In how many ways you can chose a committee of three students from a class of ten students?

The renewed interest in this old post, reminded me of a more recent event:

Question: In how many ways you can chose a committee of three students from a class of ten students?

My expected answer: {10} \choose {3} which is 120.

Alternative answer 1:(Lior)  There are various ways: you can use majority vote, you can use dictatorship (e.g. the teacher chooses); approval voting, Borda rule…

Alternative answer 2: There are precisely four ways: with repetitions where order does not matter; with repetitions where order matters; without repetitions where order matters; without repetitions where order does not matter,

Alternative answer 3: The number is truly huge. First we need to understand in how many ways we can choose the class of ten students to start with. Should we consider the entire world population? or just the set of all students in the world, or something more delicate? Once we choose the class of ten students we are left with the problem of chosing three among them.

Another way to Revolutionize Football

The angle of Victoria Beckham’s hat (here in a picture from a recent wedding) is closely related to our previous post on football

One of the highlights of the recent Newton Institute  conference on discrete harmonic analysis was a football game which was organized by Frank Barthe and initiated independently by Barthe and Prasad Tetali. There were two teams of 10 players (more or less), I was the oldest player on the field, and it was quite exciting. No spontaneous improvement of my football skills has occurred since my youth.

All the lectures at the conference were videotaped and can be found here. (The football game itself was not videotaped.) Let me mention an idea for a new version of football which occurred to me while playing. For an early suggested football revolution and some subsequent theoretical discussions see this post on football and the intermediate value theorem.

The New Football Game

There are four teams. Team L (the left team), Team R (the right team), Team D (the defense team) and team O (the offense team.)  The left team protects the left goal and tries to score the right goal, the right team protects the right goal and tries to score the left goal, the defense team tries to prevent goals on both sides of the field and the offense team tries to score as much as possible goals on both sides of the field.

In formal terms,  define X to be the number of goals scored to the right and Y the number of goals scored to the left. Then the four teams try to maximize X-Y, Y-X, -X-Y and X+Y, respectively. (I do not assume teams A and B change position at halftime, but the formula can easily be adjusted.)

We can start with five players on each team (but this is negotiable.) Continue reading

Believing that the Earth is Round When it Matters

A world map. Canada seems much bigger than Israel. Note, however, that in the map countries near the equator looks smaller than they are. Update: The round-earth hypothesis is clearer to the people of New Zealand; see the comments section.

One difficult aspect of the academic life is the requirement to fly to conferences and other academic activities all over the world. Strangely, speaking about this hardship to non- academic friends does not always elicit the sympathy we deserve.

Last month, I had to be on duty in two places outside Jerusalem. The first was  a conference in Beijing and the second was a conference and a visit in the the Los Angeles area. My solution was to make a round trip to Beijing and another round trip to LA. (I am simplifying matters since there was some interference due to additional travels, visa matters, etc..)

I discovered the following flaws I make in planning my trips:

1) I am (somewhat) biased toward round trips.

More seriously…

2) I dont take into account that the earth is round.

The book solution to this travel was to go from Jerusalem to Beijing and then from Beijing to Los Angeles and from LA to Jerusalem. I completely ignored this possibility. When I realized it, it made me wonder what this reveals about my true beliefs regarding the round earth hypothesis.

Believing that this coffee cup is a realistic model of the world suffices to prefer the Beijing-LA solution over two round-trips solution!

Continue reading

The Möbius Undershirt

“Look at this brand new undershirt,” my wife said. “I am shaking it and shaking it but still I have this twist.  Can you see what to do?” 

I gave the undershirt a good shake. And another one. And one more. And then it struck me. It was a Möbius undershirt!

What a rare case in which mathematics can come to the rescue in domestic matters

“There is no way in the world this twist can be undone,” I said. “This is a mathematical fact! It is a Möbius undershirt!” My wife listened carefully to my firm statement.

I started to day-dream about the bright future of this rare Möbius undershirt: I will show her to my colleagues!, I will display her in public lectures, and even let some selected graduate students hold her. However, Continue reading

What can the Second Prize Possibly be?

prizes

You are guaranteed to win one of the following five prizes, the letter says. (And it is completely free! Just 6 dollars shipping and handling.)

a) a high-definition huge-screen TV,

b) a video camera,

c) a yacht,

d) a decorative ring, and

e) a car.

Oh yeah, you think, a worthless decorative ring, and throw the letter away.

But once I got a letter with the following promise:

You are guaranteed to win two of the following five prizes, the letter said. 

a) a high-definition huge-screen TV,

b) a video camera,

c) a yacht,

d) a decorative ring, and

e) a car.

Now, one prize will be a worthless decorative ring, but what will the second prize be?

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

The Mystery Beeping Riddle

 

 

We came back from the airport with our daughter who has just landed after a four-month trip to India. The car was making a strange beep every so often.

Maybe it is an indicator signal that should have turned off automatically? No, this possibility was quickly eliminated.

Can it be the radio? We made sure the radio was off but the beeps continued.

I looked in the car manual. The only slightly similar symptom described there was a beeping indicating that the air bags are out of order and the air bag light warning signal is also out of order. Was this the reason? In this case there would be a 5-second beep every minute. But our beeps were once every 5 minutes and each beep was for one second. Was there some mistake in the translation of the manual to Hebrew?

I called the garage. Yes, they told me, if I bring the car they can check out what is wrong and fix it. No, they have not encountered this problem before. No, it is not dangerous to drive the car back to Jerusalem. And no, they were not familiar with translation problems in the manual.

Another breakthrough idea! Maybe the beeping came from a mobile phone in the car. Some mobile phones tend to beep when the battery is low or when there is an unread message. We turned off the two mobile phones in the car. This looked promising, Continue reading

The Simplified Pill Algorithm

 

Ok so I had to take one pill every day, and fortunately the pill package was marked, it contained 14 pills with labels Sunday, Monday and so on. It started with Sunday, which was the mark for the leftmost pill. The pharmacy gave me 30 pills every month, two packages and 2 separately, which meant that taking the pills would be out of synchronization. 

However, I had it all figured out: All I need is to remember a single number between -3 and 3 for the entire four weeks. Say the starting day is Tuesday. Then I had to remember the number -2. On Tuesday I take the leftmost pill for Sunday, on Wednesday I take the pill for Monday, on Thursday I take the pill for Tuesday,  and so on. If, for example, the starting date is Thursday, then I have to remember +3. The pill for Sunday I take on Thursday, and then the pill for Monday I take on Friday etc. Easy. (I tried to write this number of the package but this didn’t work.)

I can remember several 7 digit phone numbers so, in theory, I did not see any difficulty in remembering one small integer number for an entire month. The math itself was not too challenging.  

In practice it did not work so well. Sometimes I could not remember: either I already took the pill and the magic number is 2 or I did not and it is 3. Other times, I wondered if the magic number +2 or -2.

After several months 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?

Continue reading