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, Continue reading

Mathematics to the Rescue


It was my first day as a postdoc at MIT, and after landing at Logan Airport I took a taxi to a relative in Beverly, north of Boston, where I was going to stay for a few days while looking for a place to live for me and my family who were arriving a week later. The taxi driver had some difficulties locating the address and when we arrived the taximeter was on 34 dollars and fifty cents. “With the tip I will give you forty,” I told the driver and handed him a 100 dollar bill. “Sorry,” said the driver, “don’t you have smaller bills? I have only 40 dollars on me for change.” Hmm, I thought but could not find any small bills except for 10 notes of one dollar each. The problem seemed unsolvable. Can mathematics come to the rescue?

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