TOKYO INSTITUTE OF TECHNOLOGY

Course description and aims

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.

Student learning outcomes

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 that will be developed

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/Required learning

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.

Assessment criteria and methods

Exercises during classes and homework.
The assessment method will be adapted if the course cannot be held face-to-face.

Related courses

• None
• None

Prerequisites (i.e., required knowledge, skills, courses, etc.)

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

Other

Remarks:
- If allowed at that time, lecture will be given face-to-face and onsite attendance will be strongly recommended.
- The course schedule is probably optimistic. There may not be enough time to present all topics.