Dr. Tanya Khovanova is a mathematician whose research interests lie in recreational mathematics, combinatorics, probability, geometry, number theory. Currently, she is a Lecturer and PRIMES Head Mentor at the Massachusetts Institute of Technology (MIT).
In To Count the Natural Numbers, Emily Jia (former 2016 AWM Essay Contest winner and a recent graduate in Math and Computer Science at Harvard) writes a fantastic essay where she interviews Khovanova. I was very appreciative to read about her personal story, career path in mathematics, and the motivation behind creating her blog. In particular, the excerpt below resonated with me deeply,
“Having struggled with writers block, Tanya started a blog that changed her life. She began to take English lessons, and stopped being afraid of writing papers. When she wrote about mathematical topics on her blog, she could write 3-4 posts and have enough material for a paper. Finally, she realized, “I wasn’t successful before as a mathematician because I was always doing what people told me to do.” Gelfand gave her the problem for her first publication, and afterwards she followed her then-husbands’ interests. She had picked a job in industry that she didn’t enjoy but, finally, this blog was a chance to turn this around. For the first time, she learned to follow her heart. And her heart led her to recreational mathematics: a mix of combinatorics, geometry, probability theory, and number theory that resembles puzzles instead of abstract math” – From To Count the Natural Numbers
Her blog features a great number of neat puzzles. Some of which have been highlighted in some of the previous posts on this blog (e.g. On the mathematical wedding controversy, How Math Can Help You Avoid Talking about Politics at the Holidays, and Hot Hands and Tuesday’s Children).
In this tour, I hope to give you a glimpse of the blog’s content and review two of my favorite posts. What I love about many of her posts is that they highlight joint projects with her students from MIT’s PRIMES STEP (Solve–Theorize–Explore–Prove), a program aimed at middle schoolers who like solving challenging problems. Khovanova’s blog posts are a great segway to the articles that dive deeper into the projects.
In this post, Khovanova discusses a game that her students from the PRIMES STEP program invented where they mix the rules of two games: Penney’s game and an original game by the same group called The Non-Flippancy game. As described in the post, Penney’s game has two players, Alice and Bob, that individually select separate strings comprised of coin flip outcomes (i.e. H for heads and T for tails) of a fixed length n. They toss a fair coin repeatedly until one player’s selected string appears in the sequence of tosses and they are declared the winner.
In contrast, the non-flippancy game does not require a coin, instead, players alternately select a flip outcome deterministically according to a “flip” rule. Again, whoever’s string appears first in the sequence of choices wins. The blended game is a combination of the previous two games where now when Alice’s and Bob’s wanted outcomes coincide, that is the outcome they receive, similar to the No-Flippancy Game. If not, they flip a coin.
“For example, suppose Alice selects HHT, and Bob selects THH. Then Alice wants H and Bob wants T, so they flip a coin. If the flip is T, then they both want Hs, and Bob wins. If the first flip is H, they want different things again. I leave it to the reader to see that Bob wins with probability 3/4. For this particular choice of strings, the odds are the same as in Penney’s game, but they are not always the same.”
She concludes that this game has the interesting property of non-transitive cycle of choices of length 6. You can read more about it in the arXiv papers The No-Flippancy Game and From Unequal Chance to a Coin Game Dance: Variants of Penney’s Game. Students Co-authors: Isha Agarwal Matvey Borodin Aidan Duncan Kaylee Ji Shane Lee Boyan Litchev Anshul Rastogi Garima Rastogi Andrew Zhao.
This post brought many great memories from my time as a graduate student. The game SET was popular during our math-related outreach activities and was a favorite among my peers. In the SET game, for each of four categories of features (i.e. color, number, shape, and shading), a player must spot three cards that display said feature as all the same (or all different) to make a set.
In this post, Khovanova illustrates what is called a magic SET square , which is “a 3 by 3 table of SET cards such that each row, column, diagonal, and anti-diagonal is a set”. This square is a fantastic combination of magic squares (i.e. an arrangement of numbers in a square in such a way that the sum of each row, column, and diagonal is one constant number) and the SET game.
As she explains in the post, her students invented a version of tic-tac-toe, played on the 9 cards that form the magic SET square. It was super exciting that in this version of tic-tac-toe ties are impossible, and the first player can always win. What amazed me was the idea of combining three different games in one for a completely new experience. You can read more about it in the arXiv paper The Classification of Magic SET Squares to see an overview of the game, and its properties. Student Co-authors: Eric Chen, William Du, Tanmay Gupta, Alicia Li, Srikar Mallajosyula, Rohith Raghavan, Arkajyoti Sinha, Maya Smith, Matthew Qian, Samuel Wang.
Have an idea for a topic or a blog you would like for me and Rachel to cover in upcoming posts? Reach out in the comments below or on Twitter (@VRiveraQPhD).