The most famous thinking-outside-the-box puzzle is the Nine-Dots puzzle. This puzzle probably started the expression, “To think outside the box”. Here is the puzzle.
Puzzle. Without lifting the pencil off the paper, connect the nine dots by drawing four straight continuous lines that pass through all the dots.
Most people attempt something similar to the picture below and fail to connect all the dots.
They try to connect the dots with line segments that fit inside the square box around the dots, mentally restricting themselves to solutions that are literally inside the box.
To get to the correct solution, the line segments should be drawn outside this imaginary box.
Do you think that four line segments is the best you can do? Jason Rosenhouse showed me a solution for this puzzle that requires only three lines. Here, the outside-the-box idea is to use the thickness of the dots.
This and many other examples of thinking outside the box are included in my paper aptly titled Thinking Inside and Outside the Box and published in the G4G12 Exchange book.
A section of this paper is devoted to my students, who sometimes give unexpected and inventive solutions to famous puzzles. Here is an example of such a puzzle and such solutions that aren’t in the paper because I collected them after the paper was published.
Puzzle. Three men were in a boat. It capsized, but only two got their hair wet. Why?
The standard answer is the following: One man was bald.
Lucky for me, my creative students suggested tons of other solutions. For example,
One man was wearing a waterproof helmet.
The boat capsized on land, and two men had their hair already wet.
As my readers know, I collect Russian license plates. They are actually American plates, but the letters form words readable in Russian. This is possible because the shapes of English and Russian letters overlap. Here is my new favorite plate. It is actually not the best plate, but rather makes the best picture ever.
The plate says Moscow, the Russian capital. And the car is parked next to the Ukrainian flag. I am from Moscow too, and I too support Ukraine in its war against evil Putin, who wants to restore the Russian Empire. Did you know that now he wants Alaska?
Thinking about genders, we used to have only two options: male and female. Now we have more. I have a lot of young acquaintances who are non-binary. So I started to rethink my gender identity.
I was a typical girl: at least, I thought I was. I hated playing with boring dolls. I preferred cars, or even better, construction sets and board games. In second grade, I wanted to be a ballerina but later fell in love with Sherlock Holmes. My dreams switched to becoming a detective or a spy. In fifth grade, I signed up for rifle-shooting training. That same year, my school forced me to compete in an orienteering event, and I won.
Orienteering became my favorite sport, and I did it for many years. I was really good with maps. I would go to a competition, leisurely walk my course and win. Other kids were running around like crazy, but I always knew where to go and was overall faster than them. With time, other kids learned to read maps better, but I myself never learned to run faster. So I stopped winning, but I enjoyed the sport anyway.
It goes without saying: I loved math. Solving math problems was the best entertainment ever.
Later, to my surprise, I discovered that most girls liked shopping and wasted a lot of time on make-up. Not many girls were even interested in math. I actually liked that. I started having crushes on boys since second grade and enjoyed being the only girl in math clubs, having all these nerds to myself.
I grew up in Soviet Russia. While growing up, I wasn’t bombarded with gender stereotypes. My first eye-opening experience was when I was 17 years old. My long-time boyfriend knew that I liked mathematics, and this was okay. But when I told him that I planned to go to college to study mathematics, he didn’t approve. I broke up with him on the spot.
My mom used to tell me that most men do not like brainy women. My reply was that there are more men who like brainy women than brainy women. I got a new boyfriend the day after my breakup.
My gender identity didn’t bother me much in Russia. What bothered me was the Russian tradition for both spouses to work, but the house chores and child-rearing duties fell only on women. I read somewhere that, on average, in Russia, women worked for 4 hours a day more than men. Life was unfair to women but not to their self-esteems.
As I said, I grew up thinking I was a typical girl until I came to the US. This happened 30 years ago, and I was 30 at the time. In the US, I got bombarded with gender stereotypes: they made me feel inadequate and doubt my femininity. Just for reference: by that time, I was in my third marriage breastfeeding my second child. Still, according to those stereotypes, I was not a real woman.
For some time, I wondered what was wrong with me. Then, I was elated to find a book called Brain Sex: The Real Difference Between Men and Women by Anne Moir and David Jessel. (This was many years ago.) Among other things, this book talks about the differences between men and women with respect to the brain. According to the book, men are better on average at abstract thinking and spatial visions, aka math and maps. Women, on the other hand, are better at connecting with people and have higher social intelligence. Boys are less interested in games related to storytelling, aka dolls, preferring more concrete activities. And so on.
The book also describes situations in which girls don’t fit the paradigm. The authors attribute this variation to the mother’s hormones during pregnancy. I found myself described perfectly in the section titled “girls who have been exposed to male hormones in the womb.” I am pretty sure that my mom didn’t take any hormone supplements while pregnant with me back in Soviet Russia. On the other hand, the description in the book was spot-on.
This book classifies me as “a male brain in a female body.”
I was glad to find myself after a period of self-doubt. I was glad that I wasn’t alone and even fit into a special category with a name.
Several years later, I met Sue Katz, a writer who also has a blog Sue Katz: Consenting Adult. She made me realize how ridiculous the whole story was: I was pressured by gender stereotypes to feel bad about myself. Then I was grateful for a book based on those same stereotypes, only because it described women like me and gave me permission to exist. I liked the book because I accepted those stereotypes in the first place. If there were no stereotypes, there wouldn’t be any problems at all.
Why can’t I just be me?
Over the past few years, I have become happier than I have ever been. I do not care what society thinks about my gender. I am no longer ashamed of not feeling 100 percent female.
I like that people in the modern world embrace the idea of individuals being themselves. For example, my daughter-in-law, Robin Dahan, designed a whole line, You-Be-You, for her company, Dash of Pep.
Am I non-binary? I do not know and do not care. I am just me, proudly wearing my You-Be-You socks.
Many years ago, I wrote a blog post about an amusing fact: John Conway put Moscow, the former capital of the USSR, as a coauthor: A Math Paper by Moscow, USSR. Thus, Moscow got an Erdős number 2, thanks to Conway’s Erdős number 1. At that time, my Erdős number was 4, so I wondered if I should try coauthoring a paper with Moscow, my former city of birth, to improve my Erdős number.
This weird idea didn’t materialize. Meanwhile, my Erdős number became 2 after coauthoring a paper with Richard Guy, Conway’s Subprime Fibonacci Sequences. I relaxed and forgot all about my Erdős status. I couldn’t do better anyway, or could I?
I recently heard about a cheater who applied to grad schools. In addition to a bunch of fabricated grades, the cheater submitted an arXiv link to a phony paper. What is fascinating to me is that the cheater put real professors’ names from the university the cheater supposedly attended as coauthors. The professors hadn’t heard of this student and had no clue about the paper. So the cheater added fake coauthors to add legitimacy to their application and boost the perceived value of the cheater’s “research”. As a consequence, the cheater got a fake Erdős number.
I hope that the arXiv withdrew the paper. Cheating is the wrong way to improve one’s Erdős number.
But here is another story. Robert Wayne Thomason named as coauthor his dead friend, Thomas Trobaugh. The paper in question is Higher Algebraic K-Theory of Schemes and of Derived Categories and can be found at https://www.gwern.net/docs/math/1990-thomason.pdf. This paragraph in the paper’s introduction explains the situation.
The first author [Robert Wayne Thomason] must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom’s simulacrum remarked, “The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf.” Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing K0 obstruction to extension. I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom’s simulacrum had been so insistent, I knew he wouldn’t let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper.
This precedent gives anyone hope that they might achieve an Erdős number 1. You just need to wait for Paul Erdős to come to you in your dreams with a genius idea.
I used to be proud of my Russian math education. I am still proud of my high school one, but not so much of the one I received in college. In Soviet Russia, a student had to choose their major before applying to college. I wanted to study mathematics, and I got accepted to the best place for it in Soviet Russia: mekhmat — the math school at the Moscow State University. I used to be proud of my education there, but now I have my doubts.
I had to take, on average, four math classes per semester for five years, which totals about 40 math classes. Woo hoo! I don’t think American students could even choose to take that many. This was presumably good, but most of the courses were required, and their curriculum remained unchanged for many, many years. Obviously, the system was very rigid. The faculty members feared retaliation from the communist party and forgot how to take initiative. The bureaucracy prevented the department from adding new and exciting math to the outdated curriculum.
This post is not about my grades but about the actual subjects that we were taught then. But, in case anyone is wondering, my only B was in English; everything else was straight As.
Some of the classes listed below lasted two or more semesters, that’s why they do not sum up to the promised 40. Unfortunately, I do not remember which ones. These were the required math classes:
Analysis
Analytical Geometry
Advanced Algebra
Theoretical Mechanics
Linear Algebra and Geometry
Differential Equations
Partial Differential Equations
Functions of Complex Variables
Probability and Statistics
Differential Geometry and Topology
Numerical Methods
Introduction to Mathematical Logic
Control Theory
Analysis III
Computer Science and Programming
Programming Practice
Physics
History and Methodology of Mathematics
Thesis Work
An impressive list? But guess what — I remember nothing from most of these classes. As an exception, I remember bits of Differential Equations, taught by Vladimir Arnold, a charismatic teacher. I remember Linear Algebra well, not because of my Linear Algebra class, but because I read Gelfand’s book on the subject and loved it. I remember that the Differential Geometry and Topology class was taught by Fomenko with great pictures and boring material. By the time I took Fomenko’s class, I already knew topology from an unofficial class taught by Dmitry Fuchs, which was so much better. In fact, in order to learn what I wanted, I had to take many classes unofficially, so my total is actually way above 40.
By junior year, we were finally allowed to choose some classes which would count towards our transcripts, and this is what I picked.
Infinite-Dimensional Representations of Lie Groups
Theory of Functions of Many Complex Variables
Representations of Lie Groups
Discrete Mathematics
I remember these classes much more vividly. I also wrote a graduate thesis: “Models of Representations of Generalized Clifford Algebras.” I loved working on that paper.
We had non-math classes too: everyone had to take them.
History of the Communist Party of the USSR
Philosophy of Marxism-Leninism
Political Economy
Scientific Communism
Foundations of Scientific Atheism
Soviet Law
Foreign Language (English)
Physical Education
Foundations of Marx-Lenin’s Aesthetics
To graduate, everyone had to pass two state exams: Mathematics and Scientific Communism. Whatever the latter might mean.
Did I mention that I am no longer proud of my former Soviet college education? What a colossal waste of time!
I grew up relatively poor, but I wasn’t aware of it and didn’t care. In 7th grade, I went to a new school for children gifted in math. Looking back, I realize that most of my classmates there were privileged. My first clue about my own financial disadvantages arrived when my math teacher, Inna Victorovna, offered me several of her old dresses. I do not remember what she said to me exactly, but I remember she was tactful, so much so that I felt comfortable taking the dresses.
In an instant, I was better dressed than I had ever been. I especially loved the brown dress which I wore for my first visit to Gelfand’s seminar.
A few years passed; I went to college and married Andrey. Things got somewhat better financially, but I was still struggling. My mother-in-law, Veronika, was well-off and loved clothes. She had a habit of ordering a new dress from her tailor, every season, four times a year. In Soviet Russia, this was a lot of dresses.
One day, Veronika decided to give me some of her old dresses. Unlike my math teacher, she said something that I will never forget. She told me that she was getting rid of those dresses because they were out of fashion and made her look old. I was in my twenties at the time and didn’t want to look old either. However, I didn’t have much choice in clothes, so I wore the dresses. I hated them.
Do you know that I participated in Linguistics Olympiads in high school? They are not well-known in the US, but the Soviet Union has been running them since 1965. The first International Linguistics Olympiad was conducted in 2003, and the US joined in 2007. They are called Computational Linguistics because you are expected to discover some phenomenon in an unfamiliar language on the fly instead of knowing a lot of languages already. The problems mostly need logic and are a good fit for a person who likes mathematics. However, a feel for languages is very helpful.
I do not remember why I started attending the Olympiads, but I remember that there were two sets of problems: more difficult for senior and less difficult for non-senior years. I used to be really good at these Olympiads. When I was in 8th grade, I finished my problems before the time ran out and started the senior problems. I got two awards: first place for non-senior years and second place for senior years. In 9th grade, I got two first-place awards. I didn’t know what to do in 10th grade, which was a senior year at that time in the USSR. I couldn’t get two first-place awards, as I could no longer compete in the non-senior category. I felt ashamed that my result could only be worse than in the previous years, so I just didn’t go.
The prizes were terrific: they gave me tons of rare language books. In the picture, a guy from the jury is carrying my prizes for me. I immediately sold the books at used-books stores for a good price. Looking back, I should have gone to the Olympiad in 10th grade: my winter boots had big holes.
My friend, John Conway, had a trick to help him with tricky situations. Whenever he needed to make a non-trivial decision, he would ask himself, “What would John Conway do?” As he explained to me, he had in mind the public image he himself created. He liked the image and thought this mental trick helped him be a better, more productive, and not-to-forget, flashier person.
From time to time, I catch myself in need of a decision and ask myself, “What would John Conway do?” And he gave me the answer: I should change the question and ask myself, “What would Tanya Khovanova do?”
A familect is a portmanteau word formed by squashing together two words: family and dialect. It means words that a family uses that are not a part of a standard vocabulary. This story is about two words that my son Sergei invented, and twenty years later, my family still uses them.
As you might know, I was married three times, and I have two sons, from two different fathers. These fathers were also married several times and had other children. My two sons are half-brothers, and they also have half-siblings through their fathers. Thus a half-sister of a half-brother became a quarter-sister. Inventing this term was quite logical for a son of two mathematicians.
As you can imagine, our family tree is complicated. One day Sergei pointed out that our tree doesn’t look like a standard tree and called it a family bush.
