What are some of the best books that try to study board games in a formal mathematical way?
Below is a compact reading‑map that most researchers and advanced hobbyists follow when they want to treat board games as mathematics rather than mere recreation. I’ve grouped the titles by the role they usually play in a self‑study program and noted what each one actually does for you.
1. Bedrock texts — where the modern theory began
| Title | Why it matters | Typical level |
|---|---|---|
| On Numbers and Games — John H. Conway (2nd ed., 2001) | Introduces surreal numbers and the algebra that underlies all combinatorial game theory (CGT). A thin but dense book that defines the subject’s vocabulary. (Amazon) | Graduate‑level abstract algebra / set theory |
| Winning Ways for Your Mathematical Plays, Vols. 1 – 4 — Berlekamp, Conway, & Guy (2nd ed. 2018–2024) | The classic encyclopedia: impartial games (Nim‑type), partizan games, misère play, thermographs, hot/cold values, and dozens of concrete board‑game case studies. Readable once you’ve sampled ONAG. (Amazon, Amazon) | Upper‑undergraduate to research |
2. Friendly introductions / textbooks
| Title | What you’ll get | Level |
|---|---|---|
| Lessons in Play, 2nd ed. — Albert, Nowakowski & Wolfe (2019) | A pedagogical walk from Nim to CGT algebra with hundreds of exercises and fully worked examples. Great first book if Winning Ways feels too brisk. (Routledge) | Advanced undergrad |
| Combinatorial Game Theory — Aaron N. Siegel (AMS GSM 146, 2023) | A modern, theorem‑proof treatment that unifies the field’s last 30 years and cleans up notation. Becomes the standard reference once you want full rigor. (Amazon) | Graduate text |
3. Research‑level compilations
- Games of No Chance series (MSRI Publ. Vols. 29, 42, 56, 63, & 90; vol. 5 appeared 2019) — edited collections of cutting‑edge papers on CGT structure, new game values, and solved board‑game endgames. (Cambridge University Press & Assessment)
4. Monographs that dissect specific board games
| Game | Book | Highlight |
|---|---|---|
| Go | Mathematical Go: Chilling Gets the Last Point — Berlekamp & Wolfe (aka Mathematical Go Endgames) | Uses CGT thermography to solve real professional endgames and introduces “chilling” as a cooling technique. (Amazon, Amazon) |
| Dots‑and‑Boxes | The Dots and Boxes Game: Sophisticated Child’s Play — Berlekamp (2000) | Turns a playground pastime into a solved impartial game; shows how to read the nim‑value of any position. (Amazon) |
| (Many other titles exist for Hex, Domineering, Amazons, etc., but the two above are the gold‑standards for single‑game depth.) |
5. Algorithmic & complexity‑theoretic perspectives
| Resource | What it covers | Use‑case |
|---|---|---|
| Games, Puzzles, and Computation — Hearn & Demaine (2009) | Reduces dozens of real board games (Rush Hour, Reversi, generalized Chess/Go) to complexity‑class hard problems via Constraint Logic. (Amazon) | When you care about NP/PSPACE/EXPTIME results |
| “Algorithmic Combinatorial Game Theory” survey — Demaine & Hearn (2001; updated in Games of No Chance 3) | A readable overview of algorithmic techniques and open problems. (erikdemaine.org) | Quick orientation before diving into proofs |
| Algorithmic Game Theory (SAGT 2024 Proceedings) | Latest papers on computational aspects, including hardness proofs for new tabletop titles. (Amazon) | Keeping current with 2020‑s research |
How to read them in practice
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Start light: skim a chapter of Lessons in Play to see the main definitions and the Sprague‑Grundy theorem at work.
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Build foundations: dip into On Numbers and Games for the surreal‑number construction, then sample the early chapters of Winning Ways to see those ideas applied.
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Choose a specialty:
- If you enjoy solving concrete board games, grab the relevant monograph (Go, Dots‑and‑Boxes, etc.).
- If you’re pulled toward algorithms or CS theory, pivot to Hearn & Demaine.
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Circle back to Siegel’s textbook once you want complete proofs of structural results such as the Atomic‑Weight Lemma or canonical forms.
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Stay current by browsing the Games of No Chance volumes or SAGT proceedings whenever a new one appears (roughly every 3‑4 years).
Taken together, these works will carry you from the first Nim pile all the way to the current research frontier. Happy reading—and happy playing!
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