Introduction to Combinatorial Game Theory

Academic unit or major

Undergraduate major in Computer Science

Instructor(s)

Francois Pierre Andre Bonnet

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Intensive

Class

-

Course Code

CSC.T255

Number of credits

200

Course offered

2024

Offered quarter

2Q

Syllabus updated

Mar 17, 2025

Language

English

Syllabus

Course overview and goals

This course gives an introduction to Combinatorial Game Theory (CGT). CGT studies two-player deterministic games with perfect-information (e.g. Nim, Domineering, Grundy's game, Wythoff's game, Hex, Go, ...). In combinatorial games, the loser is typically the player who is left without legal moves. The main goal of CGT is to determine the winner of a game, while assuming perfect play of both players. The lecture introduces the fundamental notions of game outcomes and values, and explain how to compute them; first for simple games and then for sums of games. In addition to games, the closely related notion of Surreal Numbers will be presented.

Remarks: This course is not a machine learning course and does not include any programing exercises. Therefore, topics such as implementing a good AI to play games (e.g. alphaZero) are out of scope.

Course description and aims

Students will:
1) Discover the mathematical beauty of Combinatorial Game Theory,
2) Understand the notions of outcome, values, and sum of games,
3) Be able to study and solve such combinatorial games.

More generally, students will improve their ability to study complex problems.

Keywords

Combinatorial Games, Surreal Numbers, Sprague-Grundy Theorem, Subtraction games, Nim(bers), Recreational Mathematics, CGSuite.

Competencies

• Specialist skills
• Intercultural skills
• Communication skills
• Critical thinking skills
• Practical and/or problem-solving skills

Class flow

Typical classes will alternate between slide-based presentations, interactive discussions (between students and/or with teacher), class exercises. Active contribution to class discussions will be required.

Course schedule/Objectives

Study advice (preparation and review)

Textbook(s)

- Lessons in Play: An Introduction to Combinatorial Game Theory, Second Edition, by Michael H. Albert, Richard J. Nowakowski, and David Wolfe
- 組合せゲーム理論入門 －勝利の方程式－, by Michael H. Albert, Richard J. Nowakowski, and David Wolfe, translated by 川辺 治之訳

Remarks:
- Buying the textbook is not required.
- The Japanese book is the translation of the first edition of the English book.

Reference books, course materials, etc.

In addition to the textbook, the following books may be used during the course:
- Winning Ways for Your Mathematical Plays (Volumes 1-4), by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy
- Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness, by Donald E. Knuth
- On numbers and games, by John. H. Conway
- Combinatorial Game Theory, by Aaron N Siegel

Remarks:
- Students are not expected to read these books.
- The last two books are are much more advanced than this course.

Evaluation methods and criteria

Exercises during classes and homework.

Related courses

• None
• None

Prerequisites

- Interest in mathematical games and puzzles (aka recreational mathematics).
- Basic notions of Mathematics; in particular the notion of Proof by Induction.

Other

Remarks:
- The course will be given only face-to-face. There will be no remote option.
- The course schedule is optimistic. There will not be enough time to present all advanced topics.

IMPORTANT:
- This course is an intensive course given for a week in September, probably the third week of September. Exact schedule will be decided later and posted in T2SCHOLA.
- If you register and cannot attend anymore (e.g. due to conflicting internship), please contact the teacher before the beginning of the course.
- If the registration deadline has passed but you still want to register, please contact the teacher before the beginning of the course.