Archive for October 2008

## Linear Algebra for Pirates

I’ve got this puzzle from Nick Petry.

Captain Flint is dying. All his treasures are buried far away. He only has 99 pieces of gold with him. Filled with remorse at the last moments of his life, he decides that he only wants to take one piece of gold with him to his grave. The rest of the gold he will give to the families of two men that he had killed the day before.

Though Captain Flint is heavily drunk he notices that no matter which piece he takes for himself, he can divide the leftover 98 pieces into two piles of 49 pieces each of the same weight. Prove that all the gold pieces are of the same weight.

For an additional challenge, Sasha Shapovalov suggested the following generalization of the previous puzzle.

Captain Flint has N gold pieces and yesterday he killed not two but K men. He wants to take one piece with him to his grave and to divide the rest into equi-weighted piles, not necessarily of the same number of pieces. If he can choose any piece to take with him and is able to divide the rest, prove that N – 1 is divisible by K.

Both of these puzzles can be easily solved if the weight of every gold piece is an integer or even a rational number. If you don’t assume that the weights are rational numbers, then I do not know an elementary solution, but I do know a simple and beautiful solution using linear algebra for both puzzles.

Even pirates need linear algebra to divide their treasure. Hooray for linear algebra!

## Borrowing Money

To translate from a Russian joke, borrowing money is taking someone else’s: temporary; giving back your own: forever.

This is a story about my great-uncle Fred. His name is not Fred, of course, because I don’t want to reveal which one of my thirteen great-uncles created this ingenious scheme.

My great-uncle Fred asked to borrow 100 rubles from my mother. He was notorious for not returning money, but he knew how to work my mother. He whined about being sick and urgently needing to buy pills, until my mother, who has a big heart and is an easy touch, gave up. Of course, Fred wasn’t in a hurry to return the money. But 100 rubles was a lot of money for my mother and she wasn’t planning on giving up trying to get it back. My mother started bugging her uncle with increased intensity. Finally Fred promised to return the money as a gift for mom’s upcoming birthday.

Of course, it was tacky to present the money he owed as a gift, but my mom was so glad that she would finally get her money back, that she was actually looking forward to it.

During her party, as the guests sat around the table, Fred got up to give the birthday toast. Then Fred handed my mother an envelope and said, “Congratulations on your birthday! Here is a gift for you.” Everyone applauded.

My mom felt that something in this scene was not quite right. Why was the applause so enthusiastic when he was just returning a debt? After the party my mother decided to investigate. It turned out that Fred explained to my mother’s relatives that she prefered money as a birthday gift and collected the gift money from everyone. The cash he returned as his debt in the envelope was not his. Everyone else thought he was presenting the joint gift, except for my mother, who was made to believe that he was repaying his debt.

After that my mother stopped bugging her uncle Fred. It became clear she couldn’t match his superb skills in escaping his debts.

## The Right Time to Have Children

Suppose you are a woman living in the US and you would like to be a mathematician and work in academia. Suppose also that you would like to have children and spend some time with them. Let us say that you want two or three children and you would like to be with them at home for at least their first two or three years. That is, in total you need to devote 5 or 6 years to this endeavor. When would be a good time for you to have children? American women mathematicians are commonly advised to postpone having children until tenure.

Let us look at the situation more closely. Is it a good idea to have children before you earn your PhD? There are many non-mathematical reasons not to have children too early:

• You do not know yourself well enough to be able to choose a good husband.
• You might not have met the man you love yet (see my essay The Mathematical Path to the Right Husband).
• You might not have enough money to support yourself and your children.
• If you have not yet definitely decided to become a professional mathematician, you might still be clueless that your future career contradicts your plans for children.

There are also mathematical reasons not to have children too early. My former adviser Israel Gelfand liked to tell everyone that mathematicians generate all their best ideas before the age of 23. His views may be extreme. After all, we know of mathematicians who made great discoveries later in life, but it could be that they did this using their mathematical wisdom rather than the processing power of their brains. It may well be beneficial to start your first research early in life. I am not sure about math creativity, but I swear that it is much more difficult to learn new things as I age.

In addition, you might need to relocate frequently to maintain your math career and it is more difficult to do that with children. It’s hard on the children too. Besides, having children early means that while working on your tenure, you will be distracted by your kids and their problems, leaving you less time for your research.

You might think that the closer you get to your PhD, the more the situation improves. In reality, there is a very important reason not to have children while you are in grad school. When you start working on your first topic, it is very important not to be interrupted. If you take a big break someone might finish what you started. When you come back, your topic might be resolved and you would have to start all over again.

The situation after graduate school is even worse. When you apply for jobs, employers are likely to count how many papers you have published per year after your PhD. So you need that number to be high. You can’t afford to dilute your paper count per year by a several-year break. Besides, if you have an interruption in your research it might be considered as a weakness and you might lose in comparison with other applicants. Let us break down the time between PhD and tenure into three periods: postdoc, visiting positions, and tenure track. For each period there are extra reasons not to have kids:

• Right after the PhD you begin looking for postdoc positions. All of them have formal time constraints: you are not allowed to apply for these positions if you are more than 3 (or maybe 5) years past your PhD. To have children during the postdoc period is really a bad idea.
• After the postdoc you might have several visiting positions. They usually require yearly relocations and you need to produce something new every year, so that your current curriculum vitae is different from the one you sent to the same place a year ago. At this stage of your life you are much better off without a husband — forget about children.
• When you have tenure track, you are so close to tenure that you might not want to put your dream at risk after so many years of sacrifices.

For all of these reasons, advice to wait until tenure makes sense. There is one big problem with it though: you usually get tenure in your late thirties. It might be too late for children. You might not want to risk that.

You can always compromise by having one child instead of three. Or you can suppress your desire to spend a lot of time with your children by having hired help, which means that you will miss a good deal of your child’s development, and your child will miss a lot of your love. You can compromise your academic goals by taking a more stable, but less research-oriented, technical job in industry. Or you might get lucky and marry a househusband.

In short, a math career is very kids-unfriendly — there is no right time. If you’re a woman mathematician who wants to spend time with your kids, prepare for pain and disappointment.

But here is an unconventional idea you might consider.

After you finish working on your PhD, postpone your actual defense by 5 years, and have your kids in between. This way all your PhD results will be published and no one can interfere with them. At the same time, the clock that counts your publications per year after your PhD will start 5 years later.

My idea is not a good solution — you will still have many problems — but it might be better than waiting ’till tenure. I do wish there were a better way.

## Library Puzzle

Here is a logic puzzle for kids:

— John has more than a thousand books, said Pete.
— No, he has less than one thousand books, said Ann.
— Surely, he has at least one book, said Mary.

If only one statement is true, are you sure you know how many books John has?

## Four Puzzles for the Price of One

Here is a math problem from the 1977 USSR Math Olympiad:

Let A be a 2n-digit number. We call this number special if it is a square and a concatenation of two n-digits squares. Also, the first n-digit square can’t start with zero; the second n-digit square can start with zero, but can’t be equal to zero.

• Find all two- and four-digit special numbers.
• Prove that there exists a 20-digit special number.
• Prove that not more than ten 100-digit special numbers exist.
• Prove that there exists a 30-digit special number.

Obviously, these questions are divided into two groups: show the existence and estimate the bound. Furthermore, this problem can be naturally divided into two other groups. Do you see them? The puzzle about special numbers makes a special day today — you get a four-in-one puzzle.

## A Math Puzzle that Sounds like a Computer Science Puzzle

David Bernstein gave me this puzzle. He says that the puzzle was given at a Moscow math Olympiad a long time ago. At that time there were no computer science olympiads yet. I do not know why this puzzle feels to me like computer science. Maybe because the trivial solution is of order N, the easy solution is of order square root of N and the requested solution is of order logarithm of N:

Can you cut a square into N convex pieces minimizing the number of possible intersections of any straight line with your pieces?

It is easy to maximize the number of intersections. If you cut your square with N-1 parallel cuts into N equal thin rectangles, then there exists a line with N intersections.

It is easy to cut a square to guarantee no more than 2√N intersections. Can you cut your square so that any line makes no more than 2log2N intersections?

## Math Career Predictor

I am interested in a career in mathematics. How hard is it to be a woman mathematician?

Let us look at some numbers from the American Mathematical Society Survey Reports for the year 2005:

• There were 40% of women among graduating math majors.
• There were 30% of women among Math PhDs granted.
• There were 11% of women among full-time tenured or tenure-track positions.

You can’t just say that women do not like math — 40% of those choosing math as a major is quite a large number, after all.

On the other hand, the downward trend of these percentages is striking. If women’s opportunities and abilities are the same as men’s, these percentages should grow with every age step, since, as we know, the percentage of women in the population increases with age due to men dying earlier.

But the numbers go down and very fast. There are many potential explanations for this, but today we’re going to look at one of them:

Women have less ability for high-level mathematics.

Was Larry Summers right when in his speech that cost him his Harvard presidency he compared math ability to height and to the propensity for criminality, and suggested that the distribution, especially standard deviation, of math ability differs for men and women?

To answer this question, I wanted to find some other data that correlates gender with math abilities. I took the results of the American Mathematical Competitions (AMC 12) for the year 2008. Among 120,000 students who participated, 43% were females. Here are some results:

• Among students scoring 72 points or higher there were 40% of girls.
• Among students scoring 98 points or higher there were 30% of girls.
• Among students scoring 134.5 points or higher there were 11% of girls.

This picture is similar to that of the academic career: the closer you climb to the top, the smaller percentage of girls you see there. Of course, winning a competition is very different from getting tenure. People who win competitions are smart and competitive — smart and competitive enough to go for money, rather than academia. On the other hand, people who are interested in mathematics often are not interested in anything else. Why would they waste their time in competitions when the Riemann Hypothesis is still waiting to be solved?

But still, both achieving tenure and winning math competitions represent mathematical ability in some sense. If Larry Summers was right and the distribution of math ability is different among males and females, then by looking around you at the percentage of females at your level, you should be able to assess how close you are to the top of the math field.

I propose the following math career predictor: Take your results in AMC 12. If among kids who did better than you, the percentage of girls is more than 11%, you do not have a chance at tenure. If the percentage of girls is more than 30%, do not waste your time working on a math PhD. If the percentage of girls is more than 40% maybe math majoring is not for you.

I hate my math career predictor. I hate it not only because it has so many flaws that it might just deserve the Ig Nobel Prize, but because it doesn’t take people’s effort into account. You really have to work very hard to be a math professor, whether you were a winner or a loser in math competitions.

You might ask why I created a math career predictor that is so flawed. My mathematician friends, those who are more honest than polite, tell me that I have no chance at getting back to academia. On the other hand, I had the second best result at the 1976 IMO, which means I have the ability. My predictor may be my only hope.

## Linear Algebra at a Math Olympiad

A puzzle from the 1977 USSR math Olympiad can be solved naturally with linear algebra:

Seven dwarfs are sitting at a round table. Each dwarf has a cup partially filled with milk. Each dwarf in turn divides all his milk evenly between the six other cups. After the seventh dwarf has done this, every cup happens to contain the initial amount of milk. What was the initial distribution of milk?

Can you use linear algebra to intelligently solve this puzzle?

## Designing Bill Gates’ Bathroom

One of the questions from the Microsoft employment interviews for creative thinkers is: “How would you design Bill Gates’ bathroom?” I gave this question to my students at the Advanced Math and Science Academy Charter School. Most of them started by suggesting it be big and gold, but they also suggested more interesting ideas:

• Heat the floor and the toilet seat.
• Run a medical test automatically for every flush.
• Provide a shampoo dispenser with a choice of 20 smells.
• Paint the portrait of a favorite enemy inside the urinal.
• Create a shower that looks and feels like a waterfall.
• Install a face recognition system that immediately adjusts all the default settings according to who enters the bathroom.
• Build a very simple bathroom and give the leftover money to charity.

## Who Should Have Kissed Whom

I recently updated my collection of my favorite xkcd webcomics.

Today I would like to discuss the comic entitled “Regrets”. When I saw this comic, the first thing I did was go to Google to check the numbers. All my numbers were taken on September 9, 2008 at around 3:00 p.m.

Here are the Google hit counts:

• “I should have kissed her” — 10,600
• “I shouldn’t have kissed her” — 3,220

The numbers are slightly different than those in the cartoon, but the idea is the same; we regret we didn’t kiss. Does it mean that if you want to kiss someone you should go ahead, or otherwise you would contribute to this pile of regrets? The answer is coming later, but first, let’s see what happens if we change gender:

• “I should have kissed him” — 3,170
• “I shouldn’t have kissed him” — 1,240

The same story overall, but for some reason, there are fewer reports by people who either tried or didn’t try to kiss HIM. Is kissing him less interesting or important? Most probably we still expect men to take the initiative in kissing her.

Then I checked the genderless case:

• “I should have kissed you” — 15,700
• “I shouldn’t have kissed you” — 1,800

Wow. Looks like we really should start kissing each other. Right? But wait. Let’s check the point of view of a kissee, rather than a kisser:

• “You should have kissed me” — 494
• “You shouldn’t have kissed me” — 1,020

We see a completely different picture. It is easy to explain why the numbers are smaller: passive people would be less likely to discuss their feelings. But, even so, they claim that they preferred not to be kissed. Maybe it is OK that people mostly regret that they didn’t kiss. After all, if they had tried to kiss, they might not have been greeted with enthusiasm. This leaves you with a choice between your own regrets that you should have kissed and his/her regrets after you did.

I did regret that I hadn’t kissed you, but I so much prefer that I regret not kissing you than you regret being kissed by me. This small research made me feel better. I will not continue regretting any more.