{"id":5464,"date":"2020-07-29T22:27:37","date_gmt":"2020-07-30T02:27:37","guid":{"rendered":"http:\/\/blogs.ams.org\/blogonmathblogs\/?p=5464"},"modified":"2020-08-07T18:26:50","modified_gmt":"2020-08-07T22:26:50","slug":"tanya-khovanovas-math-blog-a-tour","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/blogonmathblogs\/2020\/07\/29\/tanya-khovanovas-math-blog-a-tour\/","title":{"rendered":"Tanya Khovanova’s Math Blog: A Tour"},"content":{"rendered":"

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).\u00a0<\/p>\r\n

In To Count the Natural Numbers<\/a>, <\/span><\/span><\/em>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,\u00a0 and the motivation behind creating her blog<\/a><\/span>. In particular, the excerpt below resonated with me deeply,\u00a0<\/p>\r\n

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“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, \u201cI wasn\u2019t successful before as a mathematician because I was always doing what people told me to do.\u201d Gelfand gave her the problem for her first publication, and afterwards she followed her then-husbands\u2019 interests. She had picked a job in industry that she didn\u2019t 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<\/p>\r\n<\/blockquote>\r\n

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<\/a><\/span><\/em>, How Math Can Help You Avoid Talking about Politics at the Holidays<\/a><\/em><\/span>, and Hot Hands and Tuesday\u2019s Children<\/a><\/span><\/em>).<\/p>\r\n

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\u2013Theorize\u2013Explore\u2013Prove),<\/a><\/span> 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.\u00a0<\/p>\r\n

The Blended Game<\/a><\/span><\/strong><\/p>\r\n

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<\/a><\/span><\/em>.\u00a0 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.\u00a0<\/p>\r\n

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\u2019s string appears first in the sequence of choices wins. The blended game is a combination of the previous two games where now when Alice\u2019s and Bob\u2019s wanted outcomes coincide, that is the outcome they receive, similar to the No-Flippancy Game. If not, they flip a coin.<\/p>\r\n

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“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\u2019s game, but they are not always the same.”<\/p>\r\n<\/blockquote>\r\n

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\u00a0 The No-Flippancy Game<\/a><\/span><\/em> and From Unequal Chance to a Coin Game Dance: Variants of Penney\u2019s Game<\/span><\/a><\/em>. Students Co-authors: Isha Agarwal Matvey Borodin Aidan Duncan Kaylee Ji Shane Lee Boyan Litchev Anshul Rastogi Garima Rastogi Andrew Zhao.<\/p>\r\n

Set Tic-Tac-Toe<\/a><\/span><\/strong><\/p>\r\n

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.\u00a0<\/p>\r\n

\"\"<\/a>

Figure 1. Example of the SET game cards. These three cards are considered a set since all their features are different.<\/p><\/div>\r\n

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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”.\u00a0 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.\u00a0<\/p>\r\n

As she explains in the post, her students invented a version of tic-tac-toe,\u00a0 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<\/a><\/span><\/em> to see an overview of the game, and its properties. Student Co-authors: Eric Chen,\u00a0William Du,\u00a0Tanmay Gupta,\u00a0Alicia Li,\u00a0Srikar Mallajosyula,\u00a0Rohith Raghavan,\u00a0Arkajyoti Sinha,\u00a0Maya Smith,\u00a0Matthew Qian,\u00a0Samuel Wang.<\/p>\r\n

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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<\/a><\/span>).<\/em><\/p>\r\n<\/div>\r\n

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<\/div>","protected":false},"excerpt":{"rendered":"

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).\u00a0 In To Count the Natural Numbers, … Continue reading →<\/span><\/a><\/p>\n

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