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Mathematicians have always been attracted to the field of genetics. The mathematical aspects of research on homosexuality are especially interesting. Certain studies show that male homosexuality may have a genetic component that is correlated with female fertility. Other studies show the existence of the fraternal birth order effect, that is, the correlation of homosexuality with the number of older brothers. This article is devoted to the mathematical aspects of how these two phenomena are interconnected. In particular, we show that the fraternal birth order effect implies a correlation between homosexuality and maternal fecundity. Vice versa, we show that the correlation between homosexuality and female fecundity implies the increase in the probability of the younger brothers being homosexual.

We discuss four famous card games that can help learn linear algebra. The games are: SET, Socks, Spot it!, and EvenQuads. We describe the game in the language of vector, affine, and projective spaces. We also show how these games are connected to each other. A separate section is devoted to playing Socks with the EvenQuads deck and vice versa.

I was a member of the team Death and Mayhem, which won the 2017 MIT Mystery Hunt. My team's reward was to write the 2018 MIT Mystery Hunt. By the way, some people consider it a punishment rather than a reward. As one colleague said to me on our win, ‘Congratudolences!’

I was a member of the team Death and Mayhem, which won the 2017 MIT Mystery Hunt. My team's reward was to write the 2018 MIT Mystery Hunt. By the way, some people consider it a punishment rather than a reward. As one colleague said to me on our win, ‘Congratudolences!’

When Martin Gardner first presented the Two-Children Problem, he made a mistake in its solution. Later he corrected the error, but unfortunately the incorrect solution is more widely known than his correction. In fact, a Tuesday-Child variation of this problem went viral in 2010, and the same flaw keeps reappearing in proposed solutions of that problem too. In this article, we re-visit Martin Gardner’s correction and discuss the new problem in detail.

As in many coin puzzles, we have several identical-looking coins, with one of them fake and the rest real. The real coins weigh the same. Our fake coin is special in that it can change its weight. The coin can pretend to be a real coin, a fake coin that is lighter than a real one, and a fake coin that is heavier than a real one. In addition, each time the coin is on the scale, it changes its weight in a predetermined fashion.In this paper, we seek to find our fake coin using a balance scale and the smallest number of weighings.We consider different possibilities for the fake coin. We discuss coins that change weight between two states or between three states. The 2-state coin that changes weight from lighter to real and back has been studied before, so we concentrate on the 2-state coin that changes weight from lighter to heavier and back. We also study the 3-state coin, which changes its weight from lighter to heavier to real and back to lighter.Given the total number of coins and the starting state of the fake coin, we calculate the smallest number of weighings needed to identify the fake coin. We provide an oblivious optimal strategy for this number of weighings. We also discuss what happens if the starting state is unknown or mixed. In such cases, adaptive strategies are often more powerful than oblivious ones.

Chip-firing is a combinatorial game on a graph, in which chips are placed and dispersed among its vertices until a stable configuration is achieved. We specifically study a chip-firing variant on an infinite, rooted, directed$k$ -ary tree where we place$k^n$chips labeled$0,1,\\dots, k^n-1$on the root for some nonnegative integer$n$ , and we say a vertex$v$can fire if it has at least$k$chips. When a vertex fires, we select$k$labeled chips and send the$i$ th smallest chip among them to its$i$ th leftmost child. A stable configuration is reached when no vertex can fire. In this paper, we focus on stable configurations resulting from specific firing strategies based on permutations of$1, 2, \\dots, n$ . We then express the stable configuration as a permutation of$0,1, 2, \\dots, k^n-1$and explore its properties, such as the number of inversions and descents. 20 pages, 4 figures, 1 table, v4: final version

We research a combinatorial game based on the Cookie Monster problem called the Cookie Monster game that generalizes the games of Nim and Wythoff. We also propose several combinatorial games between this Cookie Monster game and Nim, and discuss the winning positions of these games.

Consider equipping an alphabet A with a group action which partitions the set of words into equivalence classes which we call patterns. We answer standard questions for Penney’s game on patterns and show non-transitivity for the game on patterns as the length of the pattern tends to infinity. We also analyze bounds on the pattern-based Conway leading number and expected wait time, and further explore the game under the cyclic and symmetric group actions.

This is a special collection of problems that were given to select applicants during oral entrance exams to the Department of Mechanics and Mathematics of Moscow State University. These problems were designed to prevent Jewish candidates and other “undesirables” from getting a passing grade, thus preventing them from studying at MSU. Among problems that were used by the department to blackball unwanted candidate students, these problems are distinguished by having a simple solution that is usually difficult to find. Using some problems with a simple solution protected the administration from extra complaints and appeals. This collection, therefore, has mathematical as well as historical value.