Tanya Khovanova's Math Blog

My STEP students invented a coin-flipping game that doesn’t require a coin. It is called The No-Flippancy Game.

Alice and Bob choose distinct strings of length n consisting of the letters H (for heads) and T (for tails). The two players alternate selecting the outcome of the next “flip” to add to the sequence by the rule below.

The “flip” rule: Let i < n be the maximal length of a suffix of the sequence of chosen outcomes that coincides with a prefix of the current player’s string. The player then selects the element of their string with index i + 1 as the next term in the sequence.

Alice goes first, and whoever’s string appears first in the sequence of choices wins. In layman terms, the game rules mean that the players are not strategizing, but rather greedily finishing their strings.

Suppose n = 2 and Alice chose HH. If Bob chooses HT, then Bob wins. Alice has to choose H for the first flip. Then Bob chooses T and wins. On the other hand, if Bob chooses TT for his string, the game becomes infinite. On her turn, Alice always chooses H, while on his turn Bob always chooses T. The game outcome is an alternating string HTHTHT… and no one wins.

Suppose n = 4, Alice chooses HHTT, and Bob chooses THHH. The game proceeds as HTHHTHHH, at which point Bob wins.

This game is very interesting. The outcome depends on how Alice’s and Bob’s chosen strings overlap with each other. We wrote a paper about this game, which is available at math.CO arXiv:2006.09588.

2020 MIT Mystery Hunt

Every year I write about latest MIT Mystery Hunt puzzles that might be appealing to mathematicians. Before diving into mathy puzzles, I would like to mention two special ones:

• Fortune Cookies—Our team laughed at this one.
• No Clue Crossword—Our team was puzzled by this one.

Unfortunately math wasn’t prominent this year:

• Food Court—This is a probability puzzle that is surprisingly uninspiring. There is no mystery: the puzzle page contains a list of probability problems of several famous types. But this puzzles can find great use in probability classes.
• Torsion Twirl—Mixture of dancing and equations. I love it.
• People Mover—Logical deduction at the first stage.

On the other hand, Nikoli-type puzzles were represented very well:

• The Ferris of Them All—Several different Nikoli puzzles on a wheel.
• Toddler Tilt—Not exactly a Nicoli puzzle, but some weird logic on a grid, some music too.
• The Dollhouse Tour—Not exactly a Nicoli puzzle, but some weird logic on a grid, some pictures too.
• The Nauseator—The first part of the puzzle is a huge nonogram.
• Domino Maze—A non-trivial Thinkfun puzzle.
• Backlot—Finding a path on a grid with a fractal structure.
• Whale—Variation on Rush Hour.

Some computer sciency puzzles:

• Hackin’ the Beanstalk—Hidden algorithms.
• Turtle—LogoWriter.
• Bear—Origami.

Cryptography:

• The Scottish Display—You are given trigrams.
• Pig—The pigpen cypher.
• ANDARAC—You are given gibberish looking telegrams.

A couple of puzzles with the mathy side hidden:

• Tunnel of Love.
• Pied Piper—A puzzle type that I like and promoted is hidden here.

The academic year is over and my junior PRIMES STEP group finished their paper about a classification of magic SET squares. A magic SET square is a 3 by 3 square of SET cards such that each row, column, and diagonal is a set. See an example below. The paper is posted at the arXiv:2006.04764.

In addition to classifying the magic SET squares, my students invented the game of SET tic-tac-toe. It is played on nine cards that form a magic SET square. Two players take turns picking a card from the square. The first player who has a set wins.

One might think that this game is the same as tic-tac-toe, as a player wins as soon at they have cards from the same row, column, or diagonal. But if you build a magic SET square, you might notices that each magic SET square contains 12 sets. In addition to rows, columns, and diagonals, there are sets that form broken diagonals. The picture below shows all the sets in a magic SET square.

There are more ways to win in this game than in a regular tic-tac-toe game. My students proved that ties are impossible in this game. They also showed, that, if played correctly, the first player always wins.

You probably heard in the news that more men are dying from coronavirus than women. But not in Massachusetts. Here the proportion of women is about 52 percent. Why is this the case? Being a woman, should I be worried that I live in Massachusetts?

We know that coronavirus strikes older people harder than younger ones. Thus, we should take age into account. In the US more boys are born than girls. By the age of 40 the ratio evens out. Starting from 40 there are more women than men. With each next age group, the disparity increases. According to a recent US population report and for ages 85 and over there are about 4.22 million women versus 2.33 men: the proportion is almost 2 to 1.

As the coronavirus targets older people, were it gender-neutral, we would have had way more female deaths than male. This is not the case. So it hits males harder than females. But why are the ratios of female to male deaths different for different countries and states?

One simple explanation is that this is related to life expectancy and the age of the population. The older the population, the bigger the percentage of females. Which in turn increases the proportion of female deaths.

It could also be that Massachusetts has good health care making the average age of dying patients older than the average age for the country. This in turn will increase the proportion of females dying from coronavirus. No, I am not worried about living in Massachusetts.

I grew up in the USSR, where I was clueless about the race issues in the US. I have now lived in the US for 30 years, and still feel that there are many things about race that I do not understand. As a result, I am afraid to speak about it. I am worried that I’ll say something wrong. Recent events have encouraged me to say something. This is my first piece about race.

I came back to mathematics 10 year ago and started working at MIT. I love it. With some exceptions.

Many mathematicians are introverts or snobs or gender-biased. They are not usually friendly. I often walk down a corridor and people who are coming towards don’t notice me. If I say hello, they might not even reply or raise their eyes. It could be they are thinking about their next great theorem and do not notice me. It could be that I am not faculty and therefore do not deserve their attention. It could be that as a women I am not worth of their hello.

Soon after I started working at MIT, I was reminded of one of the reasons I left academia. It was this unfriendliness. But this time was different. First, I had grown a thicker skin. Second, I was working within a group. People who were working with me were nice to me. It was enough and so I stayed.

With time I adopted the same style: passing people without saying `Hello.’ Mostly I got tired of people not replying to my hello.

One day I was passing this man who, as had happened many times before, purposefully didn’t look at me. I thought my usual thought: another introverted/snobbish/gender-biased mathematician. Then I suddenly stopped in my tracks. My logic was wrong. This guy was Black. The unfriendliness of mathematicians is surely way worse for him than for me. It could be that he is looking at the floor for the same reason I do it: he is afraid that people will ignore his greeting. I failed to think about race deep enough before this realization. What happened next should have happened years earlier.

I took the initiative and the next couple of times I saw him, I said hello. This was all it took—two hellos—to change the whole feeling between us. The guy has a great smile.

It was reported last week that that 37 NYPD members died of covid-19. I assume that they were way below 65. It is known that the coronovirus death rate for people below 65 is a quarter of the total death rate. That means, 37 people in NYPD correspond to at least 150 people in general. Assuming that the mortality rate of coronavirus is 1 percent, the number of infected NYPD members a month ago was 15000.

By now, it could be that more than half of NYPD was infected.

NYPD members have to communicate with people a lot due to the nature of their work. That means they are more prone to being infected. At the same time, they transmit more than people in many other professions.

I can conclude, that about half of the people that are high transmitters in NY have antibodies by now. Assuming they are immune, the covid transmission rate in NY has to be down.

Assuming the immunity stays with people for a while, the second wave in NY can’t be as bad as the first one.

Suppose we want to pack a unit square with non-overlapping rectangles that have sides parallel to the axes. The catch is that the lower left corners of all the rectangles are given. By the way, such rectangles are called anchored. Now, given some points in the unit square, aka the lower left corners, we want to find anchored rectangles with the maximum total area.

When the given points are close to the right upper corner of the square, the total area is small. When a single point is in the bottom left corner of the square, we can cover the whole square. The problem becomes more interesting if we add one extra assumption: one of the given points has to be the bottom left corner of the square. In the 1960’s, it was conjectured by Allen Freedman that any set of points has an anchored rectangle packing with the area of at least one half. The problem is quite resistant. In 2011, Dumitrescu and Tóth showed that every set of points has a packing of area at least 0.09, which was the first constant bound found, and is the best bound currently known.

I gave this problem to my PRIMES student Vincent Bian. He wrote a paper, Special Configurations in Anchored Rectangle Packings, that is now available at the arxiv. When you look at this problem you see that the number of ways to pack depends on the relative coordinates of the points. That means you can view the points as a permutation. Vincent showed that the conjecture is true for several different configurations of points: increasing, decreasing, mountain, split layer, cliff, and sparse decreasing permutations.

An increasing permutation is easy. There are two natural ways to pack the rectangles. One way, when rectangles are horizontal and each rectangle reaches to the right side of the square (see picture above). Another way, when rectangles are vertical. When you take the union of both cases, the square is completely covered, which means at least one of the cases covers at least half of the square. The worst case scenario, that is, the case when the maximum possible area is the smallest is when your points are placed equidistantly on the diagonal.

Other cases are more difficult. For example, Vincent showed that for a decreasing permutation with n points, the worst case scenario is when the points are arranged equidistantly on a hyperbola xy = (1-1/n)ⁿ. The picture shows the configuration for 15 points. The total area is more than one half.

(I wrote this piece for La Recherche. It was translated into French by Philippe PAJOT. You can find this piece and pieces by others at John Horton Conway: a magician of maths disappears.)

Unlike many other mathematicians I know, John Conway cared a lot about the way he presented things. For example, in the puzzle he invented—known as Conway’s Wizards—the wizards had to be riding on a bus. Why was the bus so important? You see, the numbers in the puzzle were related to the age of one of the wizards, the number of the bus, and the number of the wizard’s children. It was important to John that the readers be able to use a convenient notation a, b and c for these numbers and remember which number is which.

When I give my lecture on integers and sequences, I show my students a list of different famous sequences. The first question from the audience is almost always: “What are the Evil Numbers?” As you can guess the name for this sequence was invented by John Conway. This name was invented together with the name of another sequence which is called Odious Numbers. These two sequences are complementary in the same sense as even and odd numbers are complementary: every natural number is either evil or odious. The names are good, not only because they attract, but also because they help remember what the sequences are. Evil numbers are numbers with an even number of ones in their binary representation. I assume that you can interpolate what the odious numbers are.

When he was lecturing, John used all sorts of tricks to emphasize important points: From time to time I saw him shouting or throwing his shoes. Once I remember him staring at his statement written on the blackboard for a really long time. My neighbor in the lecture hall got uncomfortable. He assumed that John, who was at that time way over 70, was blanking out and had forgotten what he wanted to say. I calmed my neighbor down. It was my fourth time listening to the same lecture, including the same pause. John Conway didn’t forget.

The picture was taken on December 21 of 2019 at Parker Life care facility right before dinner.

Every year I review MIT mystery hunt from a mathematician’s point of view. I am way behind. The year is 2020, but I still didn’t post my review of 2019 hunt. Here we go.

Many puzzles in 2019 used two data sets. Here is the recipe for constructing such a puzzle. Pick two of your favorite topics: Star Trek and ice cream flavors. Remember that Deanna Troi loves chocolate sundae. Incorporate Deanna Troi into your puzzle to justify the use of two data sets.

On one hand, two data sets guarantee that the puzzle is new and fresh. On the other hand, often the connection between two topics was forced. Not to mention that puzzle solving dynamic is suboptimal. For example, you start working on a puzzle because you recognize Star Trek. But then you have to deal with ice cream which you hate. Nonetheless, you are already invested in the puzzle so you finish it, enjoying only one half of it.

Overall, it was a great hunt. But the reason I love the MIT mystery hunt is because there are a lot of advanced sciency puzzles that can only appear there. For example, there was a puzzle on Feynman diagrams, or on characters of representations. This year only one puzzle, Deeply Confused, felt like AHA, this is the MIT Mystery hunt.

Before discussing mathy puzzles I have to mention that my team laughed at Uncommon Bonds.

I will group the puzzles into categories, where the categories are obvious.

Mathy puzzles.

• Bubbly—Combinatorial game theory.
• Rules of Order—Fractals mixed with movies.
• Cubic—A long list of cubic polynomials with integer coefficients.
• Twisty Passages—Famous number properties that remind my of my Number Gossip website.

Here are some logic puzzles, in a sense that Sudoku is a logic puzzle.

• Lantern festival—A cool mixture of Slitherlinks and Galaxies.
• Invisible Walls.
• Place Settings.
• Middle School of Mines—Minesweeper.
• Moral Ambiguity—Nonograms with a twist.
• Connect Four—Mastermind. There was a strong hint that the extraction step was also mastermind. My team spent some time trying to mastermind the ending, until we backsolved. The extraction step was not mastermind. The final grid in the puzzle had the word CODE written in red. It corresponded to letters CDEO found at that location. Given that the letters were not in alphabetical order, it gave the ordering, which didn’t exist in the puzzle. Anyway, you can see that I have a grudge against this puzzle. This could have been a great puzzle. But it wasn’t.
• Schematics—Tons of Nikoli puzzles of different types.

Now we have logic puzzles or another type, where you need to draw a grid. These are puzzles of the type: Who lives in the White House?

• Compromised.

Now we have logic puzzles or yet another type, where you need to figure out which statements are true and which are false.

• True and False².

Now some cryptography.

• Tales From The Crypt.
• You’re Gonna Need a Bigger Gravy Boat.
• The Bill.
• Helvetica Is Only an Okay Font.
• A Good Walk Spoiled—Some decyphering and Vim.
• Have You Seen Me?
• Chain of Commands.
• Picture Book.

And some programming.

• Turtle Power!

Miscellaneous.

• Hexed Adventure II: Hexed Again!—A text adventure game with a lot of logic.
• Loaded—Dice of different shapes.
• The Lives of Π—Large numbers.
• Battle of the Network Stars—A lot of actors in a grid.