Three games have shaped how computers learn to play. Chess pushed search to its limits and then handed evaluation to neural networks. Checkers became the largest game ever solved outright, when Jonathan Schaeffer's Chinook team proved in 2007 that perfect play ends in a draw. Backgammon added dice, which forced engines to reason about probability instead of certainty, and gave us TD-Gammon, one of the first programs to teach itself through reinforcement learning.
Over the past months we have written about all three games for readers in many languages. This page gathers every one of those articles in one place, sorted by game and then by language, so you can follow one idea, such as bitboards or bear-off probabilities, across languages, or read the whole series in your own.
How the three games differ
Chess has around 1043 legal positions, far too many to solve, so engines combine fast move generation with deep alpha-beta search and a learned evaluation. Modern NNUE networks update incrementally as pieces move, which is why they run fast enough on an ordinary CPU.
Checkers has about 5×1020 positions. That is small enough that a mix of forward search and backward analysis from the endgame could close the gap. The whole board fits in a handful of 32-bit integers, and the mandatory-capture rule prunes huge parts of the tree.
Backgammon is a game of chance with full information. Each turn has 21 distinct dice outcomes, so search must average over them. Late in the game, when the two sides no longer interact, the race reduces to an absorbing Markov chain that can be solved exactly. The doubling cube then turns every position into a pricing problem much like a financial option.
A note on fair dice. Online backgammon players have long suspected that dice are rigged. The cryptographic answer is a commit-reveal scheme: the server commits to a secret seed before the game by publishing its hash, the player adds their own randomness, and once the game ends the seed is revealed so anyone can recompute every roll. Boardgammon publishes how its implementation works on its provably fair backgammon dice page, and several articles below walk through the HMAC-SHA256 construction step by step.
Chess engines and chess learning 60
Bitboards, magic move generation, alpha-beta search, quiescence search, NNUE evaluation and server-authoritative online play.
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Topological State Spaces, Magic Bitboards, and Quiescence Verification in Modern Chess Engines 6ac17036e66f9.site123.me The Shannon Number and Branching Factor Bounds in Distributed Chess Architectures boardgamestheory.mataroa.blog Topological State Spaces, Magic Bitboards, and Quiescence Verification in Modern Chess Engines boardgamestheory.ucoz.net Topological State Spaces, Magic Bitboards, and Quiescence Verification in Modern Chess Engines boardgamestheory.weebly.com Zobrist Hashing, Transposition Bounds, and Monotonic State Synchronization in Distributed Chess Engines boardgamestheory.wordpress.com Topological State Spaces, Magic Bitboards, and Quiescence Verification in Chess Engines boardgamestheory.zohosites.eu Schaakengine Architectuur: 64-Bit Magic Bitboards, Alfa-Bèta Snoeien en Quiescence Search bordspellen-geschiedenis.blogspot.com Check out Boardgammon Chess online cosmos47.com High-Performance Turn-Based Game AI: Integrating Bitboard Representation with Alpha-Beta Pruning in Rust and Go dev.to Arkitektur for Moderne Sjakkmotorer: 64-bits Magic Bitboards, Alfa-Beta-Beskjæring og Quiescence Search gammontheory.vivaldi.net Algorithmic Representation of Classical Board Games: Bitboards, State Invariants, and Rollout Verification gist.github.com Combinatorial Complexity, State Reconciliation, and Move Validation in Distributed Chess Engines gitlab.com Magic Bitboards, Ray Attacks, and Quiescence Search in High-Performance Chess Engines gitlab.com Endgame Tablebase Compression, Retrograde Synthesis, and Syzygy Probing Dynamics in Chess Engines gitlab.com Combinatorial Complexity, Magic Bitboards, and Quiescence Search in High-Performance Chess Engines justpaste.it Combinatorial Complexity, Monotonic Clocks, and Distributed State Reconciliation in Modern Chess Engines poyeshsoftware.github.io Magic Bitboards, Transposition Hashing, and Quiescence Verification in Modern Chess Engines poyeshsoftware.github.io Endgame Tablebase Compression, Retrograde Synthesis, and Syzygy Probing Dynamics in Chess Engines poyeshsoftware.github.io Clock Synchronization, Monotonic Timers, and Lag Compensation in Competitive Digital Chess Protocols telegra.ph Null-Move Pruning and Zugzwang Verification in High-Depth Chess Search telegra.ph Monotonic Clocks, Network Slew, and Lag Compensation in Distributed Chess Engines workoutingym.hashnode.dev Combinatorial Complexity and Low-Latency State Synchronization in Modern Chess Architectures write.as Asynchronous Time Stepping and Delta Compression in Distributed Chess Protocols write.as Piece-Square Tables and Tapered Evaluation Functions in Modern Chess Engines write.as Boardgammon Chess — Real-Time Multiplayer, 64-Bit Bitboard Engine & Magic Bitboards www.diigo.com Monotonic Clock Synchronization, Network Slew, and Lag Compensation in Distributed Multiplayer Chess Architectures www.linkedin.com How do you approach bitboard chess engine design? www.quora.com
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Checkers and draughts 81
32-bit board representations, mandatory-capture move generation, endgame databases and the proof that 8x8 checkers is a draw.
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Combinatorial Complexity and Algorithmic State Spaces in Draughts Solvers: From SWAR Bitboards to Retrograde Tablebase Synthesis 6ac17036e66f9.site123.me Bitboard Representations Move Generation and Alpha Beta Pruning in Draughts Solvers board-games-theory.gitbook.io Why Perfect Play in 8x8 Checkers Guarantees a Draw: Analyzing the Chinook Solution boardgamestheory.bearblog.dev Algorithmic Architecture of Solved Checkers Engines: 32-Bit Bitboards, Mandatory Capture Branching, and Retrograde Analysis boardgamestheory.codeberg.page The Asymmetry of King Mobility in English Checkers vs International Draughts boardgamestheory.mataroa.blog Retrograde Value Iteration, Proof-Number Search, and Bitboard Mechanics in Solved Draughts Systems boardgamestheory.mataroa.blog Computational Draughts Architecture: 32-Bit Bitboards, Mandatory Capture Pruning, and Retrograde Value Iteration boardgamestheory.surge.sh Combinatorial Complexity and Algorithmic State Spaces in Draughts Solvers: From SWAR Bitboards to Retrograde Tablebase Synthesis boardgamestheory.ucoz.net Algorithmic Architecture of Solved Checkers Engines: 32-Bit Bitboards, Mandatory Capture Branching, and Retrograde Analysis boardgamestheory.weebly.com The Forced Jump Dilemma in Draughts: Strategic Sacrifice vs Tactical Traps boardgamestheory.wordpress.com Combinatorial Complexity and Algorithmic State Spaces in Draughts Solvers boardgamestheory.zohosites.eu Computational Solvers, 32-Bit Bitboard Topologies, and Retrograde Analysis in Classical Draughts buy-vpnservice.blogspot.com Risoluzione Computazionale della Dama 8x8: Basi di Dati Chinook e Registri a 32 Bit coda.io Computational Solvers, 32-Bit Bitboards, and Retrograde Analysis in Checkers cosmos47.com Low-Latency 2D Canvas Pipelines vs DOM Overhead in Real-Time Web Draughts dev.to Bitboard Topologies, Quiescence Search, and Transposition Networks in High-Performance Checkers Engines gamestechai.blogspot.com Retrograde Analysis and Endgame Tablebases in Solved Zero-Sum Checkers (8x8 Draughts) gist.github.com Algorithmic Architecture of Solved Checkers Engines: 32-Bit Bitboards, Mandatory Capture Branching, and Retrograde Analysis gitlab.com Computational Solvers, 32-Bit Bitboards, and Retrograde Analysis in Checkers justpaste.it Modern Draughts Architecture: Combinatorial Solvers & Bitboards nrwd9l.webmepage.com Modern Draughts Architecture: A Comprehensive Review of Mandatory Capture Mechanics and Solved Game Solvers paper.wf Algorithmic Architecture of Solved Checkers Engines payamghaderkourehpaz530787.substack.com Combinatorial State Spaces and Proof-Number Search in Solved Draughts Games payamprivate.tumblr.com Computational Draughts Architecture: 32-Bit Bitboards, Mandatory Capture Dynamics, and Chinook Tablebases peyman.voog.com Retrograde Analysis and Endgame Tablebase Verification in Solved 8x8 Draughts (Checkers) poyeshsoftware.github.io Combinatorial Divergence in Draughts: English 8x8 Short Kings vs International 10x10 Flying Kings and Majority Capture poyeshsoftware.github.io Bitboard Representation and Move Generation Efficiency in Solved 8x8 Draughts Engines poyeshsoftware.github.io Combinatorial Game Solvers: Mandatory Capture Dynamics and Bitboard Optimization in 8x8 Draughts sites.google.com Bitboard-Topologien, SWAR-Parallelismus und retrograde Endspieldatenbanken in Damesolvers spieltheorie-algorithmen.blogspot.com Combinatorial Game Solvers: Mandatory Capture Dynamics and Bitboard Optimization in 8x8 Draughts tabletop-analysis.blogspot.com The Anatomy of a Double-Corner Trap in American 8x8 Checkers telegra.ph The Mechanics of Opposition and Triangular Traps in Solved 8x8 Draughts Endgames telegra.ph Tactical Shot Combinatorics, In-and-Out Pitches, and Forcing Sequences in 8x8 Draughts telegra.ph Bitboard Topologies and Directed Acyclic Jump Chains in Draughts Solvers ultramarine-crab-1193vv9tabletopanalysis.mystrikingly.com Retrograde Analysis and Endgame Tablebase Verification in Solved 8x8 Draughts ultramarine-crab-1193vv9tabletopanalysis.mystrikingly.com Combinatorial Divergence in Draughts: English 8x8 Short Kings vs International 10x10 Flying Kings and Majority Capture ultramarine-crab-1193vv9tabletopanalysis.mystrikingly.com Retrograde Value Iteration and Topological Sort in Cyclic Draughts Endgames write.as Has our enthusiasm about playing checkers been affected by it being solved as a drawn game? www.quora.com
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Backgammon, probability and fair dice 105
Expectiminimax search, bear-off Markov chains, doubling-cube economics and commit-reveal dice that players can verify.
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Stochastic Combinatorics, Expectiminimax Bounds, and Markov Decision Processes in Classical Backgammon 6ac17036e66f9.site123.me Stochastic Markov Decision Processes and Expectiminimax Optimization in Classical Dice Games board-games-theory.gitbook.io Algorithmic State Evaluation in Classical Board Games: Minimax, Expectiminimax, and Neural Heuristics boardgamestheory.bearblog.dev Stochastic Markov Decision Processes and Expectiminimax Optimization in Classical Dice Games boardgamestheory.codeberg.page The Holding Game Paradox: Why Passive Defense Is Backgammon’s Most Mathematically Fragile Strategy boardgamestheory.mataroa.blog The Dynamics of Contact Volatility: Evaluating Market-Losing Rolls in Modern Backgammon Engines boardgamestheory.mataroa.blog Shannon Number Bounds and Computational Game Complexity: Why Backgammon, Draughts, and Chess Diverge boardgamestheory.surge.sh Modern Web Engine Architecture: WebRTC Voice Chat, CSPRNG Dice Fairness, and High-Performance 2D Canvas boardgamestheory.surge.sh Stochastic Decision Processes, Expectiminimax Horizons, and Neural Value Functions in Backgammon Solvers boardgamestheory.ucoz.net Stochastic Markov Decision Processes and Expectiminimax Horizons in Classical Backgammon boardgamestheory.zohosites.eu Minimax vs Expectiminimax: Handling Chance Nodes in Strategic Game Trees buy-vpnservice.blogspot.com Archaeological and Combinatorial Evolution of Board Games: From the Royal Game of Ur to Roman Tabula and Sasanian Nard buy-vpnservice.blogspot.com Stochastic MDPs, Expectiminimax Trees, and Provably Fair Randomness in Backgammon cosmos47.com The Resilient Geometry of Abstract Strategy: Why Backgammon, Checkers, and Chess Outlast Every Digital Trend gamestechai.blogspot.com Anchor Point Geometry, Home Board Primes, and Timing Invariants in High-Performance Backgammon Engines gist.github.com Opening Roll Combinatorics and Initial Anchor Placement in Modern Backgammon Theory gist.github.com Stochastic Markov Decision Processes and Expectiminimax Optimization in Classical Dice Games gitlab.com Stochastic State Trees, Expectiminimax Horizons, and Cryptographic Commit-Reveal Dice Protocols in Modern Backgammon Engines gitlab.com Absorbing Markov Chains, Fundamental Matrices, and Exact Analytical Bearoff Solvers gitlab.com Continuous Financial Options Pricing, Janowski Match Equity Formulas, and Doubling Cube Economics gitlab.com Stochastic MDPs, Expectiminimax, and Provably Fair Randomness in Backgammon justpaste.it Demystifying the Doubling Cube: How a 1920s Rule Transformed Backgammon into an Applied Probability Engine medium.com Absorbing Markov Chains, Fundamental Matrices, and Exact Analytical Bearoff Solvers in Stochastic Race Games medium.com Backgammon Stochastic Mathematics: Markov Chains & Cube Valuations nrwd9l.webmepage.com The Mathematics of Imperfect Certainty: Markov Chains, Doubling Cube Option Pricing, and Backgammon Decision Theory paper.wf The Mathematics of Backgammon Primes: Escaping 6-Prime Blockades and Timing Preservation payamghaderkourehpaz530787.substack.com Stochastic Markov Decision Processes and Expectiminimax Optimization in Classical Dice Games payamghaderkourehpaz530787.substack.com The Geometry of Prime Structures in Backgammon: Breaking the 6-Prime Trap payamprivate.tumblr.com Stochastic Decision Trees and Doubling Cube Economics in Modern Backgammon Engines peyman.voog.com The Convergence of Neural Network Evaluation and Monte Carlo Rollouts in Modern Backgammon Engines poyeshsoftware.github.io The Ergodicity Problem and Option Pricing of the Doubling Cube in Stochastic Games poyeshsoftware.github.io Stochastic State Trees, Expectiminimax Horizons, and Cryptographic Commit-Reveal Dice Protocols in Modern Backgammon Engines poyeshsoftware.github.io Absorbing Markov Chains, Fundamental Matrices, and Exact Analytical Bearoff Solvers poyeshsoftware.github.io Continuous Financial Options Pricing, Janowski Match Equity Formulas, and Doubling Cube Economics poyeshsoftware.github.io Stochastische Entscheidungsprozesse und Markow-Ketten in Backgammon-Rennendspielen spieltheorie-algorithmen.blogspot.com From the Royal Game of Ur to Roman Tabula and Classical Backgammon: Five Millennia of Archaeological Evidence and Combinatorial Board Design tabletop-analysis.blogspot.com Stochastic Markov Decision Processes, Expectiminimax Chance Trees, and Financial Valuation of the Doubling Cube in Modern Backgammon tabletop-analysis.blogspot.com Archaeological Foundations and Combinatorial Board Design: From the Royal Game of Ur to Roman Tabula and Sasanian Nard tabletopanalysis.wordpress.com The Opening Book in Backgammon: Roll-by-Roll Tactical Analysis of Initial Turns telegra.ph The Doubling Cube in Volatile Positions: Calculating Market Losers and Recube Vig telegra.ph Stochastic State Trees and Probability Modeling in Classical Backgammon telegra.ph Backgammon Stochastic Mathematics and Doubling Cube Option Pricing wakelet.com Stochastic Markov Processes, Expectiminimax Trees, and Fair Dice in Classical Backgammon wakelet.com The Pip Count Fallacy: Why Pure Racing Distance Fails in High-Contact Backgammon Positions write.as Do backgammon apps cheat / rig the dice to create more spending? www.quora.com
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Game theory, history and engine engineering 45
State-space complexity, the history of race and capture games, and the networking side of real-time board games.
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Mathematical Foundations of Abstract Strategy and Computational Game Engines board-games-theory.gitbook.io Combinatorial State Spaces and Topological Graph Representations of Abstract Board Games board-games-theory.gitbook.io The Mathematics of Imperfect Information and Chance Nodes in Stochastic Tabletop Systems boardgamestheory.bearblog.dev Combinatorial State Spaces and Topological Graph Representations of Abstract Board Games boardgamestheory.codeberg.page The Pip Count Paradox: Why Pure Racing Distance Fails in Contact Positions boardgamestheory.wordpress.com Building Low-Latency Turn-Based Game Clients: Canvas Rendering, Coordinate Systems, and State Reconciliation dev.to Architecture of Modern Web Game Engines: Canvas Rendering, CSPRNG Fairness, and WebRTC Voice in Distributed Turn-Based Systems dev.to Stochastic State Trees and Probability Modeling in Classical Board Games dust-polyester-5af.notion.site Neural Network Architectures for Two-Player Zero-Sum Games: From Convolutional Grids to Residual Transformers gamestechai.blogspot.com Zobrist Hashing and Transposition Table Optimization in Two-Player Zero-Sum Game Engines gist.github.com Cryptographic Randomness, Deterministic Shuffling, and Verifiable State Invariants in Digital Tabletop Systems gist.github.com Combinatorial State Spaces and Topological Graph Representations of Abstract Board Games gitlab.com Transposition Bounds, Zobrist Signatures, and State Space Search in Combinatorial Games medium.com 5,000 Years of Abstract Strategy: From Predynastic Senet and Ur to Classical Tabula nrwd9l.webmepage.com From the Royal Game of Ur to Roman Tabula: The 5,000-Year Archaeological Evolution of Tabletop Gaming paper.wf The Shannon Number of the Physical World: Why Board Geometry Defies Brute Force Computation payamghaderkourehpaz530787.substack.com Stochastic Tree Search and TD-Gammon Neural Architectures payamghaderkourehpaz530787.substack.com Combinatorial State Spaces and Topological Graph Representations of Abstract Board Games payamghaderkourehpaz530787.substack.com The 5,000-Year Archaeological Evolution of Tabletop Strategy: From the Royal Game of Ur to Roman Tabula peyman.voog.com Computational Complexity & Algorithmic State Evaluation in Classical Board Games poyeshsoftware.github.io The Geometry of Abstract Strategy & Computational Complexity sites.google.com Zobrist Hashing and Transposition Table Bounds in Combinatorial Game Solvers tabletopanalysis.wordpress.com Stochastic Game Tree Pruning: Star1 and Star2 Algorithms in Chance-Based Decision Systems tabletopanalysis.wordpress.com Evaluating Classical Board States: From Handcrafted Heuristics to Neural Network Value Functions tireless-canary-1193vvfboardgametheory.mystrikingly.com Option Pricing and Optimal Stopping Boundaries in Stochastic Board Games tireless-canary-1193vvfboardgametheory.mystrikingly.com Archaeological Foundations and Combinatorial Board Design: From the Royal Game of Ur to Roman Ludus Duodecim Scriptorum tireless-canary-1193vvfboardgametheory.mystrikingly.com Computational Combinatorics and Game Theory in Classical Board Games wakelet.com State Synchronization and Optimistic Prediction in Turn-Based Peer-to-Peer Games workoutingym.hashnode.dev Cryptographic Commit-Reveal Protocols, Verifiable Randomness, and PRNG Seed Generation in Distributed Tabletop Networks workoutingym.hashnode.dev The Ergodicity Problem in Board Games: Why Expected Value Fails Over Short Horizons write.as Backward Induction in Combinatorial Games: From Zermelo's Theorem to Modern Solvers write.as Architecting Low-Latency Turn-Based Game Engines: Why 2D Custom Canvas Bypasses the DOM Overhead www.linkedin.com
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Try the ideas yourself
Reading about search trees is one thing; watching an engine find a move is another. All three games are free to play in the browser on Boardgammon . On each one you can play against engines of increasing strength and review your games afterwards: