During the Spring 2025 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 |
| Jan 25, 2025 | 3pm | Brandon Sweeting | Polyomino Puzzles and Tic-Tac-Toe |
| Jan 25, 2025 | 4pm | Brandon Sweeting | Advanced Polyomino Challenges |
| Feb 1, 2025 | 3pm | Brandon Sweeting and Sydney Mayer | Bridges, Paths, and Puzzles – An Introduction to Graph Theory |
| Feb 1, 2025 | 4pm | Brandon Sweeting and Sydney Mayer | Graph Theory Challenges – Traversability, Trees, and Networks |
| Feb 8, 2025 | 3pm | Brandon Sweeting and Sydney Mayer | Exploring the Game of Brussel Sprouts |
| Feb 8, 2025 | 4pm | Brandon Sweeting and Sydney Mayer | Combinatorial Analysis of Brussel Sprouts |
| Feb 15, 2025 | 3pm | Brandon Sweeting and Sydney Mayer | Strategic Partitioning: The Math of Fair Division |
| Feb 15, 2025 | 4pm | Brandon Sweeting and Sydney Mayer | Algorithmic Approaches to Strategic Partitioning |
| Feb 22, 2025 | 3pm | Brandon Sweeting | Save the Sheep! |
| Feb 22, 2025 | 4pm | Silas Johnson and Michael Chen | ARML Power Contest and ARML Practice |
| Mar 1, 2025 | 3pm | Brandon Sweeting and Sydney Mayer | Exploring Fractals: Patterns in Nature and Maths |
| Mar 1, 2025 | 4pm | Brandon Sweeting and Sydney Mayer | The Mathematics of Fractals: Dimension and Computation |
| Mar 8, 2025 | 3pm | Brandon Sweeting | The Golden Ratio & Seed Distributions |
| Mar 8, 2025 | 4pm | Brandon Sweeting | Continued Fractions & the Most Irrational Number |
| Mar 15, 2025 | 3pm | Martha Precup | Eulerian Embroidery |
| Mar 15, 2025 | 4pm | No session — please consider attending the 3pm session this week! |
Save the date: Our colleague, Professor Martha Precup, is organizing the Sonia Kovalevsky Math Day at Washington Univesity in St. Louis for Middle School Girls on Saturday, April 12, 2025. Small groups of middle school girls chaperoned by parents are very welcome to attend. For more information, see the SK Day flyer or the SK Day website.
Descriptions
Jan 25, 2025
Polyomino Puzzles and Tic-Tac-Toe
Students will be introduced to polyominoes, fascinating shapes made up of connected squares. After exploring these shapes with hands-on exercises, such as filling a square grid or forming specific patterns, we’ll transition to a fun twist on tic-tac-toe. In this version, players will aim to mark grids in a way that forms a specific polyomino configuration.
Advanced Polyomino Challenges
High schoolers will take the polyomino fun further. After mastering the basics, they’ll tackle more complicated shapes and configurations, exploring strategies to solve the game under various conditions. This session encourages both strategic thinking and creative problem-solving and introduces students to an active area of research.
Feb 1, 2025
Bridges, Paths, and Puzzles – An Introduction to Graph Theory
Students will explore the classic Seven Bridges of Königsberg problem and discover how it led to the birth of graph theory! Through interactive exercises, they will learn about vertices, edges, and paths while solving fun puzzles about bridges, city maps, and networks. We’ll also play a game where students try to draw figures without lifting their pencils or retracing edges—can you figure out which ones are possible?
Graph Theory Challenges – Traversability, Trees, and Networks
Students will dive deeper into graph theory, tackling advanced problems involving Eulerian paths, bipartite graphs, and trees. We’ll explore when a path exists between certain points, prove key properties about graphs, and examine puzzles like handshake problems, friendship graphs, and shortest paths. This session encourages strategic thinking and introduces students to fundamental ideas in graph theory.
Feb 8, 2025
Exploring the Game of Brussel Sprouts
Students will be introduced to the combinatorial game Brussel Sprouts, invented by John Conway. They’ll begin by playing with two and three starting crosses, exploring patterns in game length and outcomes. Through hands-on play and guided discussion, students will develop conjectures about the structure of the game and its winning strategies.
Combinatorial Analysis of Brussel Sprouts
Building on the middle school session, high school students will engage in a deeper mathematical exploration of the game Brussel Sprouts. They will analyze the number of possible moves, derive a formula for game length, and investigate strategic implications using concepts from combinatorial game theory. This session will encourage logical reasoning and introduce students to fundamental ideas in mathematical game theory.
Feb 15, 2025
Strategic Partitioning: The Math of Fair Division
How do we divide things fairly? Whether it’s cutting a cake, splitting teams, or designing fair election districts, partitioning plays a crucial role in decision-making. In this session, students will engage in interactive challenges exploring strategic partitioning and fair division. Through hands-on games, they will examine different strategies for dividing space and explore how small changes in grouping can significantly impact outcomes.
Algorithmic Approaches to Strategic Partitioning
Building on the middle school session, high school students will dive into the mathematical and computational aspects of strategic partitioning. They will explore algorithms used to create fair (or unfair) divisions, analyze real-world challenges like gerrymandering, and experiment with optimization techniques. Students will engage in hands-on exercises that connect mathematical reasoning to decision-making and problem-solving in various contexts.
Feb 22, 2025
Save the Sheep!
Can you outsmart the wolves and save the sheep? In this interactive session, students will tackle strategic puzzles using a 4×4 or 5×5 grid. With simple rules—wolves can eat sheep in the same row, column, or diagonal—the challenge is to arrange the sheep so they stay safe. Through hands-on problem-solving, students will explore logical reasoning, spatial strategy, and pattern recognition. How many sheep can you save?
ARML Power Contest and ARML practice
This week’s high school session will be dedicated to the ARML Power Contest and ARML practice. The session will begin with the team-based Power Contest, followed by practice for the regular ARML competition. Students who need to leave at 5:00 are still welcome to participate in the Power Contest. Sign-up is not mandatory, though interested participants for the Power Contest are encouraged to register here: https://forms.gle/iYVypKUqn2Anu2wc8.
ARML (American Regions Mathematics League) is a national contest for regional teams, with a Missouri team traveling to Iowa City in late May or early June. Attending this practice doesn’t commit students to competing, and all are welcome to join and see what ARML is like. The Power Contest is a low-key, collaborative event open to everyone, regardless of prior contest experience. You can find more information about ARML here, the Power Contest here, and the Missouri ARML Team here.
Mar 1, 2025
Exploring Fractals: Patterns in Nature and Maths
What do ferns, snowflakes, and coastlines have in common? They all exhibit fractal patterns! In this interactive session, students will dive into the world of fractals, discovering how simple rules can create infinitely complex structures. Through hands-on activities like building the Sierpiński Triangle and exploring the Koch Snowflake, students will develop an intuition for self-similarity and recursion. We’ll also introduce a basic idea of fractal dimension—can something be “between” one and two dimensions? Come explore the beauty of infinite complexity!
The Mathematics of Fractals: Dimension and Computation
Fractals aren’t just visually fascinating—they have deep mathematical properties! In this session, students will take a deeper dive into fractal dimension, learning how to measure complexity using mathematical techniques like box-counting. We’ll also explore the power of recursion and algorithmic fractal generation, using Python to draw famous fractals like the Koch Snowflake and Sierpiński Triangle. No prior coding experience is required.
Mar 8, 2025
The Golden Ratio & Seed Distributions
Why do flowers and pinecones arrange their seeds in spirals? In this interactive session, students will explore how different turn angles affect seed distribution in plants and why the golden ratio plays a key role in creating the most efficient arrangements. We’ll discuss the Fibonacci sequence and how its ratios approach the golden ratio, then use a computer simulation to visualize how varying turn angles impact spacing. Through hands-on exploration, we’ll discover how nature optimizes seed placement using simple mathematical principles!
Continued Fractions & the Most Irrational Number
The turn angle that best distributes seeds is closely tied to one of the most famous numbers in mathematics—the golden ratio. But what makes this number so special? In this session, we will explore continued fractions, which provide insight into how well numbers can be approximated by rational fractions. We’ll see why the golden ratio is considered the “most irrational” number and how this property influences optimal spacing patterns in nature. Through mathematical reasoning and examples, we’ll uncover the deep connections between continued fractions, Fibonacci numbers, and number theory.
Mar 15, 2025
Eulerian Embroidery
Blackwork embroidery is a stitching technique associated with Tudor England. The goal is to embroider a pattern that is reversible, meaning it looks nice from both sides of the fabric. To attain such a product, you want to make sure no stitches are duplicated with exactly one stitch on each side of your fabric—no doubling up! We will explore which patterns can be embroidered this way with a single (possibly very long) piece of thread. Come try your hand at some mathematical arts and crafts!
This project is based on a paper The Graph Theory of Blackwork Embroidery by Joshua Holden and on the Maths & Crafts blog post “Eulerian Embroidery” by Jane Butterfield.
Note: There is no high school session this week. (But as usual, all students are welcome to attend the middle school session.)