The Mathematics in Toys and Games
Course facts
- Payment: Free, no certificate
- Credits: Non Credit
- Learning style: Self-paced, start anytime
- Resources: Activity Assignments; Design Assignments; Lecture Notes; Projects with Examples; Written Assignments
MIT OpenCourseWare details
- As taught in: Spring 2010
- Department: Experimental Study Group
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The mathematics behind games and puzzles: impartial games, surreal numbers, the Rubik’s Cube, probability, Game of Life and NP-completeness.
Many popular games, toys and puzzles hide real mathematics behind their strategies. This seminar explores that mathematics through the basics of game theory, probability, group theory and elementary programming, with a focus on open problems in combinatorial game theory. Each week pairs a lecture with discussion of a game or puzzle, among them Connect Four, the Rubik’s Cube, Monopoly, card games, the Game of Life and Dots and Boxes, and students finish with a project of their own.
What You Will Achieve
- Analyze impartial games with combinatorial game theory
- Use group theory to reason about the Rubik’s Cube
- Apply probability to strategy in board and card games
- Design, code or research a mathematical game of your own
Program Overview
Combinatorial Game Theory
The theory behind two-player games.
- Theory of impartial games
- Surreal numbers
- Dynamic programming and impartial games
- Games on graphs
Games and Puzzles
Mathematics applied to familiar games.
- AI topics: Chinese checkers and Go
- Connect Four
- Rubik’s Cube and group theory
- Probability in Monopoly and card games
Complexity and Open Problems
How hard games are to solve.
- Game of Life
- NP-completeness and constraint logic
- Dots and boxes
Course Completion
Job Outlook
These ideas are a playful entry point to mathematics, computer science, game design and algorithm research.
Good for: Students curious about the mathematics behind games who enjoy puzzles and some programming.
Keep in mind: A spring 2010 seminar with weekly course notes, project options and three example student projects; there is no textbook, though two classic books are recommended.
Bottom line: A fun, idea-packed tour of the mathematics of games, with good project prompts to take further.
Frequently Asked Questions
What project options are offered?
A research paper on a game, toy or puzzle, a coding project, inventing your own mathematical game or toy, or exploring an open problem in combinatorial game theory.
Is there a textbook?
No. Students read weekly course notes, and two optional books are suggested: Winning Ways for your Mathematical Plays and On Numbers and Games.
What did students build?
Example projects include Amoeboid, a study of strategy in Risk and a four-dimensional Rubik's Cube.
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