Tanya Khovanova's Math Blog

I was very happy to hang out with my oldest coauthor, Richard Guy, at the Gathering for Gardner conference in Atlanta in April 2018. By the way, Richard Guy is 101 years old.

I was on the writing team for the 2018 MIT Mystery Hunt. I am pleased that the hunt got very positive reviews from the participants. I spent tons of hours working on the hunt and it is good that folks liked it. I edited and tested a lot of puzzles. Here is my review of these year’s puzzles that are math-related.

I already posted an essay about the puzzles I wrote myself. Four of my five puzzles are math-related, so I am including them below for completeness. I will mention the topic of each puzzle unless it is a spoiler.

I start with Nikoli-type puzzles. Four elegant Nikoli-type puzzles were written or cowritten by Denis Auroux. In all of them the rules of the logic are stated at the beginning. That means the logic part doesn’t contain a mystery and can be solved directly.

• Good Fences Make Sad and Disgusted Neighbors (by Denis Auroux). You can guess by the title that this puzzle was in the emotions round corresponding to sadness and disgust. This is an interesting variation on the hexagonal Slitherlink. This is a relatively easy puzzle.
• Shoal Patrol (by Denis Auroux and James Douberley). Each grid is a combination of Battleship, Minesweeper, and a loop puzzle. These are difficult, but satisfying puzzles. The extraction step is not mathematical and not completely trivial.
• Submarine Patrol (by Denis Auroux and James Douberley). This is a 3D version of Shoal Patrol.
• Hashiwokakuro (Count your bridges) (by Denis Auroux). This is a mixture of Hashi and Kakuro. I enjoyed the puzzle while I tested it. The extraction is trivial.
• A Learning Path (by Tanya Khovanova and Xavid). This is a path logic puzzle that was targeted for new hunters. It contains self-referencing hints and solving techniques.

There were several puzzles that were very mathematical.

• X Marks the Spot (by Zachary Chroman). A very non-trivial puzzle about triangle centers.
• Tournament Organization (by Scott Harvey-Arnold ). A tournament reconstruction puzzle.
• Cash Cab (by Erica Newman and Justin Melvin).
• Voter Fraud (By Ben Weissmann). A voting puzzle.
• The 10,000 Puzzle Tesseract (by Charles Steinhardt, Zachary Chroman, and Scott Harvey-Arnold ). The most difficult puzzle of the hunt, by far.
• Games Club (by Tanya Khovanova and Sergei Bernstein). This puzzle is about combinatorial two-player games.
• Murder at the Asylum (by Tanya Khovanova). This is a difficult puzzle about liars and truth-tellers.
• Studies in Two-Factor Authentication (by Brandon Avila). A very elegant puzzle, that is one of my favorites.

There were also some math-related or computer-sciency puzzles.

• The Next Generation (by Colin Liotta). I enjoyed being an editor of this puzzle.
• Disorientation (by Alex Churchill). This puzzle has a beautiful visual component.
• Message in a Bottle (by Nathan Fung). The puzzle doesn’t look like it has something to do with mathematics, but my testing of it was very satisfying. I guessed from the start what it was about.
• Self-Referential Mania (by Justin Melvin). Self-referential logic puzzle, which I enjoyed editing.
• Bark Ode (by Elizabeth French, Justin Melvin, and Erica Newman). The pictures are so cute.
• Executive Relationship Commandments (by Robin Deits, John Toomey, and Michele Pratusevich). I didn’t see this puzzle until after the hunt. I wish I could have tested this puzzle with my son Alexey, who is a computer scientist.

There were also several decryption puzzles:

• Word Search (by Tanya Khovanova). A crypto word search.
• Texts From Mom (by Elizabeth French and Justin Melvin ). A text enciphered with emojis.
• Marked Deck (by Colin Liotta and Leland Aldridge). One of my favorite puzzles. Hunters received a physucal deck of cards that was laser cut. You can buy the deck at Etsy. The art in this puzzle is beautiful, but the puzzle also has a non-trivial decryption step.

Here is a famous math problem I never before wrote about:

Puzzle. Five pirates discovered a treasure of 100 gold coins. They decide to split the coins using the following scheme. The most senior pirate proposes how to share the coins, and all the pirates vote for or against it. If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the next most senior pirate making a proposal.

As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins whether he votes for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard. Assuming that all five pirates are intelligent, rational, greedy, and do not wish to die, how will the coins be distributed?

You can find the solution in many places including Wikipedia’s Pirate game. The answer is surprising: the most senior pirate gets 98 coins, and the third and the fifth pirates by seniority get one coin each. I always hated this puzzle, but never bothered to think through and figure out why. Now I know.

This puzzle emphasizes the flaws of majority voting. The procedure is purely democratic, but it results in extreme inequality.

That means a democracy needs to have a mechanism to prohibit the president from blatantly benefiting himself. With our current president these mechanisms stopped working. Given that Trump does everything to enrich himself, the pirates puzzle tells us what to expect in the near future.

We, Americans, will lose everything: money, clean air and water, national parks, future climate, health, social security, and so on, while Trump will make money.

In mathematics one of the most important questions is why. Let us consider a problem:

Problem. A number has three hundred ones and three hundred zeroes. Can it be a square?

The solution goes like this. Consider divisibility of this number by 9. The sum of the digits is 300. That means the number is divisible by 3, but not by 9. Therefore, it can’t be a square.

Why do we consider divisibility by 9? The divisibility by 9 is a very powerful tool, but why was it the first thing that came to my mind? The divisibility by 9 doesn’t depend on the order of the digits. Whenever I see a problem where the question talks about digits that can be in any order, the first tool to use is the divisibility by 9.

The why question, is very important in mathematics. But it is also very important in life. It took me many years to start asking why people did this or that. I remember my mom was visiting me in the US. Every time I came back from work, she complained that she was tired. Why? Because she did the laundry in the bath tub. She wouldn’t use my washing machine, because she didn’t have such a thing in Russia. I promised her that I’d do the laundry myself when there was a sufficient pile. However, she insisted that the dirty clothes annoyed her. I would point that my water bill went up. And so on.

We argued like this every day. We were both frustrated. Then I asked myself why. Why does she do the laundry? The answer was there. She wanted to be helpful. I calmed down and stopped arguing with her. I sucked it up and paid the water bills. Her time with me turned into the most harmonious visit we ever had. Unfortunately, it was the last.

The year is 1994. The man on the left is my first husband, Alexander Goncharov. Although we were out of touch for a decade, when I married my third husband, Joseph Bernstein (on the right), Goncharov started visiting us. It wasn’t me he was interested in: he wanted to talk mathematics with my husband. I found this situation hilarious, so I took this photo.

But that’s not all. My second husband, Andrey Radul, is not in the picture. But all four of us were students of Israel Gelfand. In short, my three ex-husbands and I are mathematical siblings — that is, we are all one big happy mathematical family.

The Best Writing on Mathematics 2016 is out. I am happy that my paper The Pioneering Role of the Sierpinski Gasket is included. The paper is written jointly with my high-school students Eric Nie and Alok Puranik as our PRIMES-2014 project.

At the end of the book there is a short list of notable writings that were considered but didn’t make it. The “short” list is actually a dozen pages long. And it includes two more papers of mine:

• Cookie Monster Plays Games—another PRIMES project written jointly with my high-school student Joshua Xiong.
• Mathematical Research in High School: The PRIMES Experience, written jointly with PRIMES Chief Research Advisor Pavel Etingof and PRIMES Program Director Slava Gerovitch.

To continue bragging, I want to mention that my paper A Line of Sages was on the short list for 2015 volume. And my paper Conway’s Wizards was included in the 2014 volume.

I married an American citizen and moved to the US in 1990. At the time I was a very patriotic Russian. It took me a year of pain to realize that some of my ideas had been influenced by Soviet propaganda. After I washed away the brainwashing, I fell in love with the US. For 25 years I thought that America was great. Not anymore.

For the last several months I’ve been worried as never before in my life. I feel paralyzed and sick. To help myself I decided to put my feelings in words.

World War. My mom was 15 when World War II started. The war affected her entire life, as well as the lives of everyone in the USSR. Every now and then my mom would tell me, “You are lucky that you are already 20 and you haven’t witnessed a world war.” I moved to the US while my mom stayed back in Russia. From time to time I tell myself something like, “I am lucky that halfway through my expected lifetime, I haven’t had to live through a world war.” It’s been more than 70 years since WWII ended. To maintain the peace is a difficult job. Everything needs to be in balance. Trump is disrupting this balance. I am worried sick that my children or grandchildren will have to witness a major war.

The Red Button. I’ve noticed that, as a true showman, Trump likes misdirecting attention from things that worry him to fantastic plot twists that he invents. What’s the best way to make people forget about his tax returns? It’s the nuclear button. Dropping a nuclear bomb some place will divert people from thinking about his tax returns. As his plot twists are escalating, is he crazy enough to push the button?

Climate. The year 2014 was the warmest on record. The year 2015 was even warmer. And last year, 2016, was even warmer than that. I remember Vladimir Arnold’s class on differential equations. He talked about a painting that had been hanging on a wall for 20 years. Then it unexpectedly fell off. Mathematics can explain how such catastrophic events can happen. I keep thinking about our Earth: melting ice, dead reefs, fish eating plastic, and so much more. My grandchildren might not be able to enjoy beaches and forests the way I did. What if, like the fallen painting, the Earth can spiral out of control and completely deteriorate? But Trump is ignoring the climate issues. Does he care about our grandchildren? I am horrified that Trump’s policies will push climate catastrophe beyond the point of no return.

The Truth. Wiretapping is not wiretapping. Phony jobs numbers stopped being phony as soon as Trump decided that he deserved the credit. The news is fake when Trump doesn’t like it. Trump is a pathological liar; he assaults the truth. Being a scientist I am in search of truth, and Trump diminishes it. I do not understand why people ignore his lies. Two plus two is four whether you are a democrat, or a republican, or whomever. Facts are facts, alternative facts are lies. I am scared that lies have become acceptable and no one cares about the truth any more.

Russia. I lived in Russia for the first unhappy half of my life, and in the US for the second happy half. I do not want to go back. There is something fishy between Putin and Trump. Whether it is blackmail or money, or both, I do not know the details yet, But Trump is under Putin’s influence. Trump didn’t win the elections: Putin won. This horrifies me. I do not want to go back to being under Russian rule.

Gender Issues. I grew up in a country where the idea of a good husband was a man who wasn’t a drunkard. That wasn’t enough for me. I dreamed of a relationship in which there would be an equal division of work, both outside and inside the home. I could not achieve that because in Russian culture both people work full-time and the wife is solely responsible for all the house chores. Moreover, Russia was much poorer than the US: most homes didn’t have washing machines; we never heard of disposable diapers; and there were very long lines for milk and other necessities.

The life in USSR was really unfair to women. Most women had a full-time job and several hours of home chores every day. When I moved to the US, I thought I was in paradise. Not only did I have diapers and a washing machine, I was spending a fraction of the time shopping, not to mention that my husband was open to helping me, and didn’t mind us paying for the occasional babysitter or cleaner.

For some time I was blind to gender issues in the US because it was so much better. Then I slowly opened my eyes and became aware of the bias. For some years it has felt like gender equity was improving. Now, with a misogynistic president, I feel that the situation might revert to the dark ages. When women are not happy, their children are not happy, and they grow up to be not happy. If the pursuit of happiness is the goal, the life has to be fair to all groups. But Trump insults not only women but also immigrants, Muslims, members of the LGBTQ community, as well as the poor and the sick. The list is so long, that almost everyone is marginalized. This is not a path towards a happy society.

Democracy. Trump attacks the press and attempts to exclude them. Trump has insulted the intelligence community and the courts. He seems to be trying to take more power to the presidency at the expense of the other branches of government. He ignores his conflicts of interest. Trump disregards every rule of democracy and gets away with it. I am horrified that our democracy is dying.

Tax Returns. Trump’s tax returns could either exonerate him or prove that he is Putin’s puppet. The fact that he is hiding the returns makes me believe that the latter is more probable. Why the Republicans refuse to demand to see his returns is beyond my understanding.

Corruption. Trump does so many unethical things. Most of his decisions as president seem to be governed by Trump trying to get richer. Let us consider his hotel in Azerbaijan—a highly corrupt country. Having lived in a highly corrupt country myself, I know how it works. For example, an Azerbaijani government official who has access to their country’s money can make a deal that involves a personal kickback. This means that their government is paying more than necessary for a service or product in order to cover that kickback. This is how national money makes its way into individual pockets. Since all the deals in Azerbaijan are reputed to be like that, I imagine that when Trump built his hotel there, the Trump organization was overpaid in order to cover the bribe to local officials. Will our country become as corrupt as Azerbaijan?

Americans. The biggest shock of the election was that so many people were so gullible and actually voted for Trump. They didn’t see that his agenda is focused on his own profit, and that he lies and makes promises he doesn’t plan to deliver. It really terrifies me that there are some people who are not gullible but still voted for Trump.

Is there hope?

I already posted a funny true story that Smullyan told me when I last visited him. Raymond Smullyan died recently at the age of 97 and my mind keeps coming back to this last visit.

The year was 2012 and I was about to drive back to Boston after my talk at Penn State. Smullyan’s place in the Catskills was on the way—sort of. I wanted to call him, but I was apprehensive. Raymond Smullyan had a webpage on which his email was invisible. You could find his email address by looking at the source file or by highlighting empty space at the bottom of the page. Making your contact information invisible sends a mixed message.

While this was a little eccentric, it meant that only people who were smart enough to find it, could access his email address. I already knew his email because he had given it to me along with his witty reply to my blog post about our meeting at the Gathering for Gardner in 2010.

In our personal interactions, he always seemed to like me, so I called Raymond and arranged a visit for the next day around lunch time. When I knocked on his door, no one answered, but the door was open, and since Smullyan was expecting me, I walked right in. “Hello? Anyone there? Hello? Hello?” As I wandered around the house, I saw an open bedroom door and inside Smullyan was sleeping. So I sat down in his library and picked up a book.

When he woke up, he was happy to see me, and he was hungry. He told me that he didn’t eat at home, so we should go out together for lunch. I was hungry too, so I happily agreed. Then he said that he wanted to drive. I do not have a poker face, so he saw the fear in me. My only other trip with a nonagenarian driver flashed in front of my eyes. The driver had been Roman Totenberg and it had been the scariest drive I have ever experienced.

I said that I wanted to drive myself. Annoyed, Raymond asked me if I was afraid of him taking the wheel. I told him that I have severe motion sickness and always prefer to drive myself. Raymond could see that I was telling the truth. I got the impression that he was actually relieved when he agreed to go in my car.

We went to Selena’s Diner. He took out playing cards with which he showed me magic tricks. I showed him some tricks too. This was probably a bad move as he abandoned me to go to the neighboring table to show his magic tricks to a couple of young girls. They were horrified at first”his unruly hair, his over-the-top energy, his ebullient behavior”but between me and the waitress, we quickly reassured them. The girls enjoyed the tricks, and I enjoyed my visit.

Like many people, I was appalled by Trump’s immigration ban. On the Internet I found many essays that explained that he did not include in the ban those majority-Muslim countries in which he has business interests. See for example, an article at Forbes with a nice map, and an article at NPR.

Now the countries that are excluded are motivated to continue to support Trump’s businesses, and to offer him bribes and good deals in exchange for staying out of the ban. The countries on the list are also motivated to approach Trump and offer him a sweet business deal.

So even if the courts stopped the ban, he has already succeeded in showing every country in the world that to be on his good side requires that they pay up. And China got the hint and granted Trump a trademark he’s been seeking for a decade.

Looks like Trump’s vision of a great America is a very rich Mr Trump.

The first time I visited the US was in 1990 at the invitation of an old friend, Joseph Bernstein. After my arrival Joseph proposed and I accepted, but my essay is not about that.

Joseph reintroduced me to his daughter, Mira, who was then in her late teens. I was struck by Mira’s charm. I had never before met teenagers like her. Of course, Joseph got points for that as I was hoping to have a child with him. When I moved to the US I met some other kids who were also incredibly charming. It was too late to take points away from Joseph, but it made me realize what a huge difference there was between Soviet and American teenagers. American teenagers were happier, more relaxed, better mannered, and less cynical than Soviet ones.

My oldest son, Alexey, was born in the USSR and moved to the US when he was eight. One unremarkable day when he was in middle school (Baker public school in Brookline), the principle invited me for a chat. I came to the school very worried. The principal explained to me that there was a kid who was bugging Alexey and Alexey pushed him back with a pencil. While the principal proceeded to explain the dangers of a pencil, I tuned out. I needed all my energy to conceal my happy smile. This was one of the happiest moments of my life in the US. What a great country I live in where the biggest worry of a principal in a middle school is the waving of a pencil! I remembered Alexey’s prestigious school in Moscow. They had fights every day that resulted in bloody noses and lost teeth. When I complained to his Russian teacher, she told me that it was not her job to supervise children during big breaks. Plus the children needed to learn to be tough. No wonder American children are happier.

I was wondering if there were any advantages to a Soviet upbringing. For one thing, Soviet kids grow up earlier and are less naive. They are more prepared for harsh realities than those American kids who are privileged.

Naive children grow up into naive adults. Naive adults become naive presidents. I watched with pain as naive Bush (“I looked the man in the eye. I found him to be very straightforward and trustworthy.”) and naive Obama (Russian reset) misunderstood and underestimated Putin.

Putin is (and, according to Forbes Magazine, has been for the last four years) the most powerful person in the world. Even though the US kept its distance from Russia, he was able to manipulate us from afar. Now that Trump wants to be close to Putin, the manipulation will be even easier. Putin is better at this game. He will win and we will lose.