Tanya Khovanova's Math Blog

Until the introduction of the Abel prize, the Fields medal was the most prestigious prize in mathematics. The medal has been awarded 48 times and all of the recipients have been men. Can we conclude that women are inferior to men when it comes to very advanced mathematics? I do not think so.

The Fields medal was designed for men; it is very female-unfriendly. It is the prize for outstanding achievement made by people under age 40. Most people start their research after graduate school, meaning that people have 10-15 years to reach this outstanding achievement. If a woman wants to have children and devote some time to them, she needs to do it before she is 40. That puts her at a big disadvantage for winning the medal.

Recently the Abel prize for mathematics was introduced. This is the math equivalent of a Nobel prize and nine people have received the prize, all of them male. The Wolf prize is another famous award: 48 people have received it so far and they too have all been male.

On the grand scale of things, women have only recently had the option of having a career in mathematics. Not so long ago it was considered quite exceptional for a woman to work in mathematics. The number of female mathematicians is increasing, but as this is a new trend, they are younger people. At the same time, Abel prizes and Wolf prizes are given to highly accomplished and not-so-young people. That means the increase in the percentage of women PhDs in mathematics might affect the percentage of females getting the prize, but with a delay of several dozen years.

There are other data covering extreme math ability. I refer to the International Math Olympiad. The ability that is needed to succeed in the IMO is very different from the ability required to succeed in math research. But still they are quite similar. The IMO data is more interesting in the sense that the girls who participate are usually not yet distracted by motherhood. So in some sense, the IMO data better represents potential in women’s math ability than medals and prizes.

Each important math medal or prize is given to one person a year on average. So the IMO champion would be the equivalent of the Fields medal or the Wolf prize winner. While no girl was the clear best in any particular year, there were several years when girls tied for the best IMO score with several other kids. For example:

• In 1995, 14 students tied for the perfect score; two of them were girls. (Maryam Mirzakhani and Chenchang Zhu)
• In 1994, 22 students tied for the perfect score; two of them were girls. (Theresia Eisenkölbl and Catriona Maclean)
• In 1991, 9 students tied for the perfect score; one of them was a girl. (Evgenia Malinnikova)
• In 1990, 4 students tied for the perfect score; one of them was a girl. (Evgenia Malinnikova)
• In 1987, 22 students tied for the perfect score; one of them was a girl. (Jun Teng)
• In 1984, 8 students tied for the perfect score; one of them was a girl. (Karin Gröger)

In one of those years, a girl might have been the best, but because the problems were too easy, she didn’t have a chance to prove it. Evgenia Malinnikova was an outstanding contender who twice had a perfect score. In 1990, she was one out of four people, and she was younger than two of them, as evidenced by the fact they they were not present in 1991. Only one other person — Vincent Lafforgue — got a perfect score in 1990 and 1991. We can safely conclude that Evgenia was one of two best people in 1990, because she was not yet a high school senior.

This might be a good place to boast about my own ranking as IMO Number Two, but frankly, older rankings are not as good as modern ones. Fewer countries were participating 30 years ago, and China, currently the best team, was not yet competing.

Girls came so close to winning the IMO that there is no doubt in my mind that very soon we will see a girl champion. The Fields medal is likely to take more time.

Due to the popularity of my previous posting of linguistics puzzles, I’ve translated some more puzzles from the online book Problems from Linguistics Olympiads 1965-1975. All of them are from the phonetics section and I’ve kept the same problem number as in the book. I’ve used the Unicode encoding for special characters.

Problem 20. In the table below there are numerals from some Polynesian languages. Note that I couldn’t find the proper English translation for one of the languages, so I used transliteration from Russian. The language sounds like “Nukuhiva” in Russian.

Your task is to find the words that should be in the empty cells. Note that wh, ‘, and ŋ denote special consonants.

Problem 21. Below you will find words in several relative languages. You can group these words into pairs or triples of words with the same origin and the same or a similar meaning.

āk, dagr, bōk, leib, fōtr, waʐʐar, buoh, dæʒ, plōgr, hām, wæter, hleifr, pfluog, eih, heimr, fuoʐ, plōʒ.

Task 1. Divide the words into groups so that the first group has words from the same language, the second group has words from another language and so on.

Task 2. (optional) List your suggestions about the meanings of the words and about the identity of the languages.

Problem 22. These words from the Aliutor language are followed by their translations. The stresses are marked by an apostrophe in front of the stressed vowel.

• t’atul — fox
• nətɣ’əlqin — hot
• nur’aqin — far away
• ɣ’əlɣən — skin
• n’eqəqin — fast
• nəs’əqqin — cold
• tapl’aŋətkən — He sews shoes
• k’əmɣətək — to roll up
• ʔ’itək — to be
• paq’ətkuk — gallop
• n’ilɣəqinat — white (they both)
• p’unta — liver
• qet’umɣən — relative
• p’iwtak — to pour
• nəm’itqin — skillful
• t’umɣətum — friend
• t’ətka — walrus
• k’əttil — forehead
• qalp’uqal — rainbow
• kəp’irik — hold (a baby in the hands)
• təv’itatətkən — I work
• p’intəvəlŋək — attack (each other)

Your task is to put the stresses in the following words: sawat ‘lasso’, pantawwi ‘fur boots’, nəktəqin ‘solid’, ɣətɣan ‘late autumn’, nəminəm ‘bouillon’, nirvəqin ‘sharp’, pujɣən ‘spear’, tilmətil ‘eagle’, wiruwir ‘red fish’, wintatək ‘to help’, nəmalqin ‘good’, jaqjaq ‘seagull’, jatək ‘to come’, tavitətkən ‘I will work’, pintətkən ‘he attacks (someone)’, tajəsqəŋki ‘in the evening’.

Note that the vowel ə is similar to many unstressed syllables in English words, such as the second syllable in the words “taken” and “pencil”. This vowel is shorter than other vowels in the Aliutor language.

Last time Sue refinanced her mortgage was six years ago. She received a 15-year fixed loan with 5.5% interest. Her monthly payment is $880, and Sue currently owes $38,000.

Sue is considering refinancing. She has been offered a 5-year fixed loan with 4.25% interest. You can check an online mortgage calculator and see that on a loan of $38,000, her monthly payments will be $700. The closing costs are $1,400. Should Sue refinance?

Seems like a no-brainer. The closing costs will be recovered in less than a year, and then the new mortgage payments will be pleasantly smaller than the old ones. In addition, the new mortgage will last five years instead of the nine years left on the old mortgage.

What is wrong with this solution? What fact about Sue’s old mortgage did I wickedly neglect to mention? You need to figure that out before you decide whether Sue should really refinance.

So many people liked the puzzles I posted in Subtraction Problems, Russian Style, that I decided to present a similar collection of multiplication and division puzzles. These two sets of puzzles have one thing in common: kids who go for speed over thinking make mistakes.

Humans have 10 fingers on their hands. How many fingers are there on 10 hands?

This one is from my friend Yulia Elkhimova:

Three horses were galloping at 27 miles per hour. What was the speed of one horse?

Here is a similar invention of mine:

Ten kids from Belmont High School went on a tour of Italy. During the tour they visited 20 museums. How many museums did each kid go to?

Another classic:

How many people are there in two pairs of twins, twice?

Can you add more puzzles to this collection?

Today I have my first invited guest blogger, J.B. He is a 2006, 2007, 2008 and 2009 USAMO qualifier. He was also selected to be on the US team at the Romanian Masters in Mathematics competition. Also, he placed 6th at the North American Computational Linguistics Olympiad. Here is his piece:

The analysis is based on the list of 2009 USAMO qualifiers.

There is a rule that if nobody naturally qualifies for the USAMO from a state, then the highest scoring individual will qualify. Unfortunately, this means that we must remove those states with only one USAMO qualifier. We have 33 states remaining. If we sort these strictly by number of USAMO qualifiers, then we find the following result.

States with at least 4 USAMO qualifiers (24 total) voted for Obama, with the following exceptions: Georgia, Texas, South Carolina, and Missouri. In addition, of the two states with 3 USAMO qualifiers, one voted for Obama and one for McCain. The remaining states with 2 qualifiers (5 total) voted Republican.

Now this is not really unexpected. States with very large populations tend to be democratic and also produce more USAMO qualifiers. The most notable exceptions are Georgia and Texas, both of which were indeed exceptions (major outliers, in fact) above. This prompts the following consideration.

States with at least 8 USAMO qualifiers per 10 million residents (25 total) voted for Obama, with the following exceptions: Florida, Wisconsin, South Carolina, Missouri, and Georgia. Of these, all but Georgia fall within 50% of the target 8 USAMO qualifiers per 10 million residents. Georgia has 18 qualifiers per 10 million residents. Note also that the entire USA has 16 qualifiers per 10 million residents.

Furthermore, if USAMO qualifiers had been used instead of population for determining electoral votes, Obama would have won with 86% of the vote rather than 68%. In general, if the Democrat can secure all those states with at least 1 qualifier per million residents (plus DC), he will win with 303 votes. He can even lose the three red states in that category (Georgia, Missouri, and South Carolina) for exactly 269.

USAMO qualifiers per 10 million residents (for states with more than one qualifier) are:

• NH — 122 (all of their qualifiers are from Phillips Exeter Academy)
• MA — 55
• ME — 30
• CA — 29
• NJ — 29
• CT — 29
• VA — 28
• MD — 25
• WA — 21
• IN — 19
• OR — 19
• NY — 18
• GA — 18
• IA — 17
• NM — 15
• MI — 15
• PA — 14
• MO — 14
• SC — 13
• IL — 13
• NC — 13
• OH — 9
• CO — 8
• MN — 8
• UT — 7
• KS — 7
• WI — 7
• KY — 7
• TX — 7
• FL — 6
• LA — 5
• AL — 4
• TN — 3

The states with only one USAMO qualifier are WY, VT, ND, AK, SD, DE, MT, RI, HI, ID, NE, WV, NV, AR, MS, OK, and AZ. The only blue one of these which falls below 8 qualifiers per 10 million is Nevada (we would expect it to have at least 2 qualifiers to fit the expected pattern). Otherwise, it is at least possible that each state fits the pattern of 8 qualifiers per 10 million residents if and only if it votes Democratic.

More than a year ago, when I had my employment benefits with BAE Systems, I called my benefits center with a general question. The customer service representative refused to answer until I gave her my password. I didn’t have a password, so she told me that they would mail my new password to me.

But I needed an answer, so I tried the website, only to be informed that my new password is in the mail and I should wait for its arrival.

In a week, a letter with a password arrived and I called the benefits center again. I happily told them my new password and opened my mouth to ask my question. However, they didn’t accept my password. Obviously, they had changed my password twice, first when I called and then again when I tried their website. Since only ten minutes passed between these two events, both passwords should have arrived on the same day. But that didn’t happen. So my valid password was still in the mail.

In the second it took me to recover from this news, the customer representative told me that they would be sending me a new password and hung up before I could tell her not to.

A new password arrived the next day. I knew that they had already reset that password, and that I’d have to wait a week for the third password to arrive.

I was tempted to call them again and try to create an infinite password resetting loop, but I actually needed to ask my question. So I threw away my freshly arrived, but no-longer-valid password and waited for a week for the next one.

I was lucky to figure it out so quickly, for otherwise my problem could have spiraled out forever. As a professional specifications writer, here are my suggestions to all benefits centers that have that kind of software on what they should do:

• Don’t send an extra password if a password was sent not long ago, for example, in the last two days.
• If two passwords are mailed to a client in the same week, make both of them valid.
• Use email rather than mail.
• Don’t request passwords for general questions.

I had to wait two weeks to ask a simple question. Now I am writing and complaining about it in the hopes that someone who can fix the problem will read this. Maybe it would have been more productive to write a program that clicks on the “I forgot my password” button every second. This would have daily generated thousands of letters with new passwords to me. Maybe then this problem would have drawn attention sooner.

My son Sergei brought back the Flip-Flop game from Canada/USA Mathcamp, and now I teach it to my students. This game trains students in the multiplication table for seven and eight. These are the most difficult digits in multiplication. This game is appropriate for small kids who just learned the multiplication table, but it is also fun for older kids and adults.

This is a turn-based game. In its primitive simplification kids stand in a circle and count in turn. But it is more interesting than that. Here’s what to say and do on your turn, and how the game determines who is next.

First I need to tell you what to say. On your turn, say the next number by default. However, there are exceptions when you have to say something else. And this something else consists of flips and/or flops.

So what are flips? Flip is related to seven. If a number is divisible by seven or has a digit seven, instead of saying this number, we have to say “flip” with multiplicities. For example, instead of 17 we say “flip” because it contains one digit seven. Instead of 14 we say “flip”, because it is divisible by seven once. Instead of 7 we say “flip-flip”, as it is both divisible by seven and has a digit seven. Instead of 49, we say “flip-flip” as 49 is divisible by the square of seven. Instead of 77 we say “flip-flip-flip” as it has two digits seven and is divisible by seven once.

Flop relates to eight the same way as flip relates to seven. Thus, instead of 16 we say “flop” as it is divisible by eight; instead of 18 we say “flop” as it contains the digit eight; and for 48 we say “flop-flop” as it is both divisible by eight and contains the digit eight.

A number can relate to seven and eight at the same time. For example 28 is divisible by seven and contains the digit eight. Instead of 28 we say “flip-flop”. The general rule is that all flips are pronounced before all flops. For example, instead of 788 we will say “flip-flop-flop-flop” as it is divisible by eight and contains the digit seven once and the digit eight twice.

The sequence of natural numbers in the flip-flop version starts as the following: 1, 2, 3, 4, 5, 6, flip-flip, flop-flop, 9, 10, 11, 12, 13, flip, 15, flop, flip, flop, 19, 20, flip, 22, 23, flop, 25, 26, flip, flip-flop, 29, 30, 31, flop, 33, 34, flip, 36, flip, flop, 39, flop, 41, flip, 43, 44, 45, 46, flip, flop-flop, flip-flip, 50, 51, 52, 53, 54, 55, flip-flop, flip, flop, 59, 60, 61, 62, flip, flop-flop, 65, 66, flip, flop, 69, flip-flip, flip, flip-flop, flip, flip, flip, flip, flip-flip-flip, flip-flop, flip, flop-flop, flop, flop, flop, flip-flop, flop, flop, flip-flop, flop-flop-flop, flop, 90, flip, 92, 93, 94, 95, flop, flip, flopflip-flip-flop, 99, 100.

So how does the turn change? Everyone stands in a circle and says their number the way explained above. We start clockwise and move to the next number. For every flip we reverse the direction and for every flop we skip a person. That means that if we have two flips, we don’t change the direction, while for two flops we skip two people. If we have flips and flops together, for example 28 corresponds to “flip-flop”, then first we change the direction and then we skip a person.

On top of that, there is an extra rule for what you do on your turn. If you say something other than a default number, you switch your position from standing to sitting and vice versa. Sometimes I skip this extra feature — not because I am too lazy to exercise, but because I usually conduct this game in a classroom, where all the desks prevent us from fully enjoying such physical activity.

There are two ways to play this game: as a competition or as practice. When we are competing, a person who makes a mistake drops out. If we’re just practicing, no one drops out. Sometimes I am particularly generous and allow my kids one mistake before making them drop out after the second mistake. So far we have played up to 100. I am curious to see if we can ever reach 700 and how long we will be able to continue the game after that.

Testing in the US is dominated by multiple-choice questions. Together with the time limit, this encourages students to stop thinking and go for guessing. I recently wrote an essay AMC, AIME, USAMO Contradiction, in which I complained about the lack of proofs in the first two rounds of math competitions.

Is there a way to improve the situation? I grew up in the USSR, where each round of the math competition had the same format: you were given several hours to write proofs for three or four difficult problems. There are two concerns with organizing a competition in this way. First, the Russian system is much more expensive, whereas the US’s multiple choice tests can be inexpensively checked by a computer. Second, the Russian system is prone to unfairness. You need many math teachers to check all these papers on the highest level. Some of these teachers might not be fully qualified, and it is difficult to ensure uniform checking. This system can’t easily be adopted in the US. I am surprised I haven’t heard of lawsuits challenging USAMO results, but if we were to start having proofs at the AMC level with several hundred thousand participants, we would get into lots of trouble.

An interesting compromise was introduced at the Streamline Olympiad. The problems were multiple choice, but students were also requested to write proofs. Students got two points for a correct multiple choice answer, and if the choice was correct the proof was checked. Students could get up to three points for a correct proof. This idea solves two issues. The writing of proofs is rewarded at an early stage and the work of the judges is not as overwhelming as it would have been, had they needed to check every proof. However, there is one problem that I discussed in previous posts that this method doesn’t solve: with multiple choice, minor mistakes cost you the whole problem, even though you might have been very close to a solution. If we want to reward thinking more than accuracy, the proof system allows us to give credit for partial solutions.

I can suggest another approach. If the Russians require proofs for all problems and the Americans don’t require proofs for any problem, why not compromise and require a proof for one problem out of the set.

But I actually have a bigger idea in mind. I think that current development in artificial intelligence may soon help us to check the proofs with the aid of a computer. Artificial intelligence is still far from ready to validate that a mathematical text a human has produced constitutes a proof. But in this particular case, we have two things working for us. First, we can use humans and computers together. Second, we do not need to check the validity of any random proof; we need to check the validity of a specific proof of a simple problem that we know in advance, thus allowing us to prepare the computers.

Let us assume that we already can convert student handwriting into computer-legible text or that students write directly in LaTeX.

Here is the plan. Suppose for every problem, we create a database of some sample right, wrong and partial solutions with corresponding scores. The computer checks the students’ solutions against the given sample. Hopefully, the computer can recognize small typos and deviations that shouldn’t change the point value. If the computer encounters a solution that is significantly different from the ones in the sample, it sends the solution to human judges. Humans decide how to score the solution and the solution and its score is added to the sample database.

For this system to work, computers should be smart enough not to send too many solutions to humans. So how many is too many? My estimate is based on the idea that we wouldn’t want the budget of AMC to go too much higher than the USAMO budget. Since USAMO has 500 participants, judges check just a few hundred solutions to any particular problem. With several hundred thousand participants in AMC, the computer would have to be able to cluster all the solutions into not more than a few hundred groups. The judges only have to check one solution in each group.

As a bonus, we can create a system where for a given solution that is not in the database, the computer finds the closest solution and highlights the difference, thus simplifying the human’s job.

In order to improve math education, we need to add proofs when teaching math. My idea might also work for SATs and for other tests.

Now that there is more money available for education research, would anyone like to explore this?

A stick has two ends. If you cut off one end, how many ends will the stick have left?

This pre-kindergarten math problem was given to me by Maxim Kazarian who lives in Moscow, Russia. That got me thinking about math education in the US. Actually, just about anything can get me thinking about education in our country. One of our math education patterns is to provide simplified templates and to train kids to plug numbers into them without thinking.

Math education should be about thinking. We need to give kids a lot of math problems that do not fit into standard templates, in order to encourage creative thinking. Here is another puzzle from Maxim:

A square has four corners. If we cut one corner off, how many corners will the remaining figure have?

I invite my readers to invent additional problems that sound as if a subtraction by one is needed, when, in fact, it is not. Here is my contribution:

Anna had two sons. One son grew up and moved away. How many sons does Anna have now?

To get to the national swimming championship, you need to win the state running championships.

What? Is that a joke? Perhaps you’re having the same reaction. Because this is exactly what is happening with math competitions. The official USA math competition has three rounds: AMC, AIME and USAMO.

AMC is a multiple-choice competition with 25 problems in 75 minutes. To be good at it, you need speed, accuracy and the ability to guess well.

AIME is 3 hours long and has 15 problems. The problems are a different level of difficulty and guessing will not help you. Though AIME is also multiple-choice, unlike AMC where you choose out of 5, in AIME you choose out of 1,000. But you still need speed and accuracy. A small arithmetic mistake will cost you the whole problem.

USAMO is a competition that runs for 9 hours and has 6 problems. The problems are much harder and you have to write proofs. Proofs? What proofs? Where are the proofs coming from? It is like you got to the national swimming championship because you are a great runner, but you do not know how to swim.

As the result of this system of selection, the USA team at the International Math Olympiad has diverse skills: these kids are good at all three types of the math competitions. It is like taking an Olympic triathlon team to the Olympic swimming event.

However, the US loses by selecting in this way. There are many kids who are great mathematicians: they may be good at difficult problems but not that good at speed-racing problems. An arithmetic mistake costs you only one point at IMO, but a whole problem at AIME. There are kids who are deep mathematicians prone to small arithmetic mistakes. They could get a gold medal at IMO, but they can’t pass AMC or AIME.

Not only that. As many AMC problems are boring and do not require ideas, AMC might discourage kids from all math competitions at an early stage.

I will write later with my ideas about how to change AMC. Right now I would like to offer a solution to a smaller problem. I am sure that the US math team organizers know many cases in which a non-senior kid does great at USAMO and is potentially a team member for the next year’s US IMO team, but, oops, next year he can’t pass AMC.

I suggest the following: USAMO participants are allowed to go to next year’s AIME no matter what their AMC scores are. USAMO winners are allowed to go to the next year’s USAMO no matter what their AIME results are. This way kids who have proved that they are great swimmers do not need to compete in running again.