Tanya Khovanova's Math Blog

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.

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?

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.

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?

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.

Tell me your own ideas.

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.

Ken Fan is running a math club for girls in Cambridge, MA, called “Girls’ Angle”. When I heard about it, my first reaction was surprise. When I was a girl, I would never have been interested in a girls-only math club.

Am I prejudiced? When I was growing up in Russia, there were not very many girls who were really good at mathematics. I would have expected that a girls’ math club would be less challenging than just a math club.

What if someone organized a boys-only math club? I would have been furious. I would have felt it was discriminatory. Should I then feel an equivalent fury about the girls’ club? But I do not feel furious, and I wonder why. Is it because deep down I think no boy would bother being interested in joining a girls’ math club? Is it because I still think a girls’ club would be weaker than a general club? I do not know.

At the same time I agree with Ken, for there are a variety of reasons why girls might prefer a girls-only club. For example, shy girls might feel more comfortable with girls or some girls might feel better able to concentrate without the distractions of boys. In some cases, the parents might have made the decision.

Obviously, since the club has students, there is a demand for it. If there is a demand, there should be a supply. I will support anything that works and helps improve American math education. I even volunteered to give a guest lecture at Girls’ Angle.

Was my lecture at the girls’ club different from my other lectures? Yes, in a way. I asked the girls to help me to finish a sequence. I started writing 1, 1, 2, 3, 5, 8, on the board and no one was shouting the next number. In great surprise I turned back to face the class and saw a forest of raised hands. They patiently waited for my permission to speak. Yes, it felt different.

I decided to check the pricing. It appears that the girls club is twice as cheap as other math clubs, like The Math Circle or S.M.Art School. I can’t help but wonder if the girls are signing up at the Girls’ Angle not because they want to study in the girls-only group, but because it is cheaper.

I am glad that Ken Fan is good at finding sponsors and that there are so many people sympathizing with his cause. However, this situation does seem unfair to boys. Should I be furious that boys are not allowed in this very affordable math club? I do not feel furious, but I decided not to give any more lectures at Girls’ Angle for free. At least not until I give a free lecture to a mixed-gender math club. I want to be fair.

This puzzle is a generalization of a problem from the 1977 USSR math Olympiad:

At the beginning of the game you are given a polynomial, which has 1 as its leading coefficient and 1 as its constant term. Two people play. On your turn you assign a real value to one of the unknown coefficients. The person that goes last wins if the polynomial has no real roots at the end. Who wins?

It is clear that if the last person’s goal is for the polynomial to have a root, then the game is trivial: in this case, he can always make 1 a root with the last move. Also, an odd degree polynomial always has a real root. Therefore, to make the game interesting we should assume that the degree of the polynomial is even.

Though I can’t imaging myself ever being interested in playing this game, figuring out the strategy is a lot of fun.

Here is a math puzzle for kids from a nice collection of ThinkFun Visual Brainstorms:

Wendy has cats. All but two of them are Siamese, all but two of them are Persian, and all but two of them are Maine Coon. How many cats does Wendy have altogether?

This puzzle has two answers: the expected answer and an unexpected answer. Can you find both?

Chris Burke gave me his permission to add his webcomics to my collection of Funny Math Pictures.

This comic doesn’t qualify as a math picture, but it is geeky enough for me to like it.