Tanya Khovanova's Math Blog

I was teaching my students the Knaster method of dividing an estate, which I learned from my friend Ingrid Daubechies. Let’s look at an example.

Problem. Alice and Bob are divorcing. They have the portrait of Alice’s grandpa and $10,000. Alice values the portrait at $10,000 because of its sentimental value. Bob values it at a market price of $2,000. How do they divide their estate?

Here is what my students initially suggest.

1. Give Alice the portrait, and give Bob all the cash.
2. Give Alice the portrait, and divide the cash in half.

Let’s look at these suggestions in greater detail. Alice values the whole estate at $20,000; Bob values it at $12,000. In the first version, Alice gets half of the estate from her point of view; Bob gets the rest, which is more than half in his point of view. The students are obviously rooting for Bob. In the second version, Bob gets half of the estate in his estimate, while Alice gets the rest, which is more than half in her estimate. The students are obviously rooting for Alice. After some discussion, the students agree that there should be a number between $5,000 and $10,000 that Bob gets, which would be a fairer division than the two initial examples. But how do we find such a number?

This is where the Knaster algorithm comes in. The main idea is that each gets the same amount of money on top of their perceived half. In other words, the Knaster method treats the estate like a sealed-bid auction and equalizes bonuses. Alice thinks that her fair half is $10,000, while Bob thinks his half is $6,000. To equalize bonuses, we want Alice and Bob to each receive their perceived half plus the same amount — call it x. Solving gives x = $2,000. Alice gets the portrait and $2,000, while Bob gets $8,000.

This is a beautiful algorithm that allows each person to be very happy, receiving more than one half. The bigger the taste difference, the more each person gets on top of their portion. The next question is: how can people cheat if they know this algorithm is used?

Alice can cheat by claiming that she values the portrait at $2,000 plus epsilon. Epsilon is needed to guarantee she gets the portrait. This way, they both value the estate the same. Alice gets the portrait and $4,000, which is $2,000 more than the honest way. Symmetrically, Bob can cheat by claiming he values the portrait at $10,000 minus epsilon. This way, he gets $10,000, which is $2,000 more than the honest way.

I’ve been teaching this topic several times now. This year, my student Ben had an out-of-the-box idea on how Alice can cheat. Alice declares that she values the portrait at $0. Bob thinks the estate is worth $12,000, while Alice pretends that she values it at $10,000. After the calculation, Bob gets the portrait and $4,500, which is $500 more than his half of the estate in his view. Alice gets $5,500, which is $500 more than half of the estate in her declaration. Then she buys the portrait from Bob for $2,000. In the end, Alice gets the portrait and $3,500, way more than she would get after an honest use of the algorithm.

The first cheating method seems more profitable than the new one. But still, I love it when my students suggest unexpected ideas.

A while ago I took writing lessons with Sue Katz. Below is my homework from 2010 (lightly edited). If I remember correctly, this piece was inspired by Sam Steingold.

—My friend Sam installed six locks on his door to protect himself from burglars.
—I know. I visited your friend. He has six very cheap locks. Any professional could open one in a second, so Sam’s door will only resist for six seconds.
—Yeah, but those locks aren’t completely identical. Three of them unlock with a clockwise motion, and three with a counterclockwise motion.
—So what? The thieves will just turn the lock mechanism whichever way it can be turned.
—Not so fast. Sam never locks all of them. Every time, he randomly picks which ones to lock.
—That might work, but what if he forgets which ones he locked?
—That’s okay, He remembers which way to turn every lock to unlock.

Here is an old joke.

A former student runs into an old calculus teacher as she is shuffling home. The student is happy to see her and says, “I recently thought about you and our classes.” “How so?” she perks up. “I was in a bit of a pickle when calculus helped me,” he says. The old lady straightens, and her face begins to glow. “Can you elaborate?”, asks the teacher in anticipation. The student proceeds with his story. “I was walking home in the pouring rain when a gust of wind snatched my hat right off my head. The hat landed in a puddle. This wasn’t just any hat; it was a gift from my dad, so I really wanted to get it back. But I didn’t exactly fancy diving headfirst into the puddle. So, I looked around and saw a piece of wire. I bent it into the shape of an integral and used it to fetch my hat.”

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.

“Follow your heart!” This is the most common advice for young people contemplating their future career path. This is not good advice. At some point in my life, the most popular aspiration among my friends’ children was to become opera singers. But the world only has room for a few opera singers. All of these children ended up doing something completely unrelated.

Aspiring to be a mathematician is a much more practical dream. There are so many professions that are friendly to mathematicians: actuary, finance, economics, teaching, computer science, cryptography, and programming, to name a few. Unlike opera singers, skilled mathematicians can find a way to get paid for their mathematical gifts.

However, most of the youngsters around me want to be research mathematicians. This is a different story. My adviser, Israel Gelfand, told everyone that if they could survive without mathematics, they should drop it. I did drop mathematics for some time to care for my children, but I couldn’t live without mathematics. I fed math to my children for breakfast and pursued math hobbies that could fit a single mother’s lifestyle. Well, that means I was mostly working with sequences for the Online Encyclopedia of Integer Sequences and building the database for my Number Gossip website. But I digress.

I agree with Gelfand. Research in math as a career choice is non-trivial. Here are some of the big issues:

• Money. An entry-level salary for MIT undergraduate alums is about twice as much as graduate students’ stipends. This discrepancy in income continues for a long time.
• Location. It is challenging to find a professorship. People looking for academic jobs are expected to send hundreds of job applications and accept a position anywhere in the country. This might be unacceptable for people who value their family and roots.
• Time. Many research mathematicians I know work 24/7. Not always by choice. Some want to secure a future tenure and need to publish papers in addition to teaching innovative courses. Some do want to do research but are too distracted with their routine and administrative tasks. As a result, their research spills out into nights and weekends.
• Gender and such. Discrimination is a separate big problem for women and minorities, which I do not want to discuss today.

So what would I suggest for young people who love math?

Many people who love math do not really love math per se. They love the way of thinking that math encourages. They love logic, generating ideas, precision, innovation, and so on. This makes their potential job search much wider. Such people might enjoy programming, cryptography, data science, actuarial science, finance, economics, computer science, engineering, etc. I know students planned to become mathematicians but tried an internship in finance and found their real passion.

For those who want to be closer to mathematics, there is always teaching: the world needs way more math teachers than research mathematicians. Plus, teaching provides a strong feeling of making an impact.

So what do I suggest for young people who love opera singing?

Many skills are less in demand than mathematics. It is important to be realistic. So here is my advice:

• Expand your skill set. If you love opera singing, working as a voice coach might be a solution.
• Explore secondary interests. If you also enjoy programming, you might find your happiness by building music software.
• Have a backup plan. You might become a lawyer to support yourself and your family and keep your love of music as a hobby, for example, by singing in a choir.
• Meet yourself halfway. You might get a half-time job to feed yourself and work on your music business for the other half of the time.

I know a former Soviet mathematician who worked as a night guard and used his quiet work time to invent theorems. He was a good mathematician but couldn’t find a research position because he was Jewish. He later immigrated to the US and found a professorship. So sometimes my advice for opera singers works for mathematicians too.

Before dividing chores, let’s divide a cake. Suppose I bought a two-layer cake. One layer is chocolate and the other is vanilla. Suppose I love chocolate and hate vanilla, My friend, Joe, is the opposite: he loves vanilla and hates chocolate. It’s very easy to divide the cake. I take all the chocolate, and Joe takes all the vanilla. I think I am lucky to get the full value of the whole cake. Joe feels lucky too.

My point is that people with different tastes can divide things so that both get more than half by their own estimates.

Let’s look at another example. Suppose Alice and Bob are divorcing. Their estate consists of one valuable thing: a portrait of Alice’s grandfather. Alice loved her grandfather and values the portrait at 10,000 dollars. Meanwhile, Bob values it the same as its market value, 2,000 dollars. There exists a division algorithm, called the Knaster procedure, which allows them to divide the portrait so that both of them end up with the same amount of money on top of their perceived half of the estate.

I will skip the calculations. The end result is that Alice gets the portrait and pays Bob 3,000 dollars. In her view, she gets a portrait worth 10,000 and loses 3,000. Her total gain is 7,000 dollars, which is 2,000 more than her estimated half. Bob gets 3,000 dollars. In his view, half of the estate is 1,000 dollars, and he gets 2,000 more than that.

The bigger the difference in perceived value, the more each person gets in addition to their expected half. Suppose this difference is D. If you trust my calculations, the Knaster procedure means each person gets D/4, in addition to one half of their estate’s perceived value.

The same idea can be applied to chores. Suppose I hate shopping while my husband hates doing the dishes. So, I can do the dishes, and he can shop. And we can live happily ever after without doing the things we hate.

So, theoretically, it is very profitable for two people to live together. Have you seen couples where each one thinks that they won the lottery by marrying their partner? Such couples benefit from dividing chores, appreciate their partners’ help, and are happy.

I have seen such couples, though not many. Actually, not many at all. If mathematics says that living together should be profitable, then why are happy couples such a rarity?

I will divide unhappy couples into four categories depending on whether they both benefit from dividing the chores and whether they both appreciate each other.

Benefit and appreciate. In this case, other parameters could affect their happiness: love, sex, children, jobs, and so on. Consider, for example, Alice and Bob. Alice relies on her husband for financial support for her and their small children and appreciates said support. Bob likes how Alice cares for the children and appreciates her for that. However, Alice doesn’t love Bob anymore, and Bob wants something special in his sex life but is afraid to request it from Alice. They are both profoundly unhappy.

Benefit and do not appreciate. It is possible that both people do not appreciate each other, or it could be just one. In addition, it could be that a person underestimates the real value of the partner’s contribution, or it could be that the appreciation is not enough for the partner. This became a more complicated paragraph than I initially expected. As Lev Tolstoy said, “All happy families are alike; each unhappy family is unhappy in its own way.” So, let me have two subcases.

Benefit and do not appreciate. Case 1. Underestimation. Such couples have a good division of labor but underestimate each others’ contribution. For example, Bob thinks that staying at home with children is a walk in the park. He thinks his job is way more difficult than his wife’s daily caring for the house and the children. He assumes that when he returns from work, the house needs to be clean and dinner ready. He is very angry when this doesn’t happen. Alice is very unhappy as she knows how much she actually works.

Benefit and do not appreciate. Case 2. No gratitude. Such couples have a good division of labor but do not express their gratitude sufficiently for the other partner. For example, Alice wants Bob to take her out to dinner as a thank you. Or to say thank you on a regular basis. But Bob brags to his friends that he is lucky in marriage and thinks that this is enough. Everyone but her knows that he feels lucky.

Do not benefit. It is usually one person who is used. There are many ways for people to force their spouses to divide the chores in their favor. There are many types of abusive relationships. I do not even want to give an example. After all, my goal was to discuss the mathematics of chores’ division, not to analyze why people do not divide their labor fairly.

Special cases. Life is complicated. Here is the case that doesn’t quite fit the cases above as the division of chores with time delays. For example, Alice worked and cared for the children while Bob went to medical school. The benefit for Alice was implied in the future. Bob promised to shower her with money when he would get rich. However, as soon as he got rich, he showered someone else.

The mathematics show that living with a partner can be extremely beneficial for both. But people’s emotions are complicated. They do not follow mathematics and often mess it up in more ways than I can imagine.

Consider a group of people in which some are friends. We assume that friendship is symmetric: if Alice is Bob’s friend, then Bob is Alice’s friend. That means we can build a friendship graph where vertices are people and edges correspond to friendships. Let’s assume that every person has at least one friend, so the friendship graph doesn’t have isolated vertices.

Darla needs to conduct research by surveying random pairs of friends. But first, she has to find those pairs. To ensure that the pairs are randomly selected, she must pick two random people from the group, contact them, and ask them whether or not they are friends. If they are, she gives them her questionnaire. If not, Darla wasted tons of time and had to keep looking.

The group she is surveying is enormous. So, when she picks two random people, they typically have never even heard of each other. Bother!

Darla decides to speed up the process. She would pick a random person, ask them for a list of friends, and then randomly pick one person from the list. Since every person has at least one friend, Darla always ends up with a filled questionnaire.

Puzzle question. Why is Darla’s method wrong? Can you describe the pairs of friends her method favors?

I often receive letters from parents of math geniuses — “My twelve-year-old is reading an algebraic geometry book: accept him to PRIMES,” or “My ten-year-old finished her calculus course: here is her picture to post on your blog,” or “My two-year-old knows the multiplication table, can you write a research paper with him?” The last letter was a sarcastic extrapolation.

I am happy to hear that there are a lot of math geniuses out there. They are potentially our future PRIMES and PRIMES STEP students. But, it is difficult to impress me. The fact that children know things early doesn’t tell me much. I’ve seen a student who didn’t know arithmetic and managed to pass calculus. I’ve also met a student claiming the full knowledge of fusion categories, which later appeared to be from half-watching a five-minute YouTube video.

There are a lot of products catering to parents who want to bring up geniuses. My grandson received a calculus book for his first birthday: Introductory Calculus For Infants. Ten years later, he still is not ready for calculus.

Back to gifted children. Once a mom brought us her kid, who I can’t forget. The child bragged that he solved 30 thousand math problems. What do you think my first thought was? Actually, I had two first thoughts: 1) Why on Earth would anyone count all the problems they solved? 2) And, what is the difficulty of the problems he solved 30 thousand of?

From time to time, I receive an email from a parent whose child is a true math genius. My answer to this parent is the same as to any other parent: “Let your child apply to our programs. We do a great job at working with math geniuses.”

Our programs’ admissions are done by entrance tests. Surprisingly, or not surprisingly, the heavily advertised kids often do poorly on these tests. It could be that the parents overestimate their children’s abilities. But sometimes, the situation is more interesting and sad: I have seen children who sabotage the entrance tests so as not to be accepted into our programs. We also had students give us hints on their application forms that they were forced to apply.

In the first version of this essay, I wrote funny stories of what these students did. Then, I erased the stories. I do not want the parents to know how their children are trying to free themselves.

Dear parents, do not push your children into our programs. If they do not want to be mathematicians, you are decreasing their chances of getting into a good college. Imagine an admission officer who reads an essay from a student who wants to be a doctor but wastes ten hours a week on a prestigious math research program. Such a student doesn’t qualify as a potential math genius, as their passion lies elsewhere. Nor does this student qualify as a future doctor, as they didn’t do anything to pursue their claimed passion. In the end, the student is written off as a person with weak character.

On the other hand, the students who do want to be in our programs, thrive. They often start breathing mathematics and are extremely successful. Encourage your children to apply to our programs if they have BOTH: the gift for mathematics and the heart for it.

I am so tired of spam emails. I keep thinking about how we can fight spam, and here is an idea.

Gmail should change its system: every email you send to me would cost 1 dollar, payable to me. We can add an exception for people on my contact list. Everyone else, pay up!

I do not often contact strangers. But if I do, it is always important. So paying 1 dollar seems more than fair. On the other hand, this system will immediately discourage mass emails to strangers. Spam would go down, and I would stop receiving emails inviting me to buy a pill to increase the size of a body part I do not have.

This idea of getting paid for reading an email is not new. It was implemented by Jim Sanborn, the creator of the famous Kryptos sculpture. Kryptos is located at the CIA headquarters and has four encrypted messages. People tried to decrypt them and would send Jim their wrong solutions. Jim got tired of all the emails and administered Kryptos fees. Anyone who wants Jim to check their solution, can do so by paying him 50 dollars. I wonder if Jim would still charge the fee if someone sent him the correct solution.

Thinking about it, I would like the payable email system to be customizable, so I can charge whatever I want. After all, I do value my time.

Gmail could get a small percentage. Either Gmail, together with me, gets rich, or spam goes away. Both outcomes would make my life easier.

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.