Tanya Khovanova's Math Blog

I decided to see the film The Oxford Murders because I loved the idea of “Frodo” being a graduate student in logic. By Frodo, I mean the actor Elijah Wood, who is very believable as a math student.

At the core of the movie are sequences of numbers and symbols. When the characters started a discussion about how to continue a sequence, I immediately tensed up. Why? Because when people ask what the next element in the sequence is, I get ready to confront them, by explaining that there are many ways to continue a sequence. For example, the sequence — 1, 2, 4 — could be powers of two, or could be Tribonacci numbers, or any of 10,000 sequences that the Online Encyclopedia of Integer Sequences spills out if you plug in 1, 2, 4. That is, if we do not count the infinity of sequences that are not in the Encyclopedia.

To my surprise and relief, the logic Professor, one of the main characters in the movie, explained that there is no unique way to continue a sequence. From that moment on, I relaxed and fell in love with the movie.

The movie is a detective story with a lot of twists and turns. The crimes are related to symbols. The first two symbols are in the picture below. Can you guess the next symbol?

I cannot. There is an irony in the film at this point, because the Professor and the student need to guess the sequence in order to solve the crimes. But the Professor has already explained that there is no unique way to continue. So illogical for a movie about logic.

And what’s worse, the sequence of symbols they finally discover doesn’t make sense. I guess I fell in love with this movie too quickly.

I am reading the book Eat to Live by Joel Fuhrman. It contains a formula that as a math formula doesn’t make any sense. But as an idea, it felt like a revelation. Here it is:

HEALTH = NUTRIENTS/CALORIES

The idea is to choose foods that contain more nutrients per calorie. The formula doesn’t make sense for many reasons. Taken to its logical conclusion, the best foods would be vitamins and tea. The formula doesn’t provide bounds: it just emphasizes that your calories should be nutritious. However, too few calories — nutritious or not — and you will die. And too many calories — even super nutritious — are still too many calories. In addition the formula doesn’t explain how to balance different types of nutrients.

Let’s see why it was a revelation. I often crave bananas. I assumed that I need bananas for some reason and my body tells me that. Suppose I really need potassium. As a result I eat a banana, which contains 800 milligrams of potassium and adds 200 calories as a bonus. If I ate spinach instead, I would get the same amount of potassium at a price of only 35 calories.

The book suggests that if I start eating foods that are high in nutrients, I will satisfy my need for particular nutrients, and my cravings will subside. As a result I will not want to eat that much. If I start my day eating spinach, that might eliminate my banana desire.

I’ve been following an intuitive eating diet. I am trying to listen to my body hoping that my body will tell me what is better for it. It seems that my body sends me signals that are not precise enough. It’s not that my body isn’t communicating with me, but it is telling me “potassium” and all I hear is “bananas.” What I need to do is use my brain to help me decipher what my body really, really wants to tell me.

As Dr. Fuhrman puts it, we are a nation of overfed and malnourished people. But Fuhrman’s weight loss plan is too complicated and time-consuming for me, so I designed my own plan based on his ideas:

I will start every meal with vegetables, as they are the most nutritious. I hope that vegetables will provide the nutrients I need. That in turn will make me less hungry by the next meal, at which time I’ll take in fewer calories. I will report to my readers whether or not my plan works. I’m off to shop for spinach. Will I ever love it as much as bananas?

Most movies related to mathematics irritate me because of simplifications. I especially do not like when a movie pretends to be intelligent and then dumbs it down. I recently watched the Spanish movie Fermat’s Room, which, as you may guess, annoyed me several times. In spite of that I enjoyed it very much.

The movie opens with people receiving invitations to attend a meeting for geniuses. To qualify for the meeting they need to solve a puzzle. Within ten days, they must guess the order underlying the following sequence: 5, 4, 2, 9, 8, 6, 7, 3, 1. Right away, at the start of the movie, I was already annoyed because of the simplicity of the question. You do not have to be a genius to figure out the order, not to mention how easy it would be to plug this sequence into the Online Encyclopedia of Integer Sequences to find the order in five minutes.

The participants were asked to hide their real names, which felt very strange to me. All famous puzzle solvers compete in puzzle championships and mystery hunts and consequently know each other.

The meeting presumably targets the brightest minds and promises to provide “the greatest enigma.” During the meeting they are given seven puzzles to solve. All of them are from children’s books and the so-called “greatest enigma” could easily be solved by kids. Though I have to admit that these were among the cutest puzzles I know. For example:

There are three boxes: one with mint sweets, the second with aniseed sweets, and the last with a mixture of the two. The boxes are labeled, but all the labels are wrong. What is the minimum number of sweets you need to taste to correctly re-label all the boxes?

Another of the film’s puzzles includes a light bulb in a room and three switches outside, where you have to correctly find the switch that corresponds to the bulb, but you can only enter the room once. In another puzzle you need to get out of prison by deciding which of two doors leads to freedom. You are allowed to ask exactly one question to one of the two guards, one of whom is a truth-teller and the other is a liar.

The other four puzzles are similar to these three I have just described. To mathematicians they are not the greatest enigmas. They are nice material for a children’s math club. For non-mathematicians, they may be fascinating. Certainly it’s a good thing that such tasteful puzzles are being promoted to a large audience. But they just look ridiculous as “the greatest enigmas.”

So what is it about this film that I so enjoyed?

The intensity of the movie comes from the fact that the people are trapped in a room that starts shrinking when they take more than one minute to solve a puzzle.

I well remember another shrinking room from Star Wars: A New Hope. When Princess Leia leads her rescuers to a room, it turns out to be a garbage compactor. The bad guys activate the compactor and two opposite walls start moving in. In contrast, Fermat’s room is shrinking in a much more sophisticated way: all four walls are closing in. Each of the walls in the rectangular room is being pressured by an industrial-strength press. The walls in the corners do not crumble, but rather one wall glides along another. I was more puzzled by this shrinking room than I was by the math puzzles. I recommend that you try to figure out how this can be done before seeing the movie or its poster.

However, the best puzzle in the movie is the plot itself. Though I knew all the individual puzzles, what happened in between grabbed me and I couldn’t wait to see what would happen next. I saw the movie twice. After the first time, I decided to write this review, so I needed to check it again. I enjoyed it the second time even better than the first time. The second time, I saw how nicely the plot twists were built.

Maybe I shouldn’t complain about the simplicity and the familiarity of the puzzles. If they were serious new puzzles I would have started solving them instead of enjoying the movie. The film’s weakness might be its strength.

Tired of the same old sudoku? Here’s an opportunity to try many variations of it. Thomas Snyder and Wei-Hwa Huang wrote a book called Mutant Sudoku. The authors are both Sudoku champions. I like the book because the authors are trying to bring everyone up to their level, rather than dumbing down their puzzles. So the book is not at all boring as are most Sudoku books.

The book contains about 180 fun puzzles. Look at the variety:

• Tight Fit Sudoku
• Extra Space Sudoku
• Tile Sudoku
• 3-D Sudoku
• Outside Sudoku
• Shape Sudoku
• Target Sum Sudoku
• Thermo-Sudoku
• Consecutive Sudoku
• Surplus Sudoku
• Deficit Sudoku
• Chimeric Sudoku

Wei-Hwa Huang kindly sent me this sample Thermo Sudoku puzzle from the book to use on my blog. The grey areas represent thermometers. Every particular thermometer has to have numbers in increasing order (not necessarily consecutive) starting from the bulb.

The second book by the same authors Sudoku Masterpieces: Elegant Challenges for Sudoku Lovers, is itself a masterpiece. With about 100 puzzles, there are fewer than in the first book, but there are more types of puzzles. As a consequence, you’ll have less practice for each particular type, but more variety. In addition, as you can see from the cover, the second book is elegantly designed.

I bought both books and immediately started scribbling in the first one. My bad handwriting would seem so out of place in the beautiful second book that I have not even started working in it yet. Maybe I will give it as a gift to someone with better penmanship.

I met Ed (Edik) Frenkel 20 years ago at Harvard when he was a brilliant math student of my now ex-husband, and a handsome young man. Now, at 42, he is a math professor at Berkeley and he is even hotter. He made a bizarre move for a mathematician: he produced and starred in an erotic short movie, Rites of Love and Math. If he wants to be known as the sexiest male mathematician alive, he just might get the title.

The movie created a controversy when Mathematical Sciences Research Institute (MSRI) withdrew its sponsorship for the first screening after a lot of objections based on the trailer. My interest was piqued by a painting that dominated the visual of the trailer’s erotica scene. The black and white amateur painting is of the integral sign with Russian letters stylized as math symbols that spell the word “Truth”. In addition, the name of the woman in the movie, Mariko, means “truth” in Japanese. Though it felt pretentious, I was hoping that the movie would be symbolic. When I heard that the actors do not talk in the movie, my expectations of symbolism grew. I love movies that are open to interpretation. So I bought the movie, watched it and wrote the following review. Before getting to the review itself I would like to thank Ed Frenkel for sending me the photos and giving me permission to use them in my frank assessment of his work.

Here is the plot:

A Mathematician, hoping to serve humanity, discovers a formula of Love. Bad guys find an evil way to use the formula to destroy humanity and are hunting for the Mathematician, who is hiding in his lover Mariko’s home. The Mathematician fears for his own life. Although it would make sense to destroy all the papers with the formula, the Mathematician loves his formula even more than his lover and himself. He wants to preserve the formula and tattoos it on her body with her consent.

There is much about the film that I like, including the slow pace and the visuals, with their minimalistic background and palette of black, white and red. The camera work is superb.

I welcomed the idea of a Love formula, because mathematics is ready to broaden the scope of its models, including venturing into love. Of course, some mathematical models of relationships already exist.

It’s great that the mathematician is portrayed against the stereotype: he’s neither introverted nor asexual. Unfortunately, the movie plays into other stereotypes of male mathematicians — being creepy and demanding sacrifices from their wives in the name of mathematics. As I mentioned, I was looking forward to the movie, hoping that it would encourage the imagination of viewers in their interpretations. To my disappointment, every scene in the movie is preceded by text that describes the plot, removing any flexibility of interpretation. Besides that, the emotions portrayed didn’t quite match the written plot, in no small part because Ed Frenkel is not a good actor.

The idea of preserving a formula by tattooing it on someone is beyond strange. He could have used a safe-deposit box. Or put the formula in an envelope and given it to the lover to keep, or just encrypted it, etc. With narcissistic lack of consciousness, the Mathematician seems unaware of the implications of his action of imprinting this dangerous secret on Mariko. She can never go swimming, or go to the gym, or be intimate with anyone else. Moreover, if the bad guys discover that Mariko is the Mathematician’s lover, her life will be in grave danger. Not to mention that tattooing is painful.

Something that could have been interesting and watchable in a historic movie, in this contemporary movie seems pointlessly cruel, dehumanizing and senseless.

I know for sure that Ed Frenkel is not stupid, so what are his reasons for constructing the plot in this way? Before investigating his reasons, I have a mathematical complaint about the movie. Every mathematician and teacher knows that when asserting a formula you need to indicate its interpretation: what its symbols refer to in the real world. For example, suppose I tell you my own great Formula of Love: C_n = (2n)!/(n+1)!n!. You may recognize C_n as the Catalan numbers, but what does this have to do with Love? To give the formula meaning I need to tell you that C_n is the number of ways you can seat n loving couples at a round table with 2n chairs, so that each couple can join hands (assuming the arms are long enough to reach across the table) without any two pairs of arms crossing. Assigning an interpretation makes the Catalan numbers part of the world’s growing body of romantic research.

Writing a formula without mentioning what the variables mean fails to preserve it for the future. Ed Frenkel knows that. Wait a minute. The formula in the movie is actually not the Formula of Love, but a real formula from Ed’s paper on instantons. It’s right there, formula 5.7 on page 74. Every variable is explained in the paper. Ah-ha! So his movie isn’t actually about art, but rather about Ed’s formula. Indeed, there is no real Formula of Love. In such situations in other movies, they have simply shown fragments of a formula. However, in Rites of Love and Math, Frenkel’s formula — which has nothing to do with Love — is shot in full view, zooming in slowly.

The movie is a commercial. Ed is using our fascination with sex to popularize his formula, and using his formula and his scientific standing to advertise his body.

I was so disappointed that the default interpretation of the movie was imposed on me by those pre-scene texts, that I decided to watch the movie for a second time, trying to ignore the text, hoping to find some new meaning.

If you decide to see the movie, you’ll probably come up with your own interpretation of the plot. I actually came up with several. I had a funny one and an allegorical one, but the most interesting task for me was to try create an interpretation matching the emotions portrayed:

Mariko knows that something is wrong in her sex life with the Mathematician. But she still loves him and writes him a love letter. The Mathematician comes to Mariko’s place. He is distant and cold. They cuddle. He explains to her that sex doesn’t bring him pleasure anymore and that moreover, he can’t even perform. He tells her that the only thing that brings him joy is mathematics and suggests that his sexual dysfunction and lack of pleasure will be fixed if they tattoo his favorite formula on her body. She agrees, but first they decide to give sex a last try. They try real hard. But he can’t relax and he doesn’t enjoy it, so she agrees to the tattoo. He does get excited during the tattooing process itself, but once he finishes his whole formula, he is no longer turned on. Mariko’s suffering has been in vain.

Suppose Romeo is encouraged by love and attention. If Juliet likes him, his feelings for Juliet grow and flourish. If she doesn’t like him, he loses his interest in her.

Juliet, on the other hand, is the opposite. If Romeo doesn’t like her, she needs to win him over and her attraction for him grows. If he likes her, she feels that her task is accomplished and she loses her interest in him. Juliet likes the challenge more than the relationship.

Steven Strogatz used differential equations to model the dynamics of the relationship between Romeo and Juliet. This is a new and fascinating area of applied mathematical research; you can read more about the roller-coaster relationship between Romeo and Juliet in Steven Strogatz’s Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering.

Mathematicians like symmetry: in math literature they switch the roles between Romeo and Juliet randomly. So in some papers they give Romeo the role of preferring a challenge over love and in some papers they give that role to Juliet.

When I teach this subject of love, Alexander Pushkin’s famous quote always pops into my mind. The quote comes from the first lines of Chapter Four of Eugene Onegin, and in Russian it is:

Чем меньше женщину мы любим,
Тем легче нравимся мы ей…

I didn’t like the English translations that I found, so I asked my son Alexey to provide a more literal translation:

The less we love a woman, the more she likes us in return…

I blame Pushkin for my tendency to always pick Juliet as the character who thrives on the challenge, even though men are often assumed to be the chasers. I’d like to ask my readers to comment on these roles: Do you think both genders play these roles equally? If not, then who is more prone to be into the chase?

Let’s return to mathematical models. In the original model, the reactions of Romeo and Juliet are a linear function of feelings towards them. I would like to suggest two other roles, in which people react to the absolute value of feelings towards them. They do not care if it is love or hate: they care about intensity.

First, there is the person, like my friend Connie, who feeds on the emotions of other people. She’s turned on by guys who love her as well as by guys who hate her. If they’re indifferent, she’s turned off.

Second, there is the opposite type, like my colleagues George, Joseph, David and many others. They hate emotion and prefer not to be involved. They lose all interest in people who feel strongly about them and they like people who are distant. I know the name for this role: it’s a mathematician!

 I love the TV series of Angel and of Buffy the Vampire Slayer. I enjoy the excitement of saving the world every 42 minutes. But as a scientist I keep asking myself a lot of questions.

Where do vampires take their energy from? Usually oxygen is the fuel for the muscles of living organisms, but vampires do not breathe. Vampires are not living organisms, and yet they have to get their energy from somewhere.

When you kill a vampire, it turns to dust. If organisms are 60% water, then a 200-pound vampire should generate 80 pounds of dust. So why, in the series, do you get just a little puff of dust whenever someone plunges a stake into a vampire? Plus 120 pounds of water apparently evaporates instantly during staking. Can someone who is less lazy than me please calculate the energy needed to evaporate 120 pounds of water in one second? Because my first reaction is that you would need an explosion, not just one stab with Buffy’s stake.

All these unscientific elements do not actually bother me that much. What does bother me are inconsistencies in logic. For example, at the end of Season One of Buffy, Angel refuses to give Buffy CPR, claiming that as a vampire he can’t breathe. But then how can Spike and other vampires smoke? If they can smoke that means they are capable of inhaling and exhaling. Not to mention that these vampires talk: wouldn’t they need an airflow through their throats to produce sounds?

It would make more sense for the show to state that vampires do not need to breathe, but are nonetheless capable of inhaling and exhaling. So Angel should have given Buffy CPR. It would have created a great plot twist: Angel saves Buffy at the end of Season One, only for her to send him to the hell dimension at the end of Season Two.

Back to breathing. I remember a scene in “Bring On the Night” in which Spike was tortured by Turok-Han holding his head in water. But if Spike can’t breathe, why is this torture?

Another thing that bothers me in the series is not related to what happens but to what doesn’t happen. For example, vampires do not have reflections. So I don’t understand why every vampire-aware person didn’t install a mirror on the front door of their house to check for reflections before inviting anyone in.

Also, it looks like producers do not care about backwards compatibility. Later in the series we get to know that vampires are cold. Watch the first season of Buffy with that knowledge. In the very first episode, Darla is holding hands with her victim, but he doesn’t notice that she is cold. Later Buffy kisses Angel, before she knows that he is a vampire, and she doesn’t notice that he’s cold either. Unfortunately, the series also isn’t forward compatible. In the second season of Angel in the episode “Disharmony”, when we already know that vampires are cold, Harmony is trying to reconnect with Cordelia. They hug and touch each other. Such an experienced demon fighter as Cordelia should have noticed that Harmony is cold and, therefore, dead.

Finally, let’s look at Spike in the last season of Angel. Spike is non-corporeal for a part of the season; we see him going through walls and standing in the middle of a desk. Yet, one time we see him sitting on a couch talking to Angel. In addition, he can take the stairs. He can go through the elevator wall to ride in an elevator instead of falling down through its floor. And what about floors? Why isn’t he falling through floors? Some friends of mine said that we can assume that floors are made from stronger materials. But, if there is a material that can prevent Spike from penetrating it, they ought to use this material to make a weapon for him.

I’ve never been involved in making a show, but these producers clearly need help. Perhaps they should hire a mathematician like me with an eye for detail to prevent so many goofs.

I am usually disappointed with American math text books. I have had an underwhelming experience with them. Often I open a book and in the next 15 minutes, I find a mistake, a typo, a misguided explanation, sloppiness, a misconception or some other annoyance.

I was pleasantly surprised when I opened the book Introduction to Algebra by Richard Rusczyk. I didn’t find any flaws in it — not in the first 15 minutes, and not even in the first hour. In fact, having used the book many times I have never found any mistakes. Not even a typo. This was disturbing. Is Richard Rusczyk human? It was such an unusual experience with an American math book, that I decided to deliberately look for a typo or a mistake. After half a year of light usage, I finally found something.

Look at problem 7.3.1.

Five chickens can lay 10 eggs in 20 days. How long does it take 18 chickens to lay 100 eggs?

There is nothing wrong with this problem. But the book is accompanied by the Introduction to Algebra Solutions Manual in which I found the following solution, that I’ve shortened for you:

The number of eggs is jointly proportional to the number of chickens and the amount of time. One chicken lays one egg in 10 days. Hence, 18 chickens will lay 100 eggs in 1000/18 days, which is slightly more than 55 and a half days.

What is wrong with this solution? Richard Rusczyk is human after all.

I like this book for its amazing accuracy and clean explanations. There are also a lot of diverse problems in terms of difficulty and ideas. Richard Rusczyk has good taste. Many of the problems are from different competitions and require inventiveness. I like that there are a lot of challenge problems that go beyond the boring parts of algebra. Also, I like that important points of algebra are chosen wisely and are emphasized.

This book might not be for everyone. It doesn’t have pretty pictures. It doesn’t have color at all. This is not a flaw for a math book. The book concentrates on ideas and problems, not entertainment. So if you’re looking for math entertainment, you’ll find it on my blog. For solid study, try Richard Rusczyk’s books.

I love Raymond Smullyan’s books , especially the trick puzzles he includes. The first time I met him in person, he played a trick on me.

This happened at the Gathering for Gardner 8. We were introduced and then later that day, the conference participants were treated to a dinner event that included a magic show. In one evening I saw more close-up magic tricks than I had in my whole life. This left me lightheaded, doubting physics and my whole scientific outlook on life.

Afterwards, Raymond Smullyan joined me in the elevator. “Do you want to see a magic trick?” he asked. “I bet I can kiss you without touching you.” I was caught off guard. At that moment I believed anything was possible. I agreed to the bet.

He asked me to close my eyes, kissed me on the cheek and laughed, “I lost.”

The book An Introduction to Diophantine Equations by Titu Andreescu and Dorin Andrica is targeted at people preparing for USAMO and IMO. It contains a lot of problems on Diophantine equations from math Olympiads used in various math Olympiads all over the world.

The first chapter discusses several methods for solving Diophantine equations: decomposition, using inequalities, using parameters, modular arithmetic, induction, infinite descent, and other miscellaneous ideas. Each sub-chapter starts with a short description of the method, accompanied by several solutions to sample problems. At the end of each sub-chapter there are a plethora of exercise problems.

The second and the third chapters are more theoretical. The former discusses some classical equations and the latter looks at Pell’s equation. These two chapters also contain problems, but the bulk of the chapters is devoted to basic theory that is essential to an understanding of Diophantine equations.

For those who are training for the Olympiads, this is an important book to own, not only because there are few other books on the subject, but because it provides so many useful problems.

I’ve long complained that most training books for math competitions leave out any discussion of how we choose a method by just looking at a problem. Andreescu and Andrica didn’t fill that gap with this book.

Perhaps in their next book they will point out clues that indicate that a particular problem might be solved by the parametric method. And explain which types of problems are best solved with induction. Let them challenge students to find those clues in a problem that help us to judge which method might be most promising, instead of randomly trying one method after another. Let me give you a sample problem from the book, which originated at the Balkan Mathematical Olympiad:

Prove that the equation x⁵ – y² = 4 has no solutions in integers.

The solution is to take the equation modulo 11, and see that it is impossible.

Is there a reason to start with the modular arithmetic method and not with other methods? If we use modular arithmetic, do we recognize why it’s best to start with 11? I’m convinced that this problem has sufficient clues to suggest starting with checking this equation modulo 11.

I wonder if you, my readers, agree with me. If so, can you explain which hints in the problem lead to taking the equation modulo 11? I believe it should be a part of competition training to learn to identify clues that suggest that one direction might be preferable to the others.