Discrete Mathematics

In this introductory lecture, Professor Benjamin introduces you to the entertaining and accessible field of discrete mathematics. Survey the main topics you'll cover in the upcoming lectures—including combinatorics, number theory, and graph theory—and discover why this subject is off the beaten track of the continuous mathematics you studied in high school.

Combinatorics is the mathematics of counting, which is a more subtle exercise than it may seem, since the question "how many?" has at least four interpretations. Investigate factorials as well as the binomial coefficient, n choose k, which shows the number of ways that k things can be chosen from n objects.

As an overview of combinatorial concepts, explore 12 different interpretations of counting by asking how many ways x pieces of candy can be distributed among b bags. The answers depend on such factors as whether the candies and bags are distinguishable, and how many candies are allowed in each bag.

Devised to calculate the payout in games of chance, Pascal's triangle is filled with beautiful mathematical patterns, all based on the binomial coefficient, n choose k. Professor Benjamin demonstrates some of the triangle's amazing properties.

How many ways can you choose three scoops of ice cream from 31 flavors, assuming that flavors are allowed to be repeated? Using the method of "stars and bars," you find 5,456 possibilities if the order of flavors does not matter. The technique also works for counting endgame positions in backgammon.

Learn how the principle of inclusion-exclusion allows you to solve problems such as these: What is the probability that a five-card poker hand has at least one card in each suit? If homework papers are randomly distributed among students for grading, what are the chances that no student gets his or her own homework back?

Proofs by induction are a fundamental tool in any discrete mathematician's toolkit. This lecture guides you through several inductive proofs and then introduces geometric proof, also known as proof without words, and combinatorial proof. You see how all three techniques can prove properties of Pascal's triangle and Fibonacci numbers.

Investigate some interesting properties of Fibonacci numbers, which are defined using the concept of linear recurrence. In the 13th century, the Italian mathematician Leonardo of Pisa, called Fibonacci, used this sequence to solve a problem of idealized reproduction in rabbits.

Starting the section of the course on number theory, explore some key properties of numbers, beginning with what you know intuitively and working toward surprising properties such as Bezout's theorem. You also prove several important theorems relating to divisibility and prime factorization.

Study the building blocks of integers and how numbers can be created additively or multiplicatively. For example, every integer can be expressed as the sum of distinct powers of 2 in a unique way. Similarly, every integer is the product of a unique set of prime numbers.

Explore fascinating examples of two ideas: the pigeonhole principle, which can be used to prove that a mathematical situation is inevitable, such as that there must be a power of 3 that ends in the digits 001; and the parity principle, which is useful for proving that certain outcomes are impossible.

Introducing the important tool of modular arithmetic, Professor Benjamin uses the example of a clock to show how practically everyone is already adept with mod 12 arithmetic. Among the technique's many applications are the ISBN codes found on books, which use mod 11 for error detection.

Exploring more applications of modular arithmetic, examine the Chinese remainder theorem, used in ancient China as a fast way to count large numbers of troops. Also learn about password protection, the mathematics behind the "perfect shuffle," and the "seed planting" technique for raising big numbers to big powers.

Use modular arithmetic to investigate more properties of prime numbers, leading to a practical way to test if an integer is prime. At the same time, meet two important figures in the history of number theory: Pierre de Fermat and Leonhard Euler.

The idea behind public key cryptography sounds impossible: The key for encoding a secret message is publicized for all to know, yet only the recipient can reverse the procedure. Learn how this approach, widely used over the Internet, relies on Euler's theorem in number theory.

This lecture introduces the last major section of the course, graph theory, covering the basic definitions, notations, and theorems. The first theorem of graph theory is yet another contribution by Euler, and you see how it applies to the popular puzzle of drawing a given shape without lifting the pencil or retracing any edge.

Use matrices to answer the question, How many ways are there to "walk" from one vertex to another in a given graph? This exercise leads to a discussion of random walks on graphs and the technique used by many search engines to rank web pages.

Apply graph theory to social networks, investigating such issues as the handshake theorem, Ramsey's theorem, and the stable marriage theorem, which proves that in any equal collection of eligible men and women, at least one pairing exists for each person so that no extramarital affairs will take place.

Discover some interesting properties of tournaments that arise in sports and other competitions. Represented as a graph, a tournament must contain a Hamiltonian path that visits each vertex once; and at least one "king chicken" competitor who has either beaten every opponent or beaten someone who beat that opponent.

When you call someone on a cell phone, you can think of yourself as a leaf on a giant "tree"—a connected graph with no cycles. Trees have a very simple yet powerful structure that make them useful for organizing all sorts of information.

Professor Benjamin introduces the concept of a planar graph, which is a graph that can be drawn on a sheet of paper in such a way that none of its edges cross. Then, encounter the two simplest nonplanar graphs, at least one of which must be contained within any nonplanar graph.

According to the four-color theorem, any map can be colored in such a way that no adjacent regions are assigned the same color and, at most, four colors suffice. Learn how this problem went unsolved for centuries and has only been proved recently with computer assistance.

Examine more problems in graph theory, including the shortest path problem, the traveling salesman problem, and the Hamiltonian cycle problem. Some problems can be solved efficiently, while others are so hard that no simple solution has yet been found.

In his final lecture, Professor Benjamin reviews areas where combinatorics, number theory, and graph theory overlap. Then he looks ahead at topics that build on the course's solid foundation in discrete mathematics. He closes with a flourish of mathematical magic, including the "four-ace surprise."