The Joy of Counting

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Visualizing Combinatorics: Art of Counting
Combinatorics deals with counting combinations of things. Discover that many such problems are really one problem: how many ways are there to arrange the letters in a word? Use this strategy and the factorial operation to make combinatorics questions a piece of cake.
Math in Your Head!
Episode 1 of Secrets of Mental Math
Dive right into the joys of mental math. First, learn the fundamental strategies of mental arithmetic (including the value of adding from left to right, unlike what you do on paper). Then, discover how a variety of shortcuts hold the keys to rapidly solving basic multiplication problems and finding squares.
Visualizing Pascal's Triangle
Keep playing with the approach from the previous lecture, applying it to algebra problems, counting paths in a grid, and Pascal's triangle. Then explore some of the beautiful patterns in Pascal's triangle, including its connection to the powers of eleven and the binomial theorem.
Visualizing Mathematical Infinities
Ponder a question posed by mathematician Georg Cantor: what makes two sets the same size? Start by matching the infinite counting numbers with other infinite sets, proving they're the same size. Then discover an infinite set that's infinitely larger than the counting numbers. In fact, find an infinite number of…
Mental Addition and Subtraction
Episode 2 of Secrets of Mental Math
Professor Benjamin demonstrates how easily you can mentally add and subtract one-, two-, and three-digit numbers. He also shows you shortcuts using the complement of a number (its distance from 100 or 1000) and demonstrates the uses of mental addition and subtraction for quickly counting calories and making change.
Mathematics and Chess
Chess is more like doing real mathematics than almost any other game out there. You'll get a quick overview of how it's is played; learn how to see connections between math and chess; explore some classic chess puzzles and problems; tap into strategies and tactics for the opening, middle, and…
Mixed Strategies and the Art of Bluffing
What happens when you're playing a game with an intelligent adversary whose goals are opposed to yours? You have a zero-sum game such as penny matching, rock-paper-scissors, and simplified poker. Discover how to use math to bluff, play unpredictably, and win these kinds of games through powerful strategies.
Go Forth and Multiply
Episode 3 of Secrets of Mental Math
Delve into the secrets of easy mental multiplication: Professor Benjamin's favorite mathematical operation. Once you've mastered how to quickly multiply any two-digit or three-digit number by a one-digit number, you've mastered the most fundamental operations of mental multiplication and added a vital tool to your mental math tool kit.
Complex Numbers in Geometry
Episode 35 of Geometry
In lecture 6, you saw how 17th-century mathematician Rene Descartes united geometry and algebra with the invention of the coordinate plane. Now go a step further and explore the power and surprises that come from using the complex number plane. Examine how using complex numbers can help solve several tricky…
Advanced Multiplication
Episode 11 of Secrets of Mental Math
Professor Benjamin shows you how to do enormous multiplication problems in your head, such as squaring three-digit and four-digit numbers; cubing two-digit numbers, and multiplying two-digit and three-digit numbers. While you may not frequently encounter these large problems, knowing how to mentally solve them cements your knowledge of basic mental…
Expert Backgammon
Mathematically trained players also have a decisive edge in backgammon, which trains you to make decisions in highly uncertain conditions. Professor Benjamin explains the rules of the game, the basic strategies for winning, the best ways to play your opening rolls, and how math constantly enters the picture--from figuring out…
The Power of Mathematical Visualization
World-renowned math educator Dr. James Tanton shows you how to think visually in mathematics, solving problems in arithmetic, algebra, geometry, probability, and other fields with the help of imaginative graphics that he designed. Also featured are his fun do-it-yourself projects using poker chips, marbles, paper, and other props, designed to…