Combinatorics is sometimes called "the art of counting without counting." This course will use visualization (e.g., trains made of colored rods!) and a structured set of problems to approach combinatorial relations in discrete mathematics. We will also look at how combinatorics and combinatorial thinking can be used to illuminate ideas from more mainstream school math courses such as algebra, arithmetic, geometry, and probability.
Look at any row of Pascal's triangle. What is the sum of the entries? What is the sum of the squares of the entries? Combinatorial proofs allow you to answer questions like these by "telling a story" - establishing a formula by showing that each side of the formula represents a different way to count the same thing. Starting from scratch, we'll learn in this course how to use combinatorial reasoning to create and establish - in very simple ways - many beautiful results.
This will be a problem-based course. During class time students will learn by working problems together, solving a carefully structured set of numerical, geometrical and algebraic problems that will stretch over the whole quarter. These problems start simply and concretely then gradually go deeper to develop general ideas. The course will be based on one of the renowned Developing Mathematics courses taught at the Park City Mathematics Institute. There will be no textbook to purchase.
The formal math background of this course does not really go beyond high school mathematics. But the course will require the willingness to tackle the more challenging problems as well as the more straightforward ones and to think about general mathematical ideas as they emerge from problems and examples.
Given the nature of the course, regular attendance will be essential. Students will work on problems and discuss them in each class session. Between classes there will be assigned some additional problems to solve and/or write-ups of work. During the quarter, there will also be occasions for formal and informal presentations. There will be no final exam, but there will be a final assignment that will include writing and a presentation. Grades will be based on all this work, including class participation.
This course will be based on one of the renowned Developing Mathematics courses taught at the Park City Mathematics Institute. There will be no textbook to purchase.