Tanya Khovanova's Math Blog

Puzzle. Alice and Bob play the following game on a regular chessboard, where all the pieces move according to the standard rules of chess. Alice has a king, and Bob has a knight. They would place their pieces on the chessboard without attacking each other. Alice starts, and they alternate moves. Alice loses if the knight checks or all available moves place her under the knight’s attack. For which initial positions of the pieces is Bob guaranteed a win?

Once, I was in Princeton and attended a lecture by Persi Diaconis. John Conway was also there. During his presentation, Persi mentioned John by referring to him as Wonderful John Conway. John couldn’t resist and said, “Thank you, you are one of not many people who remember my real first name.”

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.

I use Escher’s tessellations to teach wallpaper groups. Escher is the best painter among mathematicians and the best mathematician among painters. His fame helps energize my class. Plus, he has so many beautiful drawings to choose from.

However, there is another layer to his tessellations. Many paintings are not just a study in wallpaper groups but also in group-subgroup pairs. For example, consider these red/gray/black fishes on the left. There are three distinct points where three different reflection lines intersect. The first point is where three black fishes kiss each other. The other two points correspond to gray and red fish kisses. In orbifold notation, this symmetry group is *333.

But, the same drawing has another symmetry group. We just need to ignore color. That is, we consider all the fishes to be the same. In this case, our three distinct points where three reflection lines intersect become the same point: the point where fishes kiss, regardless of their color. The new symmetry group has an additional element: a 120-degree rotation where three fins of three different-colored fishes touch each other. Thus, the new symmetry group is 3*3.

Escher created a lot of examples of groups and their subgroups using color. But, sometimes, he was more subtle. In one of my previous posts, The Dark Secret of Escher’s Shells, I discussed my favorite Escher’s plane tessellation. In that drawing, the second group appears when we ignore the markings on one of the dark shells.

Here is another spectacular example of a group and subgroup, a tessellation of a hyperbolic plane with angels and devils. Do you see two different symmetry groups in the painting?

My PRIMES project last year, done with Rich Wang, was about violinists living in a hotel and being annoyed with each other. The project was suggested by Darij Grinberg and based on the following puzzle.

Puzzle. Consider a hotel with an infinite number of rooms arranged sequentially on the ground floor. The rooms are labeled from left to right with consecutive integers. A finite number of violinists are staying in the hotel with no more than one violinist in a room. Each night, two violinists staying in adjacent rooms get fed up with each other’s playing, and both request different rooms from the manager. The manager moves one of them to the nearest unoccupied room on the left and the other to the nearest unoccupied room on the right. This keeps happening every night as long as there are two violinists in adjacent rooms. Prove that this relocation will stop after a finite number of nights.

The project was not to solve the puzzle, and I won’t describe the solution, leaving it to my readers to ponder on their own. The project was to take this puzzle and see what we could discover. The relocation process in the puzzle is reminiscent of chip-firing, with a few differences.

Let me explain chip-firing in terms of violinists. The starting position would allow any number of violinists per room. Each night, two violinists in the same room will get annoyed with each other and request new rooms. There might be more violinists in that same room, but only two at a time are really annoyed. The manager will put one of them into the room on the left and the other into the room on the right, regardless of how crowded the rooms are.

In the chip-firing example, violinists always move to the next room. In our puzzle, they might need to go miles to a new room. And carry their luggage with them!

The original puzzle above implies that after several nights the relocations will stop, and violinists will enjoy the hotel in peace. This peaceful configuration is called a final state. Interestingly, depending on the order in which violinists are getting annoyed with each other, a starting configuration can result in different final states.

For example, consider the smallest interesting case, where only 3 violinists are staying in a hotel. We use 1 to describe a noisy room with a violinist and 0 to describe an empty room. Suppose the starting configuration is 0011100, where we ignore rooms far away from the noise. Depending on which two rooms have the complaining violinists on the first night, we can end up with a final state 0101001 or 1001010.

Initially, in the project, we looked at final states where we start with N violinists in consecutive rooms. We call such a configuration an N-clusteron. The example above describe the final states of a 3-clusteron. Here is an exercise for the reader: prove that in the final state, the gaps between occupied rooms have to be 1 or 2.

A final state can be described by the index of the leftmost occupied room and the sequence of gaps between occupied rooms. In the exercise above, it is easy to prove that the gaps must be size 1 or 2. In our paper, we proved a more refined statement: any final state has exactly one gap of size 2. Thus, a final state has two parameters: the index of the leftmost room and the location of the size 2 gap. For example, this means that in a final state, the span between the leftmost and the rightmost rooms, inclusive, is 2N.

As we continued exploring final states, we discovered something else. But first, I need to define the shadow of a final state. Let’s denote an occupied room with a one and an empty room with a zero. Then a final state corresponds to an infinite sequence of zeros and ones. But the interesting part of the final state is not infinite: it is a subsequence between the leftmost and rightmost ones, inclusive. We call this subsequence the shadow of the final state. In other words, the shadow describes the final state up to a translation and is completely defined by one parameter: the location of the 2-gap. In our example of a 3-clusteron, there are 2 final states with 2 distinct shadows. For larger clusterons, the situation is more interesting. There are 5 different possible final states for a 4-clusteron but only three distinct shadows. We discovered the following conjecture.

Conjecture. If we start from an N-clusteron and at each state uniformly select a move to perform from all possible moves, then the shadows of the final states are equiprobable.

You can find more details in our paper Ending States of a Special Variant of the Chip-Firing Algorithm. This conjecture is beautiful as it shows that there are hidden symmetries in this process of appeasing the fussy violinists. We didn’t prove this conjecture. Can you do it?

“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.

Puzzle. “I guarantee,” said the pet-shop clerk, “that this parrot will repeat every word it hears.” A customer bought the parrot but found it wouldn’t speak a single word. Nevertheless, the clerk told the truth. Explain.

The official answer:

• The parrot is deaf.

Indeed, in this case, it is not a lie that the parrot will repeat every word it hears. My students had some other ideas. The following answer differs from the official one by one letter, but the spirit of the solution is the same.

• The parrot is dead.

Another idea my students had was to introduce a time component.

• The parrot wasn’t guaranteed to say the lines immediately; it would wait 30 years before repeating any words.
• The parrot only repeated the words in the customer’s absence.

And a couple of outside-of-the-box answers.

• The parrot was a toy and did not have a battery in it.
• The customer mistook the pet-shop owner to say the statement about the parrot when in fact, the parrot was saying it.

Puzzle. Alice and Bob divide a pie. Alice cuts the pie into two pieces. Then Bob cuts one of those pieces into two more pieces. Then Alice cuts one of the three pieces into two pieces. In the end, Alice gets the smallest and the largest piece, while Bob gets the two middle pieces. Given that both want to get the biggest share of the pie, what is Alice’s strategy? How much can she get?

Many years ago, I visited a distant country and rekindled my friendship with a nice couple, Alice and Bob. I changed the names and do not remember the exact words. But here is the story.

One evening, I found myself chatting with Bob, and he decided to open up. He told me he was having an affair. After he finished his long story, I asked him whether he thought Alice might also be having an affair. He replied, “It’s impossible. She was never a passionate person. And recently, she became even more cold and distant.” I didn’t follow his logic but didn’t pursue the topic.

Several days later, I chatted with Alice, and she decided to open up. She told me that she had met someone and was having an affair. She told me that she never quite enjoyed sex with her husband, but the new guy fitted her perfectly. For the sake of symmetry, I asked her whether she thought her husband might also be having an affair. She replied, “It’s impossible. He is too busy at work, and recently, even more so. He goes on business trips twice a week and is always tired.”

I invented this puzzle. It’s a variation of something I saw on Facebook.

Puzzle. The future dinner of an anthropophagusphagusphagus met for dinner the past study object of an anthropophaguslogist. What’s for dinner?