During the Fall 2024 semester, we will have the math circle meetings in Cupples I Hall Room 199 on Saturdays, 3-5pm, broken down into two sessions:
- Saturday 3-4pm: middle school session (intended for middle school students and advanced elementary school students)
- Saturday 4-5pm: high school session (intended for high school students and advanced middle school students)
Note: Any student is welcome to attend any session. The descriptions are only guidelines.
| Date | Time | Session Leader | Topic |
| Sep 28, 2024 | 3pm | Brandon Sweeting | Toothpick Tricks |
| Sep 28, 2024 | 4pm | Alan Chang | AM-GM inequality |
| Oct 5, 2024 | 3pm | Brandon Sweeting | Cryptic Cards |
| Oct 5, 2024 | 4pm | Hao Zhuang | Convexity of functions |
| Oct 12, 2024 | 3pm | Brandon Sweeting | Chip Solitaire |
| Oct 12, 2024 | 4pm | Lily Zhang | An Introduction to Continued Fractions and the Magic Table |
| Oct 19, 2024 | 3pm | Angel Roman | Euler Characteristic of Convex Polyhedra |
| Oct 19, 2024 | 4pm | Angel Roman | Euler Characteristic of Any Polyhedra |
| Oct 26, 2024 | 3pm | Greg Knese | King Chickens |
| Oct 26, 2024 | 4pm | RJ Acuna | Markov numbers |
| Nov 2, 2024 | 3pm | Martha Precup | Map Coloring |
| Nov 2, 2024 | 4pm | Alan Chang, Brandon Sweeting | AMC 10 practice |
| Nov 9, 2024 | 3pm | Swarup Dhar, Mohao Yi | Defeating the Hydra, Part 1 |
| Nov 9, 2024 | 4pm | Swarup Dhar, Mohao Yi | Defeating the Hydra, Part 2 |
| Nov 16, 2024 | 3pm | Brandon Sweeting | Dots-and-Boxes |
| Nov 16, 2024 | 4pm | Parker Evans | Geometry and Combinatorics of Catalan Numbers |
| Nov 17, 2024 (Sunday) | 2-3pm | Michael Chen, Silas Johnson | ARML Power Contest (the target audience is high school students, but middle school students are welcome) Please fill out this form if you plan to attend |
| Nov 23, 2024 | [no meeting] | ||
| Nov 30, 2024 | [no meeting] | ||
| Dec 7, 2024 | 3-4pm | Alan Chang, Nick Danis | Language Puzzles, Part 1 |
| Dec 7, 2024 | 4-5pm | Alan Chang, Nick Danis | Language Puzzles, Part 2 |
| [The next meeting will be in mid/late-January. Stay tuned!] |
Descriptions
Sep 28, 2024
Toothpick Tricks: In this activity, students use toothpicks to explore geometry by forming squares within a grid. They begin by counting squares of different sizes and then solve puzzles, like removing two toothpicks to leave exactly two squares without loose ends. They will also explore variations, such as moving toothpicks to form images, encouraging critical thinking and the development of spatial reasoning skills.
AM-GM inequality: We will investigate the relationship between the arithmetic and geometric means, and use this to prove inequalities and study optimization problems.
Oct 5, 2024
Cryptic Cards: In this game of deduction, one player chooses a secret rule (e.g., “only black cards” or “black cards must be even”). The rest of the group tries to figure out the rule by playing cards one at a time, with the rule-setter telling them whether each card follows the rule. As students gather more clues, they can narrow down the possibilities. The rules start simple and become more challenging, promoting strategic thinking and logical reasoning. (Note: please bring a deck of cards if you have one.)
Convexity of functions: Concave up or down, that is a straightforward observation but indicating surprising quantitative relations.
Oct 12, 2024
Chip Solitaire: In this puzzle, students start with blue and red chips on a row of squares. The goal is to remove all chips, following two rules: only blue chips can be removed, and removing a chip flips the color of its neighboring chips. Students explore different starting arrangements to determine if complete removal is possible and find patterns to predict outcomes, enhancing their problem-solving and strategic thinking skills.
An Introduction to Continued Fractions and the Magic Table: Real numbers are often approximated by rational numbers. The best rational approximation of a real number x is a rational number a/b that is closer to x than any other rational number with a smaller or equal denominator. In this week’s math circle, we will explore an iterative method for finding the best rational approximation of any irrational number using continued fractions.
Oct 19, 2024
Euler Characteristic of Convex Polyhedra: Convex polyhedrons are the familiar three dimensional versions of polygons, such as cubes and pyramids. These shapes always have faces, edges, and vertices and we will call the number of faces F, the number of edges E, and the number of vertex V. The Euler characteristic is then defined as F-E+V. We’re going to build many different convex polyhedra (maybe even some nonconvex ones) and calculate the Euler characteristic. What is the value we get? How are the Euler characteristic of these polyhedra related to each other? We’ll look for patterns and then we will try to explain, that is prove the pattern we find. To do this, we will also discuss planar graphs, which are sets of vertices and sets of edges between the vertices on the plane such that the edges do not overlap.
Euler Characteristic of Any Polyhedra: We will quickly discuss the Euler characteristic of regular polyhedra, which is F-E+V, where F is the number of faces, E is the number of edges, and V is the number of vertex. We will discuss the planar graph as a way of proof for the Euler characteristic. Then we will explore the Euler characteristic of nonconvex polyhedron. The go-to example will be a polyhedral version of a torus, which is the shape of a donut. What is the Euler characteristic of such a polyhedron? Can we come up with some proof? We’ll also explore other types of shapes.
Oct 26, 2024
King Chickens: In a flock of chickens, between any two chickens there will be one chicken that always bosses (or pecks) the other. How do we decide which chicken is the king or the mega-boss chicken in a flock? We will give a reasonable notion of “king chicken” and then figure out how many kings we can end up with.
Markov numbers: We’ll introduce Markov numbers, and explain how they’re used in Diophantine Approximation.
Nov 2, 2024
Map Coloring: When creating a map, we want to make sure that regions sharing a boarder have distinct colors. An assignment of colors to the regions satisfying this rule is called a proper coloring. In this activity, we will explore proper colorings and look for patterns that explain how many colors are needed.
AMC 10 practice: We will discuss problems from the 2023 AMC 10B (AoPS page with problems and PDF version). Students should work on the problems before coming to the math circle. At the session, we will discuss problem solving strategies. Students will have the opportunity to present solutions to some questions from the 2023 AMC 10B. For more on the AMC (American Mathematics Competitions), see the Mathematical Association of America website.
Nov 9, 2024
Defeating the Hydra: We find ourselves facing off against a hydra, and not just any hydra a mathematical hydra. This mathematical creature has a tree-like body with heads branching out of its body. Whenever we defeat one of its heads it seems to grow more of them according to peculiar rules. Students go on an adventure to defeat this mathematical creature by exploring various strategies and starting configurations. Students answer questions about what kinds of hydras can be defeated and what are the optimal strategies for defeating said hydra, encouraging them to think algorithmically and challenging them to construct abstract arguments.
Nov 16, 2024
Dots-and-Boxes: Dive into the classic game of Dots-and-Boxes, where simple rules lead to complex strategy! In this interactive session, participants will learn the basics of the two-player game played on a rectangular grid of dots. Taking turns, each player draws an edge between two adjacent dots to create boxes. When a box is completed, the player claims it by marking their initial inside. The game continues until no moves are left, and the player with the most boxes wins. Through hands-on play, we’ll explore tactics, strategic planning, and ways to stay ahead of your opponent.
Geometry and Combinatorics of Catalan Numbers: In this activity, we will explore the Catalan numbers, a beautiful sequence that can be described as enumerating many different interesting geometric objets. We will focus on one such geometric perspective, and explore both a recursive way to count the Catalan numbers, as well as possible methods to explicitly count the numbers.
December 7, 2024
Language Puzzles: We will look at some questions from the North American Computational Linguistics Open (NACLO) Competition. No prior knowledge of linguistics or foreign languages is necessary. As we will see, many of these language puzzles are very similar to mathematical brainteasers. In the high school session (4pm), we will see how ideas from number theory and group theory show up in some linguistics olympiad problems.
(Note: WashU is hosting the NACLO 2025 Open Round on Thursday, January 23, 2025, 9am-12pm. Interested students can register here: https://naclo.org/ The top 10% qualify for the Invitational Round, and the top performers are invited to represent the USA in the International Linguistics Olympiad: https://ioling.org/)