The Mathematics of Games and Puzzles: From Cards to Sudoku

Course No. 1401
Professor Arthur T. Benjamin, Ph.D.
Harvey Mudd College
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Course No. 1401
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What Will You Learn?

  • Learn simple, card-counting techniques to improve your chances at winning Blackjack.
  • Explore the mathematical backbone of Texas Hold'em to make proper wagering decisions.
  • Discover a simple, eight-step tool using math that will help you master a Rubik's Cube.
  • Draw connections between math strategies and winning game strategies for chess.

Course Overview

For thousands of years, games and puzzles have been an enjoyable and rewarding aspect of human civilization. They tease our brains. They challenge our memories. They strengthen our competitive skills. And whether it’s chess, poker, or Sudoku, most games have this in common: Everything you need to win is rooted in mathematics.

Using nothing more than a simple grasp of math, you can discover optimal ways to win games and solve puzzles with the speed and accuracy of professional players—many of whom attribute their professional gaming success to mathematical strategies. By using math as a unique lens through which to explore some of the world’s most popular games, you’ll

  • improve the ways you win games and solve puzzles;
  • better understand exciting concepts in everything from algebra to probability to game theory;
  • make better decisions and take calculated risks in personal investing and other real-world situations;
  • keep your mind active and sharp at any age; and, perhaps best of all,
  • discover whole new levels of enjoyment with games you only thought you knew how to play.

Join award-winning Professor Arthur T. Benjamin—one of The Great Courses’ most popular instructors and a veritable mathemagician at making math enjoyable for everyone—for an in-depth look at new, better, more math-oriented ways to play and win games in The Mathematics of Games and Puzzles: From Cards to Sudoku. In just twelve 45-minute lessons, you’ll learn a rewarding set of skills you can apply to countless games and puzzles, whether you’re playing them in your living room or in a crowded casino. You’ll also learn that behind even the simplest games lies proof of just how beautiful and far-reaching mathematics is in your everyday life.

A Math-Centric Way to Play Games

With the same characteristic excitement that has won him the acclaim of countless of our lifelong learners, Professor Benjamin covers a range of games, puzzles, and brainteasers that we’ve all played, tried to play, or wanted to play at some point.

  • Zero-sum games: Many of the most popular games are what mathematicians and strategists call zero-sum games, or contests against an intelligent adversary whose aims oppose yours. Alongside Professor Benjamin, you’ll uncover math-centric, proven ways to succeed at backgammon, poker, and even a simple game of rock-paper-scissors.
  • Games of chance: Many of us are intimidated by casino games. And if we’re not, we’re always looking for ways to stay ahead of our opponents. In addition to revealing the surprising math behind roulette, craps, video poker, blackjack, and more, Professor Benjamin offers invaluable tips on improving the way you bluff, wager, and count cards.
  • Classic puzzles and brainteasers: Whether it’s Sudoku, peg solitaire, or even a Rubik’s Cube, puzzles can be as frustrating as they are entertaining. What math tools can help you fill a Sudoku grid without ever guessing? How can you solve a Rubik’s Cube as if it were second nature? Find out all this and more in several lectures devoted to mastering puzzling challenges.

The Mathematics of Games and Puzzles takes an elementary mathematical approach to understanding how each game and puzzle is played and won. While the mathematics in this course is detailed, Professor Benjamin always makes sure to break down the complexity into simple language that anyone eager to learn can grasp.

Discover the Tips and Tricks of Champions

These lessons are also packed with dozens of specific tips, tricks, strategies, and methodologies for getting the best of your opponents, improving your technique, increasing your chances of winning, and much more. You’ll get an invaluable toolkit to take with you every time you’re at a poker table, in front of a chess board, or reading the morning newspaper.

  • When should you hit or stand in blackjack? If the dealer’s up-card is 7 or higher, then hit until your total is 17 or higher. If the up-card is 4, 5, or 6, then take no chances—not even when your total is 12.
  • In the poker game of Texas Hold’em, being dealt an ace-king is a strong hand, but it still loses to randomly dealt hands about one-third of the time. It even loses to a pair of deuces 53% of the time.
  • Frustrated with a Sudoku grid? Look for hidden singles (numbers that can only find one place to go in a row, column, or box). When you find a number with only two possible squares in a box, lightly pencil that number in both places and use it to try to solve other trouble spots.

Professor Benjamin’s lessons are filled with strategies like these; some for an entire game, others for a particular scenario, many of them straight from champion players. In putting this course together, he consulted with experts (including some world champions) at backgammon, poker, chess, Rubik’s Cube, and Sudoku, along with professionals in the casino gaming industry.

Improve Your Odds of Becoming a Winner

With this course, you’ll find yourself in the hands of a master instructor. Professor Benjamin is a past winner of the American Backgammon Tour and a firm believer that learning math should be just as fun and enjoyable as playing games. You’ll quickly see why Professor Benjamin has won three awards from the Mathematical Association of America, and why Reader’s Digest named him “America’s Best Math Whiz.”

And with the aid of helpful exercises and problems, detailed explanations of mathematical reasoning, visual breakdowns and animations of specific techniques, and more, you’ll find yourself eager to play along with Professor Benjamin and to return to these lectures any time you get the urge to play a game.

Insightful and entertaining, The Mathematics of Games and Puzzles is a fun-filled opportunity to engage with math, strengthen your mental skills, and increase the chances that the next time someone asks you if you’re up for a game, you’ll come out a winner.

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12 lectures
 |  Average 45 minutes each
  • 1
    Let the Games Begin!
    Explore some general strategies for successfully solving simple games and puzzles. As you hone your skills at games and puzzles, including 20 Questions, Mastermind, Ghost, The Tower of Hanoi, variations of Tic Tac Toe, and Cram (a cross between checkers and dominoes), you’ll start seeing how mathematical ideas and concepts—including the strategy of working backward and the exploitation of symmetry—are essential to increasing your chances of winning. x
  • 2
    Games of Chance and Winning Wagers
    Turn now to games that rely heavily on chance and betting, such as roulette and craps, and learn how to evaluate the fairness (or unfairness) of games using the ever-important concept of expected value. Once you know if a game is to your advantage or not, you can use the Kelly criterion or the Gambler’s Ruin formula to help you decide your best wagering strategies. x
  • 3
    Optimal Blackjack and Simple Card Counting
    Depending on how you play it, blackjack can be the best or the worst game in a casino. Here, Professor Benjamin shows you how to get the best of blackjack before it gets the best of you. By taking you through scenarios of increasing difficulty to strengthen your skills, and by demonstrating the secrets of simple card counting, he reveals optimal strategies for knowing when to stand, when to hit, when to double, and when to split. x
  • 4
    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 including the equilibrium strategy (which guarantees the highest expected payoff without allowing your adversary to exploit your approach). x
  • 5
    Practical Poker Probabilities
    Investigate the mathematical backbone of Texas Hold’em, one of the most popular traditional poker games. What is the probability your hand will improve before and after more cards are revealed? How do you calculate your “pot odds” to help you determine your proper wagering decisions? As you enhance your mathematical poker-playing skills, you’ll find the answers to these and other questions about this thrilling game. You will also learn simple and effective strategies for playing video poker. x
  • 6
    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 safest way to move your checkers to the all-important doubling cube. x
  • 7
    Games You Can’t Lose and Sneaky Puzzles
    The world of games is filled with scams. The trick: knowing the strategies behind how these hustles work so you can avoid being exploited (and learn some engaging insights into math at the same time). Explore the fundamentals of nontransitive properties through penny ante, Bingo, and games involving dice and cards. Then, get a lively introduction to some simple puzzles involving cups, coins, and toothpicks—and their surprisingly sneaky answers. x
  • 8
    Solving “Impossible” Puzzles
    Try your hand at some classic puzzles that have been driving people crazy for centuries involving sliding blocks, jumping pegs, and blinking lights—each of which deals heavily with odd or even numbers. Once you’ve learned some handy mathematical concepts and tools for solving these puzzles—including mod 2 arithmetic, vector equations, and mnemonic devices—these fun and exciting games won’t seem so “impossible” anymore. x
  • 9
    Mastering Rubik’s Cube
    It’s one of the most famous puzzles ever invented. But Professor Benjamin has an easy-to-learn, eight-step method for solving this mind-bending puzzle quickly and accurately—every time. After examining the mathematics behind the cube and pondering how many different cube positions are possible, you’ll follow him step-by-step through an algorithm (taught to Professor Benjamin by a world-champion cube solver) that, with practice, will have you solving any Rubik’s Cube in less than three minutes. x
  • 10
    Solving Sudoku
    What’s the key to solving Sudoku problems when you’re at your wits’ end? Training your mind to look for patterns and to use careful logic, just like mathematicians. This lecture is packed with helpful techniques and strategies for overcoming even the most difficult Sudoku grids. Among those you’ll learn about: crosshatching, miniboxes, naked pairs and triples, and the unique rectangle rule. These tricks can even be used to solve variations on traditional Sudoku puzzles. x
  • 11
    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 the game is played; learn how to draw connections between math and chess; explore some classic chess puzzles and problems; tap into the power of strategies and tactics for the opening, middle, and end game; and get some insider tips that are sure to improve your game the next time you sit down in front of a chess set. x
  • 12
    Winning Ways—It’s Your Move!
    Finish this engaging series with a look back at the three categories that most games fall into (games where the last player to move wins, games where the goal is to be the first to create a structure, and games where the player who accumulates the most stuff wins). Cram, NIM, Chomp, Connect Four—play them all. Also, take a look at how computers play games—and how they’ve helped us become better players. x

Lecture Titles

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Video DVD
Video Download Includes:
  • Download 12 video lectures to your computer or mobile app
  • Downloadable PDF of the course guidebook
  • FREE video streaming of the course from our website and mobile apps
Video DVD
DVD Includes:
  • 12 lectures on 3 DVDs
  • 136-page printed course guidebook
  • Downloadable PDF of the course guidebook
  • FREE video streaming of the course from our website and mobile apps

What Does The Course Guidebook Include?

Video DVD
Course Guidebook Details:
  • 136-page course synopsis
  • Charts & diagrams
  • Suggested readings
  • Problems & solutions

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Your professor

Arthur T. Benjamin

About Your Professor

Arthur T. Benjamin, Ph.D.
Harvey Mudd College
Dr. Arthur T. Benjamin is Professor of Mathematics at Harvey Mudd College. He earned a Ph.D. in Mathematical Sciences from Johns Hopkins University in 1989. Professor Benjamin's teaching has been honored repeatedly by the Mathematical Association of America (MAA). In 2000, he received the MAA Deborah and Franklin Tepper Haimo National Award for Distinguished College or University Teaching of Mathematics. The MAA also named...
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Reviews

The Mathematics of Games and Puzzles: From Cards to Sudoku is rated 4.5 out of 5 by 53.
Rated 5 out of 5 by from Best. Course. Ever. I've always believed that analyzing games not only teaches you strategies that are useful for decision-making, but also teaches you to focus only on things within your control, which prepares you for the ups and downs of life. Each game has different lessons to teach you and it's clear that professor Benjamin has mastered most of them and has even found some general approaches that apply to multiple games. He's passionate about the subject and it's infectious.
Date published: 2013-08-22
Rated 4 out of 5 by from A beginne'rs course This is pretty much a beginner's course. I loved professor benjamin's secrets of mental math course and hoped this course would be just as amazing. however, this course does not deal much in strategy games, only covering how to play the game and good opening moves. almost half of this course deals with games in the casino and repeats over and over again how you lose in every game. I hope the great courses will make a follow up course or a second edition to this for more advanced strategy playing. I give this course 4 stars because the professor is so great even though the content isn't. Advice to customers: if you are a beginner, and want to learn more about strategy games, you may want to buy this course although you can probably learn about this stuff online. If you like playing at the casino, the professor will teach you how you lose on average in every single game, so buy this course. However, if you like strategy games on high levels like me, you should wait until a 2nd edition or follow up course comes through.
Date published: 2013-08-02
Rated 2 out of 5 by from subpar for Dr. Benjamin I love Dr. Benjamin, but this spent way too much time on the basics and not enough on higher math in the games. If you are a beginning gamer, then by all means, buy this. If on the other hand you're experienced and looking for deeper insights - not so much
Date published: 2013-07-11
Rated 5 out of 5 by from HOW MATHEMATICS SHOULD BE TAUGHT Dr. Benjamin's latest addition to The Great Courses is another absolute winner. He briefly explains the mathematical theory behind many games and puzzles we love to play. If you're interested, he presents just enough statistics and probability to give you the background but presents the course in such a manner that it can be ignored if you're only interested in his conclusions. His lectures on the correct strategies to use when gambling at such games as roulette, craps, blackjack, poker, video poker, and the art of bluffing are invaluable. He also has lessons on chess, backgammon, Sudoku, and various puzzles that will greatly improve your capabilities and enjoyment when engaging in these games. Professor Benjamin's techniques of presentation are as enjoyable to watch as the content of this course. His sincere enthusiasm, near-perfect presentation, preparation, sense of humor, and obvious love of mathematics make this course especially fun to watch. This course is highly recommended not only for anyone who is interested in games and puzzles but also for everyone wanting to improve their decision-making skills where risk and chance are involved.
Date published: 2013-07-09
Rated 5 out of 5 by from Entertaining, Educational and Well-Organized Professor Benjamin is the math teacher you wish you had in school. He's exciting, interesting, very well-prepared (so much so that I couldn't find a single "umm"), but most of all, he is enthusiastic. His enthusiasm spreads to featured topics, which when on paper sound very dull: probability, expected value, etc. If you like any kind of game, from Sudoku to Chess that involves chance and skill, then these lectures will cover it. If you want to go to Vegas, his first two lectures will save you a lot of money indeed, and if you bet a lot of money the savings alone will pay for the course. Presentation: Fantastic. He smiles (something many presenters do not), he is clear, he does not waste one second, nor add unnecessary detail. He has good voice variation, volume, enunciation, gestures, facial expressions, etc. Content: Very good if you're interested in the material. You will also learn some basic statistics. I studied finance and computer science as an undergrad, so I can attest that concepts like expected value are very useful for everyone (more than most people realize). Overall: Recommended, even if you're not a math fan because we all are involved in risk and chance in our daily lives.
Date published: 2013-07-03
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