The Intermediate Value Theorem Applied to Football

My idea (in my teenage years) of how to become a professional basketball player was a bit desperate. To cover for my height and my athletic (dis)abilities, I would simply practice how to shoot perfectly from every corner of the court. I would not have to run or jump. My team could pass the ball to me at the right moment and I would shoot. (I was a little worried that once I mastered this ability they would change the rules of the game.)

But this idea did not work. As much as I practiced, I could not shoot perfectly from all corners of the court, and not even from the usual places. In fact, my shooting was below average (although not as much below average as my other basketball skills.)

Next came my idea how to become a professional football (soccer) player. This idea was based on mathematics, an area where I had some advantage over other ambitious sports people; more precisely, my idea was based on the intermediate value theorem. (We had a post about this theorem.)

The idea is this: If you put a football on your head and start running the ball will fall from behind. But if you put the ball on your forehead and start running the ball will fall in front of you. By the intermediate value theorem, there must be a point, in between, such that if you run with the ball at this point, the ball will not fall at all. In fact you can find such a point for every way you would like to run. And you can even learn to adjust it if you change your route!  The plan was now simple. At the right moment I would get the ball from my team, put it on the right point on my forehead  start running and slalom my way towards the goal. (I was a little worried that once I mastered this ability they would  change the rules of the game.) I practiced it for several weeks, but the progress was insufficient and I had to give up my dream.

Update (Oct 2010) : I had some hopes that this idea will revolutionize football. After all, the fact that I did not succeed can be explained by my own poor football abilities (as a child, I was usually assigned behind the goalkeeper to catch the ball if needed and I got this responsible position mainly because I owned the ball.) However, Shmuel’s observation led to the thought that rather a revolution which will change the face of football, we may witness an impossibility result in mathematical football, showing why this idea is not feasible. However, a recent discussion with Tim Gowers gives hope that indeed football can be revolutionized by a simple extension, suggested by Tim, of the original idea. You can place the ball between the foreheads of two, three or even four players who can then run together (this will require some practice) and score a goal!

Update: Victoria Beckham demonstrates the intermediate value theorem.

Update (Oct 30. 2019):  As you may recall, Shmuel Weinberger’s raised the concern of instability of fixed points. (A partial solution of Gowers apply the original idea for three foreheads.) Sylvia Serfaty mentioned to me a possible one-player implementation based on inverted pendulum control (See this video, and this one, and this one, and this one, and for the fascinating mathematics  this lecture by Jean-Michel Coron. Please, don’t try it at home.) Implement this method in football is a notable remaining challenge (this you can try at home).

Update (Nov 2, 2019): as it turned out (by several FB commentators) football players tried to implement this forehead idea and related ideas independently and perhaps earlier than mathematicians. (The effort of mathematicians can still be of value: the stable matching algorithm was used even before it was studied by Gale and Shapley.)

Here are three examples:

In the third place: An Israeli player “Shalom Rokban”

In the second place: Cuahtemoc Blanco

In the first place: Kerlon Moura Souza master of “drible da foquinha” or seal dribble


This entry was posted in Mathematics to the rescue, Sport, Taxi-and-other-stories and tagged , , , , , . Bookmark the permalink.

20 Responses to The Intermediate Value Theorem Applied to Football

  1. shmuel says:

    Unless you have a perfectly laterally symmetric head (I never checked) you probably need 2 dimensional Brouwer.

    But you have another problem: there’s a parameter, as you run — which makes it a bit more delicate and not always possible. Using a very small real valued function g(x,t) whose 0s form an “S-curve” in a square (going from (0,0) to (1,1)), f(x,t) = x + g(x,t) can be thought of as a family of self maps of [0,1] for which there isn’t a family of fixed points. So here, it wouldn’t be your fault if you couldn’t switch equilibria.

  2. Gil Kalai says:

    Dear Shmuel, many thanks for the comment. I suppose you are right that Brouwer is needed. I am not sure I understand your argument regarding the family of maps. Anyway, I doubt if I progressed far enough that switching equilibria came to play.

  3. Vishal Lama says:

    This post was incredibly funny! 🙂

  4. Gil says:

    Thanks, Vishal!

  5. Tan says:

    What a shame! Maybe with some more practice you would have got there. I find I can do the same thing with beer bottles on a friday night.

  6. Trophy Cups says:

    Ha Ha Very funny post- thanks for that one!

  7. Hey, I just wanted to say what a excellent website. I certainly enjoyed it and found it fascinating reading. Excited to read your next post!

  8. fairy tarot says:

    Great post and a excellent read. You’ve raised some valid points. Good work, keep it up. I enjoy coming back back to this webpage and digesting the good content you always have available.

  9. Gil Kalai says:

    An important update to the post is now added.

  10. chandrasekhar says:

    What a funny post, but it’s also very much meaningful.

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  13. This is an old idea!

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