Tanya Khovanova's Math Blog

When I was in 8th grade, I was selected to be part of the Moscow math team and went to Yerevan, Armenia, to participate in the All-Soviet Math Olympiad. A group of us boarded a bus, and Alexander Karabegov paid for all of our bus tickets. He was from Yerevan himself and wanted to be a gracious host. I was impressed. The next time I met him was when I started studying at the Moscow State University. We have been friends ever since. He was even the best man at one of my weddings. Now, he lives in Texas and sends me his original puzzles from time to time. Today, he sent me a new one.

WARNING. His solution to the puzzle is also included. So if you want to solve it yourself, stop reading after the next paragraph.

Puzzle. A number c is called a fixed point of a function f, if it is a solution of the equation f(x) = x; that is, if f(c) = c. Find all solutions of the equation g(g(x)) = x, where g(x) = x² + 2x − 1; that is, find all fixed points of the function f(x) = g(g(x)). (We can assume that x is a real number.)

I gave the puzzle to my students, and they converted it to a fourth-order equation, which they solved using various methods. What I liked about Alexander’s solution is it only uses quadratic equations. I am too lazy to give his full solution. Here is just his solve path.

Solve path. If c is a fixed point of the function g(x), then it is a fixed point of f(x) = g(g(x)). Solving the equation g(c) = c gives us two fixed points. We need two more, as our equation is quartic. Suppose a is another fixed point. Let b = g(a). It follows that g(b) = a. Moreover, we can assume that a is not b, as we covered this case before. We get two equations a² + 2a − 1 = b and b² + 2b − 1 = a. Subtracting one equation from another, we get a quadratic equation that has to be divisible by a −b. As b is not a, by our assumption, we can divide the result by a − b, expressing b as a linear function of a. We plug this back into one of the two equations and get a quadratic equation for a, supplying us with the remaining two solutions. TADA!

(A small piece I wrote on Dec 29, 2009. Edited in 2024.)

They were black, very comfy, and felt like a second skin. The shoes had this shock-absorbing cushioning, so asphalt felt like carpet.

I had them for 10 years. They served me for so long that I started believing our happy relationship would last forever.

First, I noticed that they are not black anymore. They acquired a greenish color. Then, the sound changed. Steps started sounding like farts. I trusted my shoes so much that, at first, I thought I was just getting old. But I realized that I couldn’t be that perfect: I couldn’t possibly fart with such a precise rhythm. Besides, I should have run out of gas from time to time.

When I came home, I took off my shoes and looked at them. The sole of one shoe was gone. My love affair with my shoes was over. Oh well. The divorce was easy. They went to my garbage can. No tears, no broken hearts, just a lost sole.

Have you ever heard of ikigai? The Japanese concept which gives four simple requirements for a happy career:

• Do what you love.
• Do what you are good at.
• Do what impacts the world.
• Do what you can get paid for.

I often think about it for myself and for my students. Is this good advice for finding a career path?

I like that ikigai separates the first two requirements: passion and gift. Many of my students do not see the difference, as passion and gift are highly correlated. When you love something, you practice it and become better at it. When you are good at something, it becomes easy and enjoyable.

Nonetheless, passion and gift are different. Unfortunately, I’ve seen students who are good at math only because their parents push them, but they do not love it. Some of them already found their passion but are afraid to tell their parents. Some haven’t yet found their passion, but it is perfectly clear that math is not it. So, a gift doesn’t imply passion.

What about the other way around? My programs are too selective, so I haven’t seen students who are not gifted in math. I will use myself as an example. I have always passionately loved dancing, but it is obvious that my dancing career would have been a disaster. I am very happy I closed that career path in fifth grade.

Anyway, the first two ikigai requirements are not the same, and both are necessary.

The third ikigai requirement is about doing what the world needs. Impacting the world is a great motivator and makes you feel good. And yet, I see happy and successful mathematicians who only care about the beauty of what they are doing and nothing else. This requirement is important but might not be a deal breaker for everyone.

The last ikigai requirement is crucial. If you are not being paid for your efforts, it is not a career; it is a hobby. I got attracted to it because it includes an important caveat: you need to find people who want to pay you for what you can offer. I recently wrote an essay Follow Your Heart? about many young aspiring opera singers who ignored this last requirement and ended up changing careers.

Nevertheless, the whole concept of ikigai bugs me. People who find their dream job might agree to work for much less pay than they are worth. It opens them up to potential exploitation by greedy employers.

Have I reached my ikigai? Judging by my low pay, I am close.

My grandkids like playing a game while I drive. They look out the window to spot the cars they like and score points. A Jeep is 1, a convertible is 10, a Mini Cooper is 40, and a Bug is 100. If we are lucky and see a convertible Mini or Bug, we get 10 extra points for convertibility. I play with them, of course. As a result, I can recognize minis and bugs from hundreds of miles away (I am exaggerating).

Recently, a Mini annoyed me. I was driving behind one, warmly thinking about my grandchildren, when its right turn signal started flashing. The signal looked like an arrow pointing to the left. I got so confused that my grandkids flew from my mind.

When I came home, I started googling and discovered that Mini designers wanted the British symbolism on their cars. The right signal is reminiscent of the right half of the British flag.

Here is the picture from Reddit with the left turn signal on.

I am writing this essay but afraid to show my grandkids these pictures. They would be maxi-disappointed with Minis.

One of the perks of being a teacher is receiving congratulations not only from family and friends, but also from students. By the way, I do not like physical gifts — I prefer just congratulations. Luckily, MIT has a policy that doesn’t allow accepting gifts of any monetary value from minors and their parents.

Thus, my students are limited to emails and greeting cards.

One of my former students, Evin Liang, got really creative. He programmed the Game of Life to generate a Christmas card for me. You can see it for yourself on YouTube at: Conway Game of Life by Evin Liang.

This is one of my favorite congratulations ever.

I buy almost everything on Amazon. Recently, I ordered a back stretcher. It took half an hour to assemble, and after the first use, it changed shape. However, this story is not about quality but about a card that was included with the item.

In the box, I found a gift card that wasn’t a gift card but rather a promise of a $20 Amazon gift card for a 5-star review. Hmm! A bribe for a good review.

I looked at the card more closely, and it had the following text.

WARM TIPS: Please DO NOT upload gift card pictures in the review; it will affect your account.

This is not only a bribe. It contains a threat.

Initially, I assumed it was Amazon, but it makes more sense that the company making this thingy is behind it. I gave Amazon the benefit of the doubt and went to their website to leave a 1-star review. I related the card’s story to warn others that the 5-star reviews can’t be trusted. Amazon rejected my review as it didn’t comply with their guidelines. Is Amazon in on it?

I wrote a different 1-star review, which did comply. It seems I can complain about the product, but I can’t complain about the bribe and threat.

I called Amazon’s customer service, and they promised to investigate. This was three months ago. This crappy product, with a stellar average 4.6 rating, is still out there.

I am sure I was fooled many times on April Fool’s Days. But I do not remember any stories, except one. It was quite painful and had a lot of potential for humiliation.

I was in high school, and our school didn’t have enough rooms for all the classes. So that particular semester, our classes happened during the second shift: from 2 pm to 8 pm.

One day, I received a call from my friend that school was canceled the following day. My friend told me that I had to notify half of the class. We had about 40 students in our class. She would call everyone whose last name started with the letters A through N, and I would have to call everyone else.

To set the stage, I was really, really shy. Calling people was torture. On the other hand, I was a very responsible person. So, I was walking in circles around our family’s phone with my hands shaking. Then, suddenly, a light bulb went off in my head: the day was April 1st.

It was a great excuse: if I told people not to come to school, they might not believe me. Hooray! I can procrastinate until the next day. Because we wouldn’t start until 2 pm, I would have enough time to notify everyone in the morning.

When I woke up the next day, the other light bulb went off: I didn’t need to call anyone; but I definitely had to have a chat with my “friend.”

I am deeply fascinated with my family’s history, so I took it upon myself to enlighten my children about our family tree. One day my son retorted, “This is not a family tree; this is a family bush.”

He was right. I was married three times and had a child from two of my three husbands. My ex-husbands had other children. So our family “tree” branches out in chaotic and weird ways. My two sons are half-brothers, and each of them has other half-siblings. With mathematics all around them, my sons decided to quantify their family connections: they named a half-sister of a half-brother a quarter-sister.

I didn’t have any children with my first husband, Alexander. But he remarried after our divorce and had two children. I’ve never met Alexander’s offspring, but my children hang out with them. Go figure! This is not just because the world is too small; rather, my exes are mathematicians and work with each other. So, theoretically, if Alexander and I had a common child, Alexander’s children would have been quarter-siblings to my sons. In real life, we didn’t have a child. Still, can my sons call Alexander’s children virtual quarter-siblings?

Every summer when I was little, my mom would take us on vacation to a rusty village far away from Moscow. I do not remember much, mostly just cows grazing on the grassy fields. However, one particular memory is really special and vivid.

I was five years old, tired of another day in the fields, lying in bed about to fall asleep. I started counting. I do not remember what. I am sure it wasn’t sheep; it could have been cows. Then, I got bored of small numbers and jumped to a thousand, counting from there. Then, I jumped to another even bigger number. After a few jumps, I realized that I could always add one to a previous number. The number of numbers must be infinite. Wow!

I will always remember the feeling I had. It was like touching eternity, being one with the whole universe.

You can imagine why I became a mathematician. From time to time, I am touching eternity and getting paid for the bliss.

As my readers know, I am devoted to my students. When I need something I can’t buy, I try to make it. That is why I crocheted a lot of mathematical objects. One day, I resolved to have in my possession a Seifert surface bounded by Borromean rings (a two-sided surface that has Borromean rings as its border).

However, my crocheting skills were not advanced enough, so I signed up for a wet and needle felting workshop. When I showed up, Linda, our teacher, revealed her lesson plan: a felted soap with a nice pink heart on top. It looked cool to have soap inside a sponge, not to mention that wool is anti-bacterial. But I had bigger plans than soap and eagerly waited for no one else to show up.

When my dream materialized, and, as I had hoped, no one else was interested in felt, I asked Linda if we could drop the hearty soap and make my dream thingy. She agreed, but my plan didn’t survive for long. As soon as Linda saw a picture of what I wanted, she got scared. Seifert surfaces were not in the cards, so soap it was. I told her that there was no way I was going to needle-felt a pink heart onto my felted soap. I ended up with a blue Sierpiński gasket.

We had a great time. Linda was teaching me felting, and I was teaching her math. I am a good teacher, so even felters working on a farm enjoy my lessons.

After the workshop, I went online and found my dream surface on Shapeways. In the end, I was happy to just buy it and not have to make it.

But my felting workshop wasn’t a waste of time: tomorrow I will wash myself with a gasket.