A central idea behind the mathematical field of combinatorics is finding why seemingly unrelated mathematical expressions are equal, without needing to dive into messy algebra. One particularly elegant and intuitive way to do so is double counting: we tell a story, look at it from two different perspectives, allowing us to represent the same thing in two different ways mathematically. We'll puzzle out how to interpret sums, products, and more as tangible stories and images: learn what taking a walk in Manhattan has to do with Pascal's Triangle and how assembling a team of cats and dogs can simplify a scary-seeming sum. On a personal note, taking a class like this when I was in middle school helped me realize the creativity hidden in math and contributed to combinatorics being one of my favorite subjects. I hope to present a similar experience to others in an accessible manner.

In this class, we’ll dive into mathematical trees—simple graph-theoretic structures that appear everywhere. We’ll start from scratch with fundamental properties, then explore different perspectives that unlock powerful problem-solving tricks. Along the way, we’ll see how and why trees arise so naturally across mathematics.

Do you love games and puzzles? Nim is a simple, ancient game with a surprising mathematical twist! In this interactive class, we’ll dive into the secrets of winning Nim every time. First, we’ll learn the basic rules and play a few rounds. Then, we’ll discuss some foundational strategies -- after defining what a "strategy" even means. Finally, we'll explore a beautiful connection between Nim and algebra, which will give us tools to solve a wider variety of more complicated games. Whether you're a math enthusiast, a strategic gamer, or just curious, this class will give you new tools to outsmart your friends and see games in a whole new light. Come play, think, and conquer Nim!