball’s mass is given by m. We might expect that for a smaller ball or a lighter ball the amount it will curl will be different, so it is good to see these things are represented in the equation – sort of a sanity test if you will. The air that the ball is moving through is also important and this is represented by r, which is the density of the air. It will be pretty constant unless it’s a particularly humid or dry day.

Now, what else do you think might have an effect on how much the ball will curl? Well, surely it will depend on how hard the ball is kicked… correct. The velocity of the ball is given by v. The distance the ball has moved in the direction it is kicked is given by x, which is important as the ball will curl more over a long distance than it will if kicked only 1 metre from the goal. For a penalty this distance will be fixed at 12 yards or about 11m. The final variable is w – the angular velocity of the ball. This represents how fast the ball is spinning and you can think of it as how much ‘whip’ has been put on the ball by the player. Cristiano Ronaldo loves to hit them straight so w will be small, but for Beckham – aka the king of curl- w will be much larger. He did of course smash that one straight down the middle versus Argentina in 2002 though…

So there you have it. The maths equation that tells you how much a football will curl based on how hard you hit it and how much ‘whip’ you give it. Footballers often get a bad reputation for perhaps not being the brightest bunch, but every time they step up to take a free kick or a penalty they are pretty much doing this calculation in their head. Maybe they’re not quite so bad after all…