The Art of Counting: An Introduction to Combinatorics

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This course is under review for 2021. Course registration will open to accepted students once courses are confirmed.

Course Description

Have you ever wondered how many crayons you need to color a map of the US so that no state is the same color as its neighbor? Or how many people in your high school are guaranteed to have the same class schedule? Though these questions seem elementary, only those who wield the power of combinatorics may answer them. Learn how to unpack the most difficult problems without the use of advanced calculus or endless calculations - learn the joy of counting!

In this course, students will learn about methods and questions in combinatorics, working both individually and in groups to delve into combinatorial problems. First, students will build on the fundamentals of permutations and combinations to tackle increasingly ambitious problems. Then, they will learn how to construct a combinatorial proofs using a variety of techniques, such as double counting, bijection, and the inclusion-exclusion principle. They will even come up with their own proofs of the famous five color theorem. Students will add certain significant number sequences to their mathematical toolkits - for instance Catalan numbers, and their many avatars, and explore the secret significance of the numbers in Pascal’s Triangle.

During this course, we will focus on particularly “neat” problems. Students can expect to do some light problem sets outside of class to prepare them for the topic of the next class, and we will focus on interesting applications of that topic during class time. As a final project, students will take an in-depth look at a topic in this field, and present results of interest; then, they will take the class through a (sufficiently complex) proof of one such result.

By the end of the course, students can expect to be familiar with some of the basic methods in the combinatorialist toolkit, becoming pigeonhole principle professionals and binomial coefficient buffs. The course will also refine their ability to construct an airtight proof, preparing them for rigorous math classes in the future. It will be instructive for students considering entering math competitions, and particularly those who plan to take advanced math courses in the future. Additionally, it will introduce them to problems that they may encounter in physics and computer science at the college level.

Prerequisites: Algebra 1 is required; geometry is helpful but not required. The only additional prerequisite is a mathematically curious mind!

Course Information

  • Course Code: CEMA0926
  • Length: 2 weeks

Program Information

STEM for Rising 9th and 10th Graders

Two-week, non-credit residential program focused on STEM subjects and taught on Brown’s campus. For students completing grades 8-9 by June 2021; minimum age of 14 and maximum age of 15 by the start of the program.

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