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Mastering Board Games by External and Internal Planning with Language Models

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Language models can reach grandmaster-level chess by planning with search, without a game engine at play time.

desk verdict A genuinely useful empirical study of LLM planning on board games, with a real (if noisy) result: search over a learned world model improves play, and the internal-search distillation works; the headline Elo numbers need more caveats than the abstract gives them. read the letter →

arxiv 2412.12119 v3 pith:22HF37HR submitted 2024-12-02 cs.AI cs.CLcs.LG

John Schultz , Jakub Adamek , Matej Jusup , Marc Lanctot , Michael Kaisers , Sarah Perrin , Daniel Hennes , Jeremy Shar
show 8 more authors
Cannada Lewis Anian Ruoss Tom Zahavy Petar Veličković Laurel Prince Satinder Singh Eric Malmi Nenad Tomašev
This is my paper · ORCID
classification cs.AIcs.CLcs.LG
keywords languagemodelsplanningMonteCarloTreeSearchworldmodelchessinternalexternalboardgames
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that search-based planning can turn language models into strong game players, provided the model is first trained to act as a reliable world model. The authors introduce the Multi-Action-Value (MAV) model, a transformer pretrained on textual game data to track board states, predict legal moves, and estimate action values in a single call. Embedding MAV inside a Monte Carlo Tree Search controller raises its chess strength from roughly 2923 external Elo to about 3209 at 2000 simulations, and a variant fine-tuned to emit an internal search trace reaches about 2930 external Elo in a single call. The central message is that planning, not just next-token prediction, is a direct lever on LLM competence in sequential decision tasks.

What carries the argument

The load-bearing object is the Multi-Action-Value (MAV) model, a decoder-only transformer trained to answer a command header such as `%state %top_5 %best_action` by emitting legal moves, bucket-encoded win probabilities, and the engine's preferred move in one token stream. The external-search mechanism is an AlphaZero-style MCTS whose transition, legal-move, value, and terminal checks are all taken from MAV's output rather than from a game engine, with a parser that discards malformed responses by assigning them $-\infty$. The internal-search mechanism is a fine-tuned MAV (MAV-IS) trained on linearized depth-first minimax trees, so that a single model call alternates between evaluating a node, expanding it, and selecting a move, effectively executing a search algorithm in language. Two auxiliary mechanisms carry the performance: mean scoring over the 64 value buckets, which distinguishes near-tied moves better than taking the mode, and an asynchronous MCTS with dynamic virtual counts, which lets batches of LLM evaluations proceed in parallel without collapsing exploration.

What would settle it

Take a set of chess positions whose best move wins only through a tactic of depth five or more, run MAV-MCTS with 2000 simulations on each, and record how often the parser's $-\infty$ fallback removes the true best move from the tree; if that frequency grows with tactical depth, the learned-transition assumption fails precisely where search is supposed to add the most value.

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Extended reading notes

Core claim

Stated on the paper's own terms, the discovery is that one language model can simultaneously serve as a world model, a value function, and a policy for several perfect-information games, and that this model supports two modes of planning. In external search, MAV replaces the game engine inside an AlphaZero-style MCTS: it predicts the next state from a state-action pair, returns legal moves with their values, and detects terminal states, with any unparsable output assigned $-\infty$ to keep the search tree clean. In internal search, MAV is fine-tuned on linearized minimax trees and learns to produce a search trace and a final choice within one model call, occasionally correcting its own first impulse (in the paper's example, finding a winning move that the initial value ranking missed). The result is grandmaster-level chess with a search budget closer to human thinking than to engine search: MAV-MCTS with 2000 simulations reaches internal Elo 1707, mapped to roughly 3209 external Elo, and MAV-IS at breadth 4 and depth 2 reaches roughly 2930 external Elo. The same framework, with game-specific caveats, lifts performance across Chess960, Connect Four, and Hex.

Load-bearing premise

External search without a game engine relies on MAV's learned transition function being accurate on the states that search visits, because a single hallucinated transition silently drops that node from consideration with value $-\infty$ and the paper does not measure how such errors accumulate with depth.

Editorial extensions

If this is right

  • Search-based planning adds roughly 300 Elo to the searchless MAV model in chess, with strength growing logarithmically in the number of MCTS simulations.
  • MAV-IS shows that a model can be trained to emit a structured search trace and a final decision in a single call, with playing strength scaling with the search budget measured in tokens.
  • Because `%best_action` training teaches the model to convert decisive advantages, MAV and its search variants can play complete games without any external game engine at inference time.
  • The same architecture and training format transfer across four different games, indicating the recipe is not tied to chess-specific features.
  • Depth and breadth of internal search provide a test-time compute knob: more tokens per call buy more playing strength, analogous to adding simulations in external search.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • My inference: if the learned-transition assumption holds outside board games, the same "one model as world model plus parser-enforced search" recipe should transfer to any domain whose state transitions can be written as text and checked for well-formedness, such as code execution or formal proof search; the paper does not test this.
  • My inference: the external/internal distinction is essentially a choice of where to spend the compute budget, many short model calls versus one long call, and the paper leaves open whether the two budgets compose additively at a fixed total token cost.
  • My inference: the parser's $-\infty$ fallback is where hallucination risk is concentrated; directly measuring how often the true best move is dropped as search depth grows would quantify the ceiling of engine-free search, which the paper does not report.
  • My inference: internal-search training data could plausibly be regenerated from the external search's own visit statistics, letting the model learn from its own search rather than only from engine annotations; this bootstrapping loop is an extension the paper leaves implicit.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper introduces MAV, a 2.7B-parameter decoder-only Transformer trained on textual game data to act as a policy, value function, and world model for chess, Chess960, Connect Four, and Hex. It then presents two planning methods: external search, where MAV guides an AlphaZero-style MCTS without an external game engine, and internal search, where a fine-tuned model (MAV-IS) generates a linearized minimax tree in a single call. The paper reports internal Elo gains from MCTS and maps them to external Elo, claiming Grandmaster-level chess performance at roughly 100-1000 considered moves per decision, and it reports that internal search strength increases smoothly with search budget. The appendices contain dataset statistics, tournament details, hyperparameters, algorithm pseudocode, and annotated example games.

Significance. If the central claims hold, the paper would be a meaningful contribution: it demonstrates that a language model can serve as a learned transition and value function inside MCTS in deterministic board games, and that a distilled search procedure can be executed in a single model call with scaling behavior in the search budget. The paper is strong in its breadth: four games, large tournament sizes (about 15k chess games, 6k Hex games, and 2k+ Connect Four games), detailed pseudocode, and an explicit discussion of limitations. It also ships a playable artifact (MAV-small) and openly discloses the dependence on engine annotation for training data. The main weaknesses are the reliability of the external Elo calibration and the lack of evidence that learned transition errors do not silently corrupt the search tree.

major comments (4)
  1. [§5.1, Table 1] The headline external Elo values are derived from a linear fit of internal Elo to externally reported Stockfish ratings, but the paper gives no details of the fit, no anchor points, and no confidence intervals. Table 1's own footnote states that the external Elo estimates for MAV-MCTS agents are 'not accurate' because those agents would not be able to play at blitz time controls, which is precisely the rating scale used for the anchors. Since the abstract and introduction rely on 'Grandmaster-level performance' in chess, the external Elo values (e.g., 3209 for MAV-MCTS with 2000 simulations) should either be replaced by relative/internal Elo comparisons, or accompanied by a defensible calibration with uncertainty quantification and a clear statement of the time-control mismatch.
  2. [§3.1, Table 2] The engine-free external search claim rests on MAV's reliability as a transition function. Section 3.1 handles hallucinated outputs by assigning -infinity to the affected node and removing it from future consideration. Table 2 reports one-step FEN accuracy of 99.6-100%, but this is not sufficient evidence for the search setting: a per-node failure rate of 0.4% can accumulate over 2000 simulations, and a failure at a high-value child permanently prunes that line, potentially removing the true best move before Final Move Selection. The paper does not report parse-failure rates during actual MCTS searches, does not measure how transition errors grow with search depth, does not analyze whether failures correlate with sharp tactical positions, and does not provide an oracle-transition ablation. Without these analyses, the +340 internal Elo improvement of MAV-MCTS(2000) over searchless MAV could be partly an artifact of searching over a silently biased subset of the game tree. I would like to see these measurements, or, failing that, a substantially weakened claim about the engine-free variant.
  3. [§5.4 and footnote 3] The internal search results are evaluated under a different protocol from the other agents: games are stopped once either side reaches a decisive 1200+ centipawn advantage. The external Elo axis in Figure 5 (right) is therefore built from truncated games, and the magnitudes such as roughly 2930 Elo for MAV-IS(b=4,d=2) are not directly comparable to the completed-game Elo of the other agents. The qualitative claim that internal search strength scales with the search budget may survive, but the quantitative Elo levels are not supported under a common protocol. Please either report a consistent evaluation protocol for all agents or analyze how the truncation threshold changes the resulting ratings.
  4. [§5.3] The paper states that for Hex, 'only a game engine is used to determine legal actions, state transitions, and terminal states' because the final model was not adequately trained for state tracking in Hex. Consequently, the Hex entries in Table 1 for MAV-MCTS do not demonstrate Contribution 2, the engine-free external search method. The abstract and Contribution 1 say the model reliably captures transition and value functions in 'the respective environments,' which is misleading for Hex. The claim should be qualified to the games where state-tracking MAV was actually used as the transition model.
minor comments (6)
  1. [Appendix A.2] The sentence 'The average amount of time taken per search is shown in Table 1' appears to refer to Table 3 in the same appendix; please correct the cross-reference.
  2. [§5.1] The phrase 'As an additional, we include the Ext-BoN model' should be 'As an additional baseline, we include the Ext-BoN model.'
  3. [Table 2] The table is titled 'Error rate analysis' but the reported values are accuracies (e.g., 0.999, 0.996). Please rename the table or invert the metrics to avoid confusion.
  4. [Appendix E] There is a typo in 'Language Agent Tree Seatch (LATS)' in the related work; it should be 'Search.'
  5. [§5.3, Appendix A.3] The external MCTS hyperparameters are numerous (prior temperature, top-k, epsilon, batch size, timeout, dynamic virtual counts), and the paper reports only final values without a sensitivity analysis. At minimum, please state whether the simulation-count trend in Figure 5 (left) is stable across reasonable variations of these hyperparameters.
  6. [§5.2] The generalization analysis in Appendix D is useful, but the claim that '10% of the positions played by MAV in evaluation appear in its training data' would be easier to interpret if the paper reported how the overlap rate varies across game phases, which the appendix partially does; please make that connection explicit in the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the search improvements are measured against held-out external engines, and no claim reduces to its own training input by construction.

full rationale

The central claims of the paper are empirical and externally benchmarked. MAV is trained on engine-annotated data (Stockfish, Fhourstones, neurobenzene), but the claimed strength of MAV and the +340 Elo gain of MAV-MCTS over searchless MAV are measured in head-to-head games against Stockfish levels from held-out TCEC openings (Table 1), not on training positions. The external Elo numbers are an explicitly caveated linear calibration anchored to CCLR Blitz ratings, with a footnote stating that the estimates are not accurate for slower agents; this is estimation, not a fitted prediction passed off as a scientific result. External MCTS uses MAV as transition, value, and policy, and Section 3.1 assigns -infinity to malformed transitions; this is a robustness mechanism, and any concern about cumulative transition error is a correctness limitation, not a circular reduction, since Table 2 measures held-out single-step FEN accuracy and the Elo gains are not defined by that accuracy. Internal search (MAV-IS) is explicitly presented as distillation of Stockfish-annotated minimax trees (Section 4), so improved play with larger breadth and depth is an expected supervised-learning outcome rather than a first-principles derivation, and it is tested against external engines on held-out positions. The self-citations present (Ruoss et al. 2024a for mean scoring and the Ext-BoN baseline; Veličković et al. 2024 for temperature adaptation; Lanctot et al. 2019 for OpenSpiel) are prior-work baselines or minor design choices, not load-bearing premises, and none of them makes a central claim true by definition. No step of the derivation chain is equivalent to its own input.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central results rest on engine-provided training labels, several hand-tuned search hyperparameters, and an evaluation rule that stops internal search games at a fixed centipawn threshold. No new physical or mathematical entities are introduced; MAV and the search controllers are trained artifacts, not postulated entities requiring independent evidence.

free parameters (6)
  • MCTS prior temperature tau = 0.1
    Tuned via manual and head-to-head comparisons in Appendix A.3; controls the sharpness of the softmax prior in Equation 1.
  • top-k and epsilon in prior = k=5, epsilon=0.05
    Tuned in Appendix A.3; the prior is an epsilon-greedy softmax over top-k values, so these shape all external search policies.
  • dynamic virtual count parameters = n_min=2; n_max from 8 to 32 depending on M
    Tuned for Async MCTS in Appendix A.3; balances exploration and exploitation in parallel search.
  • Async MCTS batch size and timeout = b=16, t0=60s
    Chosen for throughput; the paper notes lower batch sizes were preferred in performance comparisons, so this is a hand-tuned trade-off.
  • internal search evaluation stopping threshold = 1200+ centipawns
    Games were stopped once a side reached a decisive advantage to avoid aimless play in winning positions; this post-hoc rule affects all MAV-IS Elo values in Table 1 and Figure 5.
  • value bucket discretization = 64 buckets
    Action values are mapped to 64 discrete win-probability buckets in Section 2; a modeling choice that affects value resolution and classification difficulty.
assumptions (5)
  • domain assumption Stockfish, Fhourstones, and neurobenzene evaluations are sufficiently accurate ground-truth action values to train MAV.
    Used to label all training data in Section 2 and Appendix B; if these labels are systematically wrong, the distilled value function inherits the errors.
  • domain assumption The Lichess win-probability mapping from centipawns is a valid proxy for game-theoretic win probability.
    Section 2 uses this formula to convert Stockfish centipawn evaluations to win probabilities and then to bucket tokens.
  • domain assumption The Gemini architecture and tokenizer can be trained from scratch on game-only text and still generalize.
    Section 2 states MAV is a randomly initialized decoder-only Transformer using the Gemini architecture; no proof is given that game-only pretraining is sufficient, only empirical results.
  • domain assumption Linearized depth-first minimax traces teach an algorithmic search procedure, not just imitation of memorized trees.
    Section 4 frames internal search as an algorithmic execution task; the claim that MAV-IS can self-correct and scale with budget depends on this.
  • domain assumption MCTS with a learned transition model and -infinity fallback for parser failures is a good proxy for exact game-tree search.
    Section 3.1 replaces the game engine with MAV's predicted states; the results are conditional on the fallback not discarding important lines.

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Cite this review

Pith. "Pith review of Mastering Board Games by External and Internal Planning with Language Models." pith.science (2026). https://pith.science/paper/22HF37HR

@misc{pith2026241212119,
  author       = {Pith},
  title        = {Pith review of: Mastering Board Games by External and Internal Planning with Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/22HF37HR}},
  note         = {Machine review of arXiv:2412.12119}
}
read the original abstract

Advancing planning and reasoning capabilities of Large Language Models (LLMs) is one of the key prerequisites towards unlocking their potential for performing reliably in complex and impactful domains. In this paper, we aim to demonstrate this across board games (Chess, Fischer Random / Chess960, Connect Four, and Hex), and we show that search-based planning can yield significant improvements in LLM game-playing strength. We introduce, compare and contrast two major approaches: In external search, the model guides Monte Carlo Tree Search (MCTS) rollouts and evaluations without calls to an external game engine, and in internal search, the model is trained to generate in-context a linearized tree of search and a resulting final choice. Both build on a language model pre-trained on relevant domain knowledge, reliably capturing the transition and value functions in the respective environments, with minimal hallucinations. We evaluate our LLM search implementations against game-specific state-of-the-art engines, showcasing substantial improvements in strength over the base model, and reaching Grandmaster-level performance in chess while operating closer to the human search budget. Our proposed approach, combining search with domain knowledge, is not specific to board games, hinting at more general future applications.

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Forward citations

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Reference graph

Works this paper leans on

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    White will further rapidly develop its pieces by attacking the exposed Black’s queen.Qxc3+ 13

    d6MAV–MCTS sacrificed the second pawn to paralyze Black’s queenside. White will further rapidly develop its pieces by attacking the exposed Black’s queen.Qxc3+ 13. Kf2 Nf6

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    Queue b new evaluations

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    for the state𝑠0 in the root nodeN0; notice that the state evaluator is only necessary for the root node – (𝑠𝑡, 𝒂𝐿 𝑡, 𝑸(𝑖)(𝑠𝑡, 𝒂𝐿 𝑡)) = M AV(𝑠𝑡−1,𝑎𝑡−1): a state-action evaluator; for a given state-action parent tuple(𝑠𝑡−1,𝑎𝑡−1) returns the child state𝑠𝑡 =𝑇(𝑠𝑡−1,𝑎𝑡−1), a list of...

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    Bxb5+ Bc6 21

    Ngxe6MAV–IS sacrifices a knight to keep Black’s king in the center.Nxe6 20. Bxb5+ Bc6 21. Nxe6 Bxb5 22. Re1 Bd6 23. Nxg7+ Kd7 24. Rxe5 Bxe5 25. Qxe5MAV–IS correctly estimated that the vulnerable Black’s king and dark squares are worth the sacrificed exchange. Qc5 8 0Z0Z0Z0s 7 ...

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    Bb5 a4 22

    Rac1 Nb6 21. Bb5 a4 22. a3 Ra5 23. Bxc6 bxc6 24. Nge4 h6 25. Rd6 Rb5 26. Rd2 Kh7 27. Nd6 Rd8 28. Nce4 Rd5 29. Rxc6 Rxd2 30. Nxd2 35 Mastering Board Games by External and Internal Planning with Language Models Na8 31. N6c4 Bb7 32. Rxe6 Rd4 33. f3 Bd5 8 nZ0Z0Z0Z 7 Z0Z0Z0ok 6 0Z0...

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    Rd3 f6 23

    Re1 Qxf5 22. Rd3 f6 23. Bb2 Kd8 24. Rxd6 b6 25. Bxe5 fxe5 26. Rd5 Qf6 27. Rexe5 Kc7

  14. [22]

    Ne2 Rd2 24

    Bb5+ Kf8 23. Ne2 Rd2 24. Rbc1 Rxh5MAV regains the sacrificed pawn with interest after 17 moves; despite the even material, White is losing. 25. Rc8+ Rd8 26. Rxd8+ Bxd8 27. Nc3 Rxh2 28. Rf1 Bb6 29. Nd1 Ke7 30. a3 f5 31. Kb1 Kf6 32. Kc2 a6 33. Bc4 Bd4 34. b4 b5 35. Bd3 e5 36. Kd...

  15. [23]

    Qd2 h6 5

    e3 Bg7 4. Qd2 h6 5. Bh4 c5 6. f4 Qb6 32 Mastering Board Games by External and Internal Planning with Language Models 8 rmbZkZns 7 opZpo0a0 6 0l0Z0Zpo 5 Z0o0ZpZ0 4 0Z0O0O0A 3 Z0Z0O0Z0 2 POPL0ZPO 1 SNZ0JBMR a b c d e f g h

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    Nc3 A prelude to a pawn sacrifice.cxd4

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    exd4 Bxd4MAV–MCTS sacrifices a central pawn and likely loses castling rights for a compensation that is unclear to a human eye

  18. [26]

    Qd3 Nc6 11

    Nf3 Be3 10. Qd3 Nc6 11. Nd5 Qxb2 12. Qxe3 8 rZbZkZns 7 opZpo0Z0 6 0ZnZ0Zpo 5 Z0ZNZpZ0 4 0Z0Z0O0A 3 Z0Z0LNZ0 2 PlPZ0ZPO 1 S0Z0JBZR a b c d e f g h 12...Qxa1+ MAV–MCTS proceeds with sacrific- ing an exchange with a check.13. Kf2 Kf8 8 rZbZ0jns 7 opZpo0Z0 6 0ZnZ0Zpo 5 Z0ZNZpZ0 4 ...

  19. [27]

    Rf1 Qf8 29

    Re2 Re7 28. Rf1 Qf8 29. Qb4 Nd7 30. Bh3 d3 31. cxd3 Bc5 32. Rxf5 Bxb4 8 0Z0Z0l0j 7 Z0Zns0op 6 pZ0ZPZ0Z 5 Z0Z0ZRO0 4 0a0Z0Z0O 3 Z0ZPZ0ZB 2 PO0ZRZ0Z 1 ZKZ0Z0Z0 a b c d e f g h

  20. [28]

    fxg5 dxc6 8 rZbZ0jns 7 opZ0o0Z0 6 0ZpZ0Z0o 5 Z0ZNZpO0 4 0Z0Z0Z0A 3 Z0Z0LNZ0 2 PZPZ0JPO 1 Z0Z0Z0Zq a b c d e f g h

    Bxc6 g5 16. fxg5 dxc6 8 rZbZ0jns 7 opZ0o0Z0 6 0ZpZ0Z0o 5 Z0ZNZpO0 4 0Z0Z0Z0A 3 Z0Z0LNZ0 2 PZPZ0JPO 1 Z0Z0Z0Zq a b c d e f g h

  21. [29]

    Bxe7+ Kg7 19

    g6 A nail in the coffin – Black is lost despite being two rooks up.Qa1 18. Bxe7+ Kg7 19. Bd6 cxd5 20. Qe8 Be6 21. Qxe6 (game terminated by early termination).MAV–MCTS’s playing style resembles games of players such as GM Mikhail Tal and IM Rashid Nezhmetdinov who sacrificed ma...

    2024

  22. [30]

    f4 exf4 3

    e4 e5 2. f4 exf4 3. Nf3 g5 4. Nc3 Nc6 5. g3 g4

  23. [31]

    d4 Bb4 8

    Nh4 f3 7. d4 Bb4 8. d5 Qe7

  24. [32]

    bxc3 Qa3 8 rZbZkZns 7 opopZpZp 6 0Z0Z0Z0Z 5 Z0ZPmNZ0 4 0Z0ZPZpZ 3 l0O0ApO0 2 PZPZ0Z0O 1 S0ZQJBZR a b c d e f g h

    Nf5 Bxc3+ 11. bxc3 Qa3 8 rZbZkZns 7 opopZpZp 6 0Z0Z0Z0Z 5 Z0ZPmNZ0 4 0Z0ZPZpZ 3 l0O0ApO0 2 PZPZ0Z0O 1 S0ZQJBZR a b c d e f g h

  25. [33]

    Rfe5 Rxe5 35

    exd7It was likely better to grab the queen, but the position remain unclear.Qd8 34. Rfe5 Rxe5 35. Rxe5 Be7 36. d4 h6 37. g6 a5 38. Re1 Kg8 39. d5 Qb6 40. h5 Kf8 41. a3 a4 42. Rc1 Qa5 43. d6 Bd8 44. Rc8 Qe1+ 45. Ka2 Qe6+

  26. [34]

    Bd4 Qc6 8 rZbZkZ0s 7 opZpZpZp 6 0Zqo0m0Z 5 Z0Z0mNZ0 4 0Z0APZpZ 3 Z0ZBZpO0 2 PZPZ0J0O 1 S0ZQZ0ZR a b c d e f g h

    Bd3 cxd6 15. Bd4 Qc6 8 rZbZkZ0s 7 opZpZpZp 6 0Zqo0m0Z 5 Z0Z0mNZ0 4 0Z0APZpZ 3 Z0ZBZpO0 2 PZPZ0J0O 1 S0ZQZ0ZR a b c d e f g h

  27. [35]

    a4 At the first glance, this move aims to stop b5 and queenside development, but the move also has a much deeper idea.Rg8

  28. [36]

    Qd2 Rxf5A knight paralyzes the entire Black’s position, so Black desperately sacrifices the rook for some activity

    Ra3 Now the real idea behind 16.a4 becomes apparent; MAV–MCTS wants to activate its rook via c3!Rg5 18. Qd2 Rxf5A knight paralyzes the entire Black’s position, so Black desperately sacrifices the rook for some activity. 19. exf5 Ne4+ 20. Bxe4 Qxe4

  29. [37]

    Qb6 Qd7 39

    Re3 Rc7 38. Qb6 Qd7 39. Qb5 Qh3 40. c4 Qxh4 41. c5 Qf4 42. Rd3 h4 43. Qxa5 Rc8 44. Qa6 Rc7 45. Qb6 Rc8 46. Qb7 Rf8 47. c6 h3 48. c7 h2 49. Rd8 h1=Q 50. c8=Q Rxd8 51. Qxd8 8 0Z0L0Z0Z 7 ZQZ0Zpak 6 0Z0Z0ZpZ 5 Z0Z0o0O0 4 PZ0ZNl0Z 3 Z0Z0ZPZ0 2 KO0Z0Z0Z 1 Z0Z0Z0Zq a b c d e f g h 51...

  30. [38]

    Rc3+ Kb8 30

    Re3 h6 29. Rc3+ Kb8 30. a5 Qe7 31. Qf4+ d6 32. Re3 Qc7 33. Rxd6 Kb7 34. Qe4+ Kb8

  31. [39]

    Rxc7 Kxc7 37

    Rc6 Bd7 36. Rxc7 Kxc7 37. Qxa8 bxa5 38. Qxa7+ Kc6 39. Rc3+ Kd6 40. Rd3+ Kc6 41. Qxd7+ Kb6 42. Rd6+ Kc5 43. Rc6+ Kb4 44. Qb7+ Ka3 45. Qb3# 1 – 0 Game 3 Precise pawn endgame Date: 2024-11-12 White: MAV–MCTS(𝑀 = 2000) - max scoring Black: Stockfish–L18 Result: 1 – 0 Online PGN: h...

    2024

  32. [40]

    e4 Nc6 2. d4 d5 3. Nc3 Nf6 4. e5 Nd7 5. Nce2 e6 6. c3 f6 7. Nf4 Qe7 8. exf6 Nxf6 9. Be2 Qd7 10. Nf3 Bd6 11. O-O O-O 12. Nd3 Ne4 13. Nfe5 Bxe5 14. Nxe5 Nxe5 15. dxe5 a5 16. Be3 b6 17. Qc2 Bb7 18. Bd3 h6 19. Rad1 Qe8 20. f3 Nc5 21. Bh7+ Kh8 22. Bg6 Qe7 23. Qd2 Kg8

  33. [41]

    b5 Rd2+ 43

    Bc4 Be3 42. b5 Rd2+ 43. Ke1 Rb2 44. b6 Bxb6 45. Kd1 Ke5 46. Ke1 Be3 47. Kd1 f4 48. Ke1 f3 49. Kd1 Kd4 50. Bf7 Kc3 51. Ba2 Rxa2

  34. [42]

    Qc2 g5 26

    Bb1 Ba6 25. Qc2 g5 26. Bxc5 bxc5 27. Qg6+ Qg7 28. Qxe6+ Qf7 8 rZ0Z0skZ 7 Z0o0ZqZ0 6 bZ0ZQZ0o 5 o0opO0o0 4 0Z0Z0Z0Z 3 Z0O0ZPZ0 2 PO0Z0ZPO 1 ZBZRZRJ0 a b c d e f g h

  35. [43]

    Qxh6 MAV–MCTS correctly sacrifices an exchange to ruin Black’s kingside. Bxf1 30. Kxf1 c6 31. Re1 Rae8 32. Bg6 Qg7 33. Qxg7+ Kxg7 34. Bxe8 Rxe8 35. e6 Kf6 36. e7 c4 37. b3 cxb3 38. axb3 Rxe7 39. Rxe7 Kxe7 40. Ke1 Kd6

  36. [44]

    h4 gxh4 43

    g3 c5 42. h4 gxh4 43. gxh4 Kd7 44. h5 Ke8

  37. [45]

    Kd1 Kf8 34 Mastering Board Games by External and Internal Planning with Language Models 8 0Z0Z0j0Z 7 Z0Z0Z0Z0 6 0Z0Z0Z0Z 5 o0opZ0ZP 4 0Z0Z0Z0Z 3 ZPO0ZPZ0 2 0Z0Z0Z0Z 1 Z0ZKZ0Z0 a b c d e f g h

  38. [46]

    The idea is to timely set up the f5-h5 pawn formation

    f4 MAV–MCTS plays the only winning move in a seemingly simple pawn endgame. The idea is to timely set up the f5-h5 pawn formation. Kg7 47. f5 Kg8 48. Kc2 Kh7 49. f6 Kh6 50. Kb1 Kh7 51. Kc2 Kh6 52. Kd3 Kh7 53. Ke3 Kh8 54. Kf4 Kg8 55. Kf5 d4 56. cxd4 Kh7

  39. [47]

    Bg5 Ne4 3

    d4 Nf6 2. Bg5 Ne4 3. Bf4 c5 4. f3 Qa5+ 5. c3 Nf6 6. d5 Qb6 7. e4 Qxb2 8. Nd2 Qxc3 9. Bc7 g6 10. Rc1 Qe3+ 11. Ne2 d6 12. Rb1 Nfd7

  40. [48]

    Rb3 Qxb3 MAV–MCTS sacrifices a queen hoping its position is solid enough to hold back White’s pieces activity.15

    f4 8 rmbZka0s 7 opAnopZp 6 0Z0o0ZpZ 5 Z0oPZ0Z0 4 0Z0ZPO0Z 3 Z0Z0l0Z0 2 PZ0MNZPO 1 ZRZQJBZR a b c d e f g h 13...Na6 Prelude to a queen sacrifice.14. Rb3 Qxb3 MAV–MCTS sacrifices a queen hoping its position is solid enough to hold back White’s pieces activity.15. axb3 Nxc7 16. ...

  41. [49]

    Qc1 a6 23

    Rh7 Bd4 22. Qc1 a6 23. Nc4 Nf6 24. Rh3 Bxc3+ 25. Rxc3 Nxe4 26. Re3 f5 27. Nxb6 Nxd5

  42. [50]

    Qxe3 Bxa8 30

    Nxa8 Nxe3 29. Qxe3 Bxa8 30. g4 Rh8 31. gxf5 gxf5 32. b4 cxb4 33. Qb6 Kd7 34. Qd4 Rc8

  43. [51]

    Ke2 Nc5 37

    Bxa6 Rc1+ 36. Ke2 Nc5 37. Bb5+ Bc6 38. Qxb4 Ra1 39. Bc4 Ra4 40. Qc3 Kc7 41. Kd1 Kb6

  44. [52]

    Rxf3 exf3 53. Ke1 Ra1# 0 – 1 Game 1 The first win against Stockfish Unclear long-term attack Date: 2024-05-31 White: MAV–MCTS, development version Black: Unoptimized Stockfish (50K-90K nps) Result: 1 – 0 Online PGN: https://lichess.org/gVNnuSWn

    2024

  45. [53]

    Qb2+ Kc7 44

    Qc1 e6 43. Qb2+ Kc7 44. Qc3 Bd7 45. Ke1 Ne4 46. Qc1 Ra5 47. Bf1+ Bc6 48. Be2 Nc5 49. Qc3 Ra4 50. Qg3 Ra1+ 51. Bd1 Kb6 52. Qc3 Rb1 53. Qd4 Kc7 54. Qg7+ Kb6 55. Qe7 Ba4

  46. [54]

    g3 Ba3 56

    Rxe7 Kxe7 55. g3 Ba3 56. Kg2 Kf7 57. Rb8 Kf6 58. Rb5 Ke7 59. Kh3 Kd7 60. Rb7+ Kc6 61. Rb3 Be7 62. f4 Bd8 63. g5 Ba5 64. Kg4 Kd5 65. Rb5+ Ke6 66. Rxa5 Kd6 67. Ra7 Ke6 68. Rb7 Kd6 69. g6 Ke6 70. a4 Kd6 71. g7 Kd5 72. a5 Kd6 73. g8=Q Kc5 74. f5 Kd4 75. Qg5 Kc5 76. Qe3+ Kc6 77. Rh...

    2024

  47. [55]

    Re7+ Kc6 57

    Re1 Na6 56. Re7+ Kc6 57. Qa8+ Kb5 58. g7 Nb8 59. c4+ Kxc4 60. Re1 Kb5 61. g8=Q Kxc5 62. Rg1 Na6 63. Kd1 Kb5 64. Rh1 Nc7 65. Rh5 Nxa8 66. Qd5+ Kb6 67. Qc4 Kb7 68. Qd5+ Kc8 69. Rxf5 Nb6 70. Qc5+ Kb7 71. Qc2 Nd5

  48. [56]

    Qd8+ Kb7 58

    Qxd6+ Bc6 57. Qd8+ Kb7 58. Qf8 Nd7 59. Qa3 Kb6 60. Qd3 Rc1 61. Kd2 Rc5 62. Qb3+ Kc7 63. Be2 Bd5MAV–MCTS builds a fortress

  49. [57]

    Ke6 c4 59

    f7 Kg7 58. Ke6 c4 59. Ke7 Kh6 60. f8=Q+ Kh7 61. Kf6 cxb3 62. Qg7# 1 – 0 Game 4 Building a fortress Date: 2024-11-12 White: Stockfish–L18 Black: MAV–MCTS(𝑀 = 2000) - max scoring Result: 1/2 – 1/2 Online PGN: https://lichess.org/a5Rqsg8s

    2024

  50. [58]

    e4 e6 2. d4 d5 3. e5 c5 4. c3 Ne7 5. Nf3 Nec6

  51. [59]

    O-O Nd7 8

    Bd3 Be7 7. O-O Nd7 8. Re1 f6 9. exf6 Nxf6 10. dxc5 a5 11. c4 O-O 12. Nc3 dxc4 13. Bxc4 Qxd1 14. Rxd1 Bxc5 15. Na4 Ba7 16. Be3 Nd5 17. Bxa7 Nxa7 18. Nc5 Nc6 19. Ng5 Re8

  52. [60]

    Rd6After dominating the key squares by masterful maneuvers, MAV–MCTS correctly evaluates that entering a rook versus minor pieces endgame is decisive. Rxc4 35. Nxc4 Bxc4 36. Rc6 Bd5 37. Ra6 Nc7 38. Rxa4 Kg8 39. Ra7 Ne6 40. b4 Kf8 41. b5 Ke8 42. b6 Nd8 43. Rxg7 Nf7 44. a4 Kd7 4...

  53. [61]

    Rf6 Bc4 49

    h4 h5 48. Rf6 Bc4 49. Rf5 Nd8 50. Rxh5 Bf7

  54. [62]

    Rh7 Ne5 53

    Rh8 Nc6 52. Rh7 Ne5 53. f4 Nd7 54. Rxf7 Kc6 55. Rxd7 Kxd7 56. a6 Kd6 57. b7 Kc7 58. h5 Kd6 59. b8=Q+ Kc6 60. a7 Kd5 61. a8=Q+ Kc5 62. Qa4 Kd5 63. Qe5# 1 – 0 Game 6 Stunning piece sacrifice Date: 2024-11-14 White: MAV–MCTS(𝑀 = 100) Black: Stockfish–L18 Result: 1/2 – 1/2 Online ...

    2024

  55. [63]

    Nf3 d6 3

    e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Be3 e6 7. f3 Be7 8. Qd2 b5 9. g4 Nfd7

  56. [64]

    Ba6 Bc6 66

    Qb4 Bh1 65. Ba6 Bc6 66. Ke2 Kd8 67. Bc4 Bd5 68. Ba6 Bc6 69. Kd2 Bh1 70. Ke3 Bd5 71. Bb5 Rc7 72. Ba4 Ra7 73. Qd4 Rc7 74. Qh8+ Ke7 75. Kf2 Kd6 76. Qg7 Bh1 77. Bb5 Bc6 78. Qd4+ Ke7 79. Qb4+ Kd8 80. Bc4 Bd5 81. Bd3 Ke8 82. Ke3 Rb7 83. Qd4 Be4 84. Qh8+ Ke7 85. Bc4 Rc7 86. Qg7+ Kd6 ...

  57. [65]

    g5 b4 12

    O-O-O Ne5 11. g5 b4 12. Na4 O-O 13. f4 Ned7 14. Qxb4 Nc5 15. h4 Bd7 8 rm0l0skZ 7 Z0Zbapop 6 pZ0opZ0Z 5 Z0m0Z0O0 4 NL0MPO0O 3 Z0Z0A0Z0 2 POPZ0Z0Z 1 Z0JRZBZR a b c d e f g h

  58. [66]

    f5 d5 It is still unclear what MAV–MCTS gained for a sacrificed piece.18

    Nxe6MAV–MCTS sacrifices a knight on a square defended by three pieces.Nxe6 17. f5 d5 It is still unclear what MAV–MCTS gained for a sacrificed piece.18. Qb3 Bxa4 19. Qxa4 Bc5 20. Kb1 Bxe3 21. fxe6 d4 22. exf7+ Kh8 23. e5 Rxf7 24. Bg2 Raa7 25. Rhe1 Bf2 26. e6 Rf5

  59. [67]

    Qg7 f2 69

    Nc5 gxf3 68. Qg7 f2 69. Qd7+ Kf8 70. Ne6+ Kg8 71. Qg7# 1 – 0 Game 12 Exploiting opponent’s aggressive play Date: 2024-11-14 White: MAV–MCTS(𝑀 = 2000) Black: Stockfish–L17 Result: 1 – 0 Online PGN: https://lichess.org/0vDAbIW8

    2024

  60. [68]

    Bg4 Qg5 48

    Ka1 Qf6 47. Bg4 Qg5 48. Bf5 Qf6 49. Be6 Qf1+ 50. Ka2 Qa1+ 51. Kxa1 (stalemate). 36 Mastering Board Games by External and Internal Planning with Language Models 8 0ZRa0j0Z 7 Z0ZPZ0o0 6 0Z0OBZPo 5 Z0Z0Z0ZP 4 pZ0Z0Z0Z 3 O0Z0Z0Z0 2 0O0Z0Z0Z 1 J0Z0Z0Z0 a b c d e f g h Stunning end ...

    2024

  61. [69]

    Nc3 Nc6 3

    e4 c5 2. Nc3 Nc6 3. Nf3 g6 4. d4 cxd4 5. Nxd4 Bg7 6. Be3 d6 7. h3 Nf6 8. g4 O-O 9. g5 Ne8 10. h4 Nc7 11. f4 e5 12. Nde2 f5

  62. [70]

    Qd2 d5 15

    h5 fxe4 14. Qd2 d5 15. O-O-O d4 16. hxg6This is a start of an attack where a cost of a mistake is an immediate loss.dxc3 17. Qxc3 Qe8 18. Qb3+ Be6 19. Qxb7 Nd5 20. Bc5 Rb8 21. gxh7+ Kh8 8 0s0Zqs0j 7 oQZ0Z0aP 6 0ZnZbZ0Z 5 Z0Ano0O0 4 0Z0ZpO0Z 3 Z0Z0Z0Z0 2 POPZNZ0Z 1 Z0JRZBZR a b...

  63. [71]

    Qxg7+White sacrifices a queen as the last resource. Kxg7 23. Bxf8+ Qxf8 8 0s0Z0l0Z 7 o0Z0Z0jP 6 0ZnZbZ0Z 5 Z0Zno0O0 4 0Z0ZpO0Z 3 Z0Z0Z0Z0 2 POPZNZ0Z 1 Z0JRZBZR a b c d e f g h

  64. [72]

    Rxh8 Rxh8

    h8=R White finds an opportunity to underpromote to a rook!Qxh8 25. Rxh8 Rxh8

  65. [73]

    fxe6 Nxd1 28

    f5 Ne3 27. fxe6 Nxd1 28. Kxd1 Rh1After the dust has settled, MAV–MCTS has a decisive advantage. 29. Ng3 Rg1 30. Nf5+ Kg6 31. e7 Kf7 32. g6+ Ke8 33. Ke1 Nd4 34. Ne3 Kxe7 35. c3 Nf3+ 36. Ke2 Rxg6 37. Nc4 Kf6 38. Ne3 Kg5

  66. [74]

    Be2 Nh4 41

    Kf2 Rd6 40. Be2 Nh4 41. b4 Rd2 42. a4 Nf5

  67. [75]

    Ke3 Rc2 45

    Nxf5 Kxf5 44. Ke3 Rc2 45. c4 Ra2 46. c5 Rxa4 47. b5 Ra3+ 48. Kd2 e3+ 49. Kc2 Ra2+

  68. [76]

    c6 Rd2+ 52

    Kd3 Kf4 51. c6 Rd2+ 52. Kc3 Rd8 53. Kb4 Kg3 54. Ka5 Kf2 55. Bc4 Rd4 56. b6 Rxc4 57. bxa7 e2 58. a8=Q e1=Q+ 59. Kb6 Qb4+ 60. Ka7 Qa5+ 61. Kb7 Rb4+ 62. Kc8 Qxa8+ 63. Kd7 Qg8 64. Ke7 Qb8 65. Kf6 e4 66. c7 Qg8 67. Kf5 Qf7+ 68. Kg5 Qd7 69. c8=Q Qxc8 70. Kh5 Rb1 71. Kh4 Rg1 72. Kh5 ...

    2024

  69. [77]

    g3 Nf6 3

    d4 f5 2. g3 Nf6 3. Bg2 e6 4. Nf3 Be7 5. c4 O-O 6. O-O d6 7. Nc3 Qe8

  70. [78]

    Qc2 Nbd7 10

    b4 Kh8 9. Qc2 Nbd7 10. a4 c6 11. a5 e5 12. b5 a6 13. bxc6 bxc6 14. dxe5 dxe5 37 Mastering Board Games by External and Internal Planning with Language Models 8 rZbZqs0j 7 Z0Zna0op 6 pZpZ0m0Z 5 O0Z0opZ0 4 0ZPZ0Z0Z 3 Z0M0ZNO0 2 0ZQZPOBO 1 S0A0ZRJ0 a b c d e f g h

  71. [79]

    Qxf5 MAV–MCTS grabs a pawn without fearing numerous discovered attacks. Nd5

  72. [80]

    Qxc3 e4 18

    Qc2 Nxc3 17. Qxc3 e4 18. Ng5Already on move 15, MAV–MCTS needed to assess the exchange sacrifice correctly.Bf6 19. Qa3 Bxa1

  73. [81]

    Ba3 Rg8 22

    Qxa1 Nf6 21. Ba3 Rg8 22. Bb2 h6 23. Bxf6 hxg5 24. Bxg5 Bg4 25. Re1 Qh5 26. h4 Bxe2 27. Bxe4 Bxc4 28. Qd4 Bb3 29. Bxc6 Rab8 30. Kh2 Qg6 31. Bd7 Rgf8 32. Re5 Qc2 33. Be3 Qd1 34. Rg5 Qxd4 35. Bxd4 8 0s0Z0s0j 7 Z0ZBZ0o0 6 pZ0Z0Z0Z 5 O0Z0Z0S0 4 0Z0A0Z0O 3 ZbZ0Z0O0 2 0Z0Z0O0J 1 Z0Z0...

  74. [82]

    Kg3 Rf6 43

    Bc4+ Kf8 42. Kg3 Rf6 43. Bd5 a5 44. Rh5 Ra6 45. Rh8+ Ke7 46. h5 Rd6 47. Be4 Rd8 48. Rh7 Kf6 49. f4 Rg8 50. g5+ Ke6 51. h6 gxh6

  75. [83]

    Ra6 a4 54

    Rxh6+ Ke7 53. Ra6 a4 54. g6 Be2 55. Rxa4 Kf6 56. Rb4 Rxg6+ 57. Bxg6 Kxg6 58. Rb1 Bc4

  76. [84]

    Rb1 Kf6 61

    Rb4 Be6 60. Rb1 Kf6 61. Rb4 Bc8 62. Rb1 Kf5 63. Rb8 Bd7 64. Rb7 Ba4 65. Re7 Kg6 66. Re2 Kf5 67. Ra2 Bb3 68. Ra7 Be6 69. Rb7 Bc4

  77. [85]

    Rb7 Bc4 72

    Rg7 Be6 71. Rb7 Bc4 72. Rd7 Kg6 73. Kf3 Kf5 74. Rg7 Bd3 75. Rh7 Be4+ 76. Ke3 Bc6 77. Rh3 Be8 78. Rh7 Bg6 79. Re7 Kf6 80. Rc7 Be8

  78. [86]

    Rc8 Bd7 83

    Rc4 Bb5 82. Rc8 Bd7 83. Ra8 Bh3 84. Ra2 Kf5 85. Kf3 Kf6 86. Ra8 Ke6 87. Rc8 Kf6 88. Rc4 Kf5 89. Kf2 Ke6 90. Rc2 Bf5 91. Rc7 Kd6

  79. [88]

    b5 Ke1 90

    b4 Ke2 89. b5 Ke1 90. Bg2 Ke2 91. d8=Q Ke1 92. Qe8# 1 – 0 Game 20 Endgame masterclass Date: 2024-12-01 White: MAV–MCTS(𝑀 = 2000) Black: Stockfish–L19 Result: 1 – 0 Online PGN: https://lichess.org/WjsKWo1E

    2024

  80. [89]

    Qa1 Rc5 91

    Ba6 Be4 90. Qa1 Rc5 91. Kf2 Bd5 92. Be2 Bc6 93. Kg3 Rd5 94. Qa3+ Rc5 95. Bf1 Kc7 96. Be2 Bd5 97. Kh4 Kc8 98. Qb2 Kc7 99. Kg3 Bc4

  81. [92]

    Kg2 Kc5 94

    Rb7 Kd5 93. Kg2 Kc5 94. Rf7 Bd3 95. Kf3 Kc6 96. Kg4 Be2+ 97. Kh4 Bf1 98. f5 Bb5 99. Re7 Bc4 100. Kg5 Bg8 101. Kg6 Kc5 102. Kg7 Bb3 103. Rd7 Kc6 104. Ra7 Kd5 105. Ra6 Ke4

  82. [93]

    d4 Nf6 2. c4 c5 3. d5 e5 4. Nc3 d6 5. e4 Be7

  83. [94]

    g3 O-O 8

    Nf3 Nbd7 7. g3 O-O 8. Bg2 8 rZbl0skZ 7 opZnapop 6 0Z0o0m0Z 5 Z0oPo0Z0 4 0ZPZPZ0Z 3 Z0M0ZNO0 2 PO0Z0OBO 1 S0AQJ0ZR a b c d e f g h 38 Mastering Board Games by External and Internal Planning with Language Models 8...b5 MAV–MCTS plays a pawn sacrifice similar to Benko gambit.9. c...

  84. [95]

    Re2 Kh8 22

    Re3 Nb4 21. Re2 Kh8 22. Nf1 Bg4 23. Rae3 Qe8 24. Qb3 8 rZ0Zqs0j 7 Z0Z0a0op 6 0Z0o0m0Z 5 ZNoPZ0Z0 4 PmBZpObZ 3 ZQZ0S0O0 2 0O0ZRZ0O 1 Z0A0ZNJ0 a b c d e f g h 24...Qh5 For two moves in a row MAV–MCTS estimates that the bishop is a stronger piece thantherookanddeclinestowinanexchange

  85. [96]

    Bd2 Nbxd5 27

    Re1 Rfb8 26. Bd2 Nbxd5 27. Rxe4 Bf8 28. R4e2 Nb6 29. Bf7 Qh3 30. Be6 Nxa4 31. Bxg4 Qxg4 32. Qf7 Qg6 33. Qxg6 hxg6 34. Nc7 Ra7

  86. [97]

    Bc3 Nc4 37

    Ne6 Nxb2 36. Bc3 Nc4 37. Bxf6 gxf6 38. Rc1 d5 39. Rd1 Rd7 40. Rc2 Nb6 41. Rb1 Bd6 42. Rcb2 Re8 43. Rxb6 Rxe6 44. Kf2 c4 45. Rc6 Rc7

  87. [98]

    Rxc6 Kg7 48

    Rbb6 Rxc6 47. Rxc6 Kg7 48. Kf3 Kf7 49. Ne3 Bb8 50. Rc8 Rb6 51. Nxd5 Rb3+ 52. Ke4 f5+ 53. Kd4 c3 54. Kc4 Rb7 55. Nxc3 Ba7 56. Nd5 Bg1 57. h3 Bh2 58. g4 fxg4 59. Rh8 Kg7 60. Rh4 g3 61. Rg4 Rb2 62. Kc3 Rf2 63. Ne3 Rf3

  88. [99]

    Nxg2 Rxh3 66

    Kd2 g2 65. Nxg2 Rxh3 66. Ke1 Bxf4 67. Kf2 g5 68. Nh4 Rxh4 69. Rxh4 gxh4 70. Ke1 h3 71. Kf2 Bh2 72. Kf3 Bd6 73. Kf2 Bh2 74. Kf3 Bd6

  89. [100]

    Bd1 Rc4 102

    Bf3 Bd5 101. Bd1 Rc4 102. Bh5 Bc6 103. Qg7 Rc1 104. Qe7 Rc3+ 105. Kf2 Rc2+ 106. Kg1 (threefold repetition).MAV–MCTS proves its decision to sacrifice a queen on move 14 was correct. 1/2 – 1/2 Game 5 Masterful middlegame maneuvering Date: 2024-11-12 White: MAV–MCTS(𝑀 = 2000) - m...

    2024

  90. [101]

    Kf2 Bh2 (threefold repetition).This is a well known draw because the queening square is the opposite color of the bishop.1/2 – 1/2 Game 10 Exemplary advantage conversion Date: 2024-11-14 White: MAV–MCTS(𝑀 = 2000) Black: Stockfish–L19 Result: 1 – 0 Online PGN: https://lichess.o...

    2024

  91. [102]

    d4 d5 2. c4 c6 3. Nf3 Nf6 4. e3 e6 5. Nc3 a6

  92. [103]

    Be2 Bg7 10

    b5 g6 9. Be2 Bg7 10. O-O h5 11. Rb1 O-O 12. Na4 Qe7 13. Ne5 Nxe5

  93. [104]

    Nb6 Rb8 16

    dxe5 Ng4 15. Nb6 Rb8 16. f4 Bd7 17. Bxg4 hxg4 18. Qxg4 cxb5 19. Nxd7 Qxd7 20. Bb2 b6 21. cxb6 Rxb6The material is equal, but MAV–MCTS has much more active bishop and upper-hands for a kingside attack with the minimal resources. 22. h4 Rc6 23. h5 b4 24. Rf3 Rfc8 25. hxg6 fxg6 2...

  94. [105]

    Kf2MAV–MCTS finds a beautiful maneuver which activates all its pieces at the cost of a 39 Mastering Board Games by External and Internal Planning with Language Models pawn. Rc2+ 31. Kf3 R8c4 32. Rh1+ Kg8 33. Bd4 Rxa2 8 0Z0Z0ZkZ 7 Z0Z0Z0a0 6 0Z0Z0Z0Z 5 o0ZpOpZ0 4 0orA0O0Z 3 Z0Z...

  95. [106]

    Rd6 Ke5 108

    Rb6 Bd5 107. Rd6 Ke5 108. Rxd5+ Kxd5

  96. [107]

    Ra8+ Kh7 64

    Kf5 d2 63. Ra8+ Kh7 64. Ra7+ Kh8 65. Rd7 d1=B 66. Rxd1 Kh7 67. Rh1# 1 – 0 Game 11 Controlling key squares Date: 2024-11-12 White: MAV–MCTS(𝑀 = 2000) - max scoring Black: Stockfish–L18 Result: 1 – 0 Online PGN: https://lichess.org/l2F7vJ7q

    2024

  97. [108]

    Nf3 d6 3

    e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 g6 6. Be3 Bg7 7. f3 O-O 8. Qd2 Nc6 9. O-O-O d5 10. Qe1 e5 11. Nxc6 bxc6 12. exd5 Nxd5 13. Bc4 Be6 14. Kb1 Rb8

  98. [109]

    Kg5 Kd5 111

    Kh6 Ke4 110. Kg5 Kd5 111. f6 Ke6 112. Kg6 Kd6 113. f7 Ke7 114. Kg7 Kd7 115. f8=Q Kc7 116. Qe7+ Kb6 117. Kg6 Kb5 118. Qa3 Kc4

  99. [110]

    Bc5 Rfd8 17

    Ne4 Qc7 16. Bc5 Rfd8 17. g4 a5 18. a4 h6 19. Bb3 Kh8 8 0s0s0Z0j 7 Z0l0Zpa0 6 0ZpZbZpo 5 o0Ano0Z0 4 PZ0ZNZPZ 3 ZBZ0ZPZ0 2 0OPZ0Z0O 1 ZKZRL0ZR a b c d e f g h

  100. [111]

    Additionally, it protects b2 square from any possible attack along the b line.Nf4 21

    Ba3 MAV–MCTS vacates c5 square for a knight and estimates that the control over a3-f8 diagonal is much more important than the control over g1-a7 diagonal. Additionally, it protects b2 square from any possible attack along the b line.Nf4 21. Bxe6 Nxe6 22. h4 Rxd1+ 23. Qxd1 c5 ...

  101. [112]

    Qc4 Qd7 28

    g5 h5 27. Qc4 Qd7 28. Ka2 Rd8 29. Rf1 Qc8

  102. [113]

    Qb5 Qd8 32

    Rf2 Rd4 31. Qb5 Qd8 32. Qc6 Rd1 33. Bxc5 Nxc5 34. Qxc5 Kh7 35. Re2 Rd5 36. Qc6 Rd7

  103. [114]

    Nf3 Nc6 3

    e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. O-O Be7 6. Re1 b5 7. Bb3 O-O 8. a4 b4 9. a5 d5

  104. [115]

    dxc6 Black sacrifices a knight and relies on aggressive piece play with a possible kingside attack.Bd6 12

    exd5 e4 11. dxc6 Black sacrifices a knight and relies on aggressive piece play with a possible kingside attack.Bd6 12. d4 MAV–MCTS decides to immediately activate its pieces by returning the sacrificed knight. exf3 13. Qxf3 h6 14. h3 Rb8 15. Nd2 Rb5 16. Nc4 Re8 17. Rxe8+ Qxe8 ...

  105. [116]

    Bxh6 MAV–MCTS enters a scary tactical sequence. Rf5 19. Nxd6 cxd6 20. Qg3 Nh5 21. Qe3 Qxc6 22. g4 Rf3 23. d5 Rxe3 24. dxc6 Rxh3

  106. [117]

    Bxf6 gxf6 27

    Bg5 Nf6 26. Bxf6 gxf6 27. Ra4MAV–MCTS finishes the resulting winning endgame with a nice rook lift.Kf8 28. Rxb4 Rh8 29. Rb8 Ke7

  107. [118]

    f3 Rg5 32

    Bd5 Rg8 31. f3 Rg5 32. c4 Kd8 33. b4 Re5

  108. [119]

    Qc1 Ke4 121

    Kf7 Kd4 120. Qc1 Ke4 121. Qg5 Kd4 122. Qb5 Kc3 123. Kf6 Kd2 124. Qf1 Ke3 125. Kf5 Kd4 126. Qc1 Kd5 127. Qh6 Kc4 128. Qb6 Kd5

  109. [120]

    Ra8 f5 36

    Kf2 Kc7 35. Ra8 f5 36. f4 Rxd5 37. cxd5 Kd8

  110. [121]

    Rxc8 f6 40

    g5 Ke7 39. Rxc8 f6 40. Ke3 fxg5 41. fxg5 Kf7

  111. [122]

    Rg8 Kf7 44

    Kf4 Ke7 43. Rg8 Kf7 44. Rc8 Kg7 45. Kxf5 Kh7 46. c7 Kg7 47. Rb8 Kf7 48. c8=Q Ke7 49. Qb7# 1 – 0 Game 13 Well executed caveman attack Date: 2024-11-14 White: MAV–MCTS(𝑀 = 100) Black: Stockfish–L19 Result: 1 – 0 Online PGN: https://lichess.org/gvDiGfcP

    2024

  112. [123]

    d4 Bg7 3

    e4 g6 2. d4 Bg7 3. Nc3 d6 4. Be3 a6 5. Qd2 b5 6. h4 h6 7. h5 b4 8. Nd5 c6 9. Nxb4 g5

  113. [124]

    Nd3 Nf6 12

    Ne2 a5 11. Nd3 Nf6 12. f3 Qc7 13. Ng3 c5

  114. [125]

    Be2 O-O 8 rmbZ0skZ 7 Z0l0opa0 6 0Z0o0m0o 5 Z0o0Z0oP 4 pZ0OPZ0Z 3 Z0ONAPM0 2 PO0LBZPZ 1 S0Z0J0ZR a b c d e f g h

    c3 a4 15. Be2 O-O 8 rmbZ0skZ 7 Z0l0opa0 6 0Z0o0m0o 5 Z0o0Z0oP 4 pZ0OPZ0Z 3 Z0ONAPM0 2 PO0LBZPZ 1 S0Z0J0ZR a b c d e f g h

  115. [126]

    h6 Bh8 18

    Bxg5MAV–MCTS sacrifices a bishop for a devastating attack.hxg5 17. h6 Bh8 18. Qxg5+ Kh7 19. Nf5 Nh5 20. Rxh5 f6 8 rmbZ0s0a 7 Z0l0o0Zk 6 0Z0o0o0O 5 Z0o0ZNLR 4 pZ0OPZ0Z 3 Z0ONZPZ0 2 PO0ZBZPZ 1 S0Z0J0Z0 a b c d e f g h

  116. [127]

    hxg7+ Kg8 23

    Qg7+ A stunning queen sacrifice – the cherry on top.Bxg7 22. hxg7+ Kg8 23. Rh8+ Kf7 24. Rxf8+ Ke6 25. Nxc5+ Qxc5 26. g8=Q+ Kd7 27. Rd8+ Kc7 28. Rxc8+ Kb6 29. dxc5+ Kb7 30. Bb5 Ra6 31. O-O-O Ka7 32. g3 a3 33. Nxe7 d5 34. b3 d4 35. f4 Re6 36. Re8 d3 37. 41 Mastering Board Games ...

  117. [128]

    Nc8+ Kb7 43

    Rh1 Rxc4 42. Nc8+ Kb7 43. Qg7+ Nd7 44. bxc4 Kc6 45. Kb1 Kxc5 46. Kc1 Nb8 47. Qd4+ Kc6 48. Rf8 Nd7 49. c5 Nxf8 50. Re1 Nd7 51. Re2 Kc7 52. g5 Kxc8 53. Qh8+ Kc7 54. g6 Nb8

  118. [129]

    Qa6 Kb4 131

    Qh6 Kc5 130. Qa6 Kb4 131. Qa2 Kc3 132. Ke4 Kb4 133. Kf3 Kb5 134. Qa7 Kc6 135. Ke4 Kb5 136. Qa3 Kb6 137. Kd5 Kb7 138. Qa5 Kb8

  119. [130]

    Re5 Kb7 74

    Rxd5 Kb8 73. Re5 Kb7 74. Qf2 Kc7 75. Kc2 Kb8 76. f5 Kb7 77. Qe3 Kc7 78. Qg1 Kd8 79. Qe3 Kc7 80. f6 Kc6 81. Kd2 Kb7 82. Kc2 Ka6

  120. [131]

    f7 Kc7 85

    Qg1 Kb7 84. f7 Kc7 85. Kd3 Kd8 86. Qf1 Kc7 87. f8=Q Kb7 88. Kd2 Kc7 89. Qe2 Kc6 90. Qe7 Kb6 91. Qb5#Note that MAV–MCTS was inefficient in delivering forced mates because of the extremely low simulation budget.1 – 0 Game 14 King in the center Date: 2024-11-21 White: MAV–IS(𝑏 = ...

    2024

  121. [132]

    e4 e6 2. d4 d5 3. Nd2 Nf6 4. e5 Nfd7 5. Bd3 c5 6. c3 b5

  122. [133]

    a3 Nc6 9

    Ne2 Ba6 8. a3 Nc6 9. Nf3 cxd4 10. cxd4 Qc8 11. b4 Bb7 12. O-O a5 13. bxa5 Rxa5 14. Ng5 Be7 15. Bd2 Rxa3 16. Rxa3 Bxa3 17. Nf4 MAV–IS starts a sequence that sacrifices two central pawns.Nxd4 18. Qh5 Nxe5 8 0ZqZkZ0s 7 ZbZ0Zpop 6 0Z0ZpZ0Z 5 ZpZpm0MQ 4 0Z0m0M0Z 3 a0ZBZ0Z0 2 0Z0A0O...

  123. [134]

    h3 MAV–IS plays a quite prophylactic move which resolves a potential back rank issues before proceeding with the attack.Ra8

  124. [135]

    Nh5 Rc8 29

    Bf4 Bd3 28. Nh5 Rc8 29. Be3 d4 30. Nf6+ Kc6 31. Bxd4 Qd6 32. Qe3 Kb7 33. Be5 Qe6

  125. [136]

    Qb2+ Ka6 36

    Qd4 Rc4 35. Qb2+ Ka6 36. Qa3+ Kb6 37. Qb3+ Kc5 38. Qb8 Rc1+ 39. Kh2 Rc2 40. Ng4 Bf5 41. Ne3 Rxf2 42. Qa7+ Kb5 43. Qb7+ Kc5

  126. [137]

    Qb7+ Ka4 46

    Qa7+ Kb4 45. Qb7+ Ka4 46. Qa7+ Kb4

  127. [138]

    Qa7+ (threefold repetition)

    Qb7+ Kc5 48. Qa7+ (threefold repetition). 1/2 – 1/2 Game 15 Dominating the center Date: 2024-11-21 White: Stockfish–L10 Black: MAV–IS(𝑏 = 4,𝑑 = 2) Result: 0 – 1 Online PGN: https://lichess.org/8a0RINHx

    2024

  128. [139]

    Qc7#Note that MAV–MCTS was inefficient in delivering forced mates becauseofaverylowsimulationbudget

    Kc6 Kc8 140. Qc7#Note that MAV–MCTS was inefficient in delivering forced mates becauseofaverylowsimulationbudget. 1 – 0 Game 9 Material quality over quantity Date: 2024-11-14 White: Stockfish–L19 Black: MAV–MCTS(𝑀 = 500) Result: 1/2 – 1/2 Online PGN: https://lichess.org/Mka8BrUO

    2024

  129. [140]

    Nf3 Nc6 3

    e4 c5 2. Nf3 Nc6 3. d4 cxd4 4. Nxd4 e5 5. Nb5 a6 6. Nd6+ Bxd6 7. Qxd6 Qf6 8. Qd1 Qg6

  130. [141]

    f4 d5 11. f5 42 Mastering Board Games by External and Internal Planning with Language Models 8 rZbZkZ0s 7 ZpZ0mpop 6 pZnZ0ZqZ 5 Z0ZpoPZ0 4 0Z0ZPZ0Z 3 Z0M0Z0Z0 2 POPZ0ZPO 1 S0AQJBZR a b c d e f g h 11...Nxf5 To avoid a passive defense, MAV–IS sacrifices a piece for two central ...

  131. [142]

    c4 Bg4 22

    Qxe2 h5 21. c4 Bg4 22. Qe1 Rd1 23. Be3 Rxe1 24. Bxc5 Rxa1 25. Rxa1 h4 26. Re1 Bf5

  132. [143]

    b3 hxg2 29

    Kg1 h3 28. b3 hxg2 29. Rd1 Bh3 30. Rd5 f6

  133. [144]

    Bd6 Rd3 33

    Nxh3 Rxh3 32. Bd6 Rd3 33. Kxg2 Rxd5 34. cxd5 Nd4 35. Kf2 Nf5 36. Bb4 Kd7 37. Ba3 Kc7

  134. [145]

    a4 Kc7 40

    Bc5 Kd7 39. a4 Kc7 40. Bb4 Kd7 41. Bf8 g6

  135. [146]

    Bxe7 Kxe7 44

    b4 Ne7 43. Bxe7 Kxe7 44. a5 Kd6 45. Ke1 Kxd5 46. Ke2 Kd4 47. b5 axb5 48. Kf2 (game terminated by early termination). 0 – 1 Game 16 The power of zwischenzug Date: 2024-11-21 White: MAV–IS(𝑏 = 4,𝑑 = 2) Black: Stockfish–L10 Result: 1/2 – 1/2 Online PGN: https://lichess.org/cHJzIu8F

    2024

  136. [147]

    f4 exf4 3

    e4 e5 2. f4 exf4 3. Nf3 d6 4. d4 g5 5. g3 g4 6. Nh4 f3 7. Nc3 Nf6 8. Bg5 Be7 9. Qd2 Nc6 10. O-O-O h6 11. Bxh6 Nxd4 8 rZblkZ0s 7 opo0apZ0 6 0Z0o0m0A 5 Z0Z0Z0Z0 4 0Z0mPZpM 3 Z0M0ZpO0 2 POPL0Z0O 1 Z0JRZBZR a b c d e f g h

  137. [148]

    Qxd4 Rxg7 14

    Bg7 Instead of an immediate recapture, MAV–IS complicates the game by playing an in-between move.Rg8 13. Qxd4 Rxg7 14. e5 Nh5 15. Bc4 Kf8 16. Kb1 d5 17. Nxd5 c6 18. Nxe7 Qxd4 19. Rxd4 Kxe7 20. Re1 b5 8 rZbZ0Z0Z 7 o0Z0jps0 6 0ZpZ0Z0Z 5 ZpZ0O0Zn 4 0ZBS0ZpM 3 Z0Z0ZpO0 2 POPZ0Z0O ...

  138. [149]

    Nf5+ Kf6 23

    e6Instead of retreating the attacked piece, MAV–IS plays a strong in-between move that sacrifices a piece and leads to a dynamic equality.bxc4 22. Nf5+ Kf6 23. Nxg7 Nxg7 24. Rxg4 Bxe6 25. Rf4+ Kg5 26. Rxf3 f5 27. b3 Rh8

  139. [150]

    bxc4 a5 30

    Rf2 Rh3 29. bxc4 a5 30. Re5 a4 31. Rc5 Bd7

  140. [151]

    Rxa4 Ne8 34

    Ra5 Be6 33. Rxa4 Ne8 34. Re2 Rh6 35. c5 Ng7 36. Ra7 Kf6 37. Ra8 Bc4 38. Rf8+ Kg6 39. 43 Mastering Board Games by External and Internal Planning with Language Models Rf2 Rh7 40. a4 Ne6 41. R8xf5 Ra7 42. Rf6+ Kg7 43. h4 Bd5 44. h5 Nxc5 45. Rg6+ Kh7 46. Rff6 Be4 47. Rg5 Bd5 48. R...

  141. [152]

    Kc1 Nxa4 52

    Re5 Rb7+ 51. Kc1 Nxa4 52. g4 Nb6 53. g5 Ra7 54. Kd1 Nc4 55. Re8 Kg7 56. h6+ Kh7 57. Ke1 Rb7 58. Kf2 Rd7 59. Ke2 Na3 60. Kf2 Nxc2

  142. [153]

    Kg3 Nb4 63

    Re5 Bg8 62. Kg3 Nb4 63. Re8 Bf7 64. Rb8 c5 65. Rc8 Re7 66. Rxc5 Nd3 67. Rcc6 Re3+ 68. Kh4 Ne5 69. Rc7 Kg8 70. Rc8+ Kh7 71. Rc7 Kg8 72. Rc8+ Kh7 73. Rc7 1/2 – 1/2 Game 17 Execution is stronger than a threat Date: 2024-12-01 White: MAV–MCTS(𝑀 = 2000) Black: Stockfish–L20 Result:...

    2024

  143. [154]

    d4 Bg7 3

    e4 g6 2. d4 Bg7 3. Nc3 d6 4. f4 Nc6

  144. [155]

    Nf3 a6 7

    Be3 e6 6. Nf3 a6 7. Bd3 b5 8. a4 b4 9. Ne2 Nf6 10. O-O a5 11. f5MAV–MCTS does not hesitate to start an immediate pawn break.exf5 12. exf5 O-O 13. Bg5 Ne7 14. Ng3 Ba6 15. Bxa6 Rxa6

  145. [156]

    Nd2 Ned5 18

    Qd3 Ra8 17. Nd2 Ned5 18. c3 bxc3 19. bxc3 Qd7 20. Nde4 Nxe4 21. Nxe4 Rfe8 22. h4 Rab8

  146. [157]

    Qf3 Qd7 25

    Rf2 Qc6 24. Qf3 Qd7 25. Qd3 h6 26. Bd2 c5

  147. [158]

    fxg6 fxg6 8 0s0ZrZkZ 7 Z0ZqZ0a0 6 0Z0o0Zpo 5 o0ZnZ0Z0 4 PZ0oNZ0O 3 Z0OQZ0Z0 2 0Z0A0SPZ 1 Z0Z0ZRJ0 a b c d e f g h

    Raf1 cxd4 28. fxg6 fxg6 8 0s0ZrZkZ 7 Z0ZqZ0a0 6 0Z0o0Zpo 5 o0ZnZ0Z0 4 PZ0oNZ0O 3 Z0OQZ0Z0 2 0Z0A0SPZ 1 Z0Z0ZRJ0 a b c d e f g h

  148. [159]

    Nf6+ MAV–MCTS does take time for the preparatory moves, but jumps with a knight to a square that is defended twice to open the diagonals and lines around Black’s king.Nxf6

  149. [160]

    Qxg6 Rxf6 32

    Rxf6 Re6 31. Qxg6 Rxf6 32. Rxf6 Qe7 33. Rf3 Qe8 34. Qg4 Kh8 35. Rg3 Be5 36. Bf4 dxc3

  150. [161]

    Qg6 Qd4+ 39

    Bxe5+ Qxe5 38. Qg6 Qd4+ 39. Kh1 Qxh4+

  151. [162]

    Kh2 Qe5+ 42

    Rh3 Qe1+ 41. Kh2 Qe5+ 42. Kg1 Qe1+ 43. Kh2 Qe5+ 44. Kg1 Qe1+ 45. Kh2 (threefold repetition). Perpetual check saves Black from getting mated. 1/2 – 1/2 Game 18 The dragon bishop Date: 2024-12-01 White: Stockfish–L19 Black: MAV–MCTS(𝑀 = 2000) Result: 1/2 – 1/2 Online PGN: https:...

    2024

  152. [164]

    Nb3 Qd8 13

    Kb1 Rfc8 12. Nb3 Qd8 13. Be2 Ne5 14. h4 a5 15. Nd4 a4 16. g4 a3 17. b3 Qa5 18. Ncb5 Qxd2 19. Bxd2 Nc6 20. Bc1 Nxd4 21. Nxd4 Rc3 22. Rhe1 h5 23. g5 Ne8 This seemingly unimportant knight will play a pivotal role in what is to come. 8 rZ0ZnZkZ 7 ZpZbopa0 6 0Z0o0ZpZ 5 Z0Z0Z0Op 4 0...

  153. [165]

    Bxa3 Prelude to a very unusual tactic. Rxa3 44 Mastering Board Games by External and Internal Planning with Language Models 8 0Z0ZnZkZ 7 ZpZbopa0 6 0Z0o0ZpZ 5 Z0Z0Z0Op 4 0Z0MPZ0O 3 sPs0ZPZ0 2 PZPZBZ0Z 1 ZKZRS0Z0 a b c d e f g h

  154. [166]

    forking" two rooks. Rxa2+ 26. Kxc3 8 0Z0Z0ZkZ 7 Zpmbopa0 6 0Z0o0ZpZ 5 Z0Z0Z0Op 4 0Z0MPZ0O 3 ZPJ0ZPZ0 2 rZPZBZ0Z 1 Z0ZRS0Z0 a b c d e f g h 26...Nc7 27. b4White makes a “luft

    Kb2 A lone king is "forking" two rooks. Rxa2+ 26. Kxc3 8 0Z0Z0ZkZ 7 Zpmbopa0 6 0Z0o0ZpZ 5 Z0Z0Z0Op 4 0Z0MPZ0O 3 ZPJ0ZPZ0 2 rZPZBZ0Z 1 Z0ZRS0Z0 a b c d e f g h 26...Nc7 27. b4White makes a “luft” to escape a pin from the dark-squared bishop.Ne6 The knight enters the game with a...

  155. [167]

    Kd3 Bc3 33

    Rc1 Rc8+ 32. Kd3 Bc3 33. Rf1 Bxb4 34. c3 Ba5 35. Bd1 Bb5+The light-squared bishop takes over the role of a pinner.36. c4 8 0ZrZ0ZkZ 7 ZpZ0o0Z0 6 0Z0opZpZ 5 abZ0Z0Op 4 0ZPZPZ0O 3 Z0ZKZPZ0 2 0Z0Z0Z0Z 1 Z0SBZRZ0 a b c d e f g h 36...d5 MAV–MCTS is pinning White from every angle!3...

  156. [168]

    Bc2 Kf7 49

    Kd2 Bf2 48. Bc2 Kf7 49. Ke2 Rb2 50. Kxf2 Rxc2+ 51. Kg3 Ke7 52. Kf3 Rc1 53. Kg2 d4 54. Ra4 Rd1 55. Kf3 Rh1 56. Rxd4 Rxh4 57. Ke3 e5

  157. [169]

    Kd2 Rf3 60

    Re4 Rh3+ 59. Kd2 Rf3 60. Rxe5+ Kf8 61. Re6 Kf7 62. Rf6+ Kg7 63. Ke1 h4 64. Ke2 Rg3

  158. [170]

    Kf2 Rg4 67

    Re6 h3 66. Kf2 Rg4 67. Re7+ Kg8 68. Ra7 h2 69. Ra8+ Kg7 70. Ra7+ Kg8 71. Ra1 Kf7 72. Re1 Rh4 73. Rh1 Rh3 74. Kf1 Ke6 75. Kg2 Re3

  159. [171]

    Ra1 Re8 78

    f5+ Kxf5 77. Ra1 Re8 78. Kh1 Re7 79. Ra5+ Re5 80. Ra1 Re8 81. Kxh2 Re7 82. Ra4 Re4 83. Ra3 Kg4 84. Ra1 Re3 85. Rg1+ Kf4 86. Rb1 Re2+ 87. Kh1 Re3 88. Rc1 Kg3 89. Kg1 Rb3 90. Kh1 Kh3 91. Kg1 Rb4 92. Kf2 Rf4+ 93. Ke1 Rf5

  160. [172]

    Kf1 Rg4 96

    Rc2 Rxg5 95. Kf1 Rg4 96. Re2 g5 97. Re6 Kh2 98. Rh6+ Kg3 99. Rb6 Rc4 100. Rb3+ Kh2

  161. [173]

    Kf2 Kh2 103

    Rb2+ Kh1 102. Kf2 Kh2 103. Kf1+ Kh1

  162. [174]

    Kf1+ (threefold repetition)

    Kf2 Kh2 105. Kf1+ (threefold repetition). Apeacefuloutcomefoawildgame. 1/2 – 1/2 Game 19 Taming the dragon bishop 45 Mastering Board Games by External and Internal Planning with Language Models Date: 2024-12-01 White: MAV–MCTS(𝑀 = 2000) Black: Stockfish–L19 Result: 1 – 0 Onlin...

    2024

  163. [175]

    Nf3 d6 3

    e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 g6 6. Be3 Bg7 7. f3 O-O 8. Qd2 Nc6 9. Bc4 Qa5 10. O-O-O Bd7

  164. [176]

    Bh6 MAV–MCTS immediately exchanges the favorite piece of every Dragon player.Bxh6 13

    Bb3 MAV–MCTS chooses the different approach to the Sicilian Dragon than Stockfish in Game 18.Ne5 12. Bh6 MAV–MCTS immediately exchanges the favorite piece of every Dragon player.Bxh6 13. Qxh6 Rac8 14. Nd5 Nxd5 15. exd5 Qc5 16. a3 a5 17. h4 f6 18. Rhe1 a4 19. Ba2 b5 20. Qd2 Rfe...

  165. [177]

    Rh5 Ne5 30

    Ka1 Nc4 29. Rh5 Ne5 30. c3 Nxf3 31. Nxf3 Bxg4 8 0Z0Z0skZ 7 Z0Z0o0sp 6 0Z0o0o0Z 5 ZplPZ0ZR 4 pZ0Z0ZbZ 3 O0O0ZNZ0 2 BO0Z0Z0Z 1 J0ZRL0Z0 a b c d e f g h

  166. [178]

    Qh1MAV–MCTS finds a beautiful geometry to secure a huge advantage.Qe3 8 rmblkans 7 opopopop 6 0Z0Z0Z0Z 5 Z0Z0Z0Z0 4 0Z0Z0Z0Z 3 Z0Z0Z0Z0 2 POPOPOPO 1 SNAQJBMR a b c d e f g h

  167. [179]

    Notice that a symmetric move Rf1 is the only alternative which secures the advantage

    Rh3 Another beautiful move by MAV– MCTS. Notice that a symmetric move Rf1 is the only alternative which secures the advantage. Kh8 34. Re1 Qf4 35. Bb1 Bxh3 36. Qxh3 Qg4 37. Qh1 f5 38. Rg1 Qxf3 39. Qxf3 Rxg1 40. Qh5 Re1 41. Qh4 Re2 42. Bd3 Re5 43. Bxb5 Rf6 44. Qf4 Re4 45. Qf3 R...

  168. [180]

    Be8 Rfg7 49

    Qe2 Rf7 48. Be8 Rfg7 49. Qe6 f4 50. Bf7 Kh7 51. Qf5+ R4g6 52. Qxh5+ Rh6 53. Qf5+ Kh8 54. Qc8+ Kh7 55. Qf5+ Kh8 56. Qxf4 Rf6

  169. [181]

    Bh5 Kg7 59

    Qh4+ Rh7 58. Bh5 Kg7 59. Qg4+ Kh8 60. Qxa4 Kg8 61. Qe8+ Rf8 62. Qg6+ Kh8 63. a4 Rg8 64. Qf5 Rh6 65. a5 Rf6 66. Qh3 Kg7 67. Qe3 Kf8 68. a6 e5 69. a7 Ke7 70. Qb6 Rff8 71. Qc7+ Kf6 72. Qxd6+ Kf5 73. Qe6+ Kf4 74. d6 Kg5 75. Bf7 Ra8 76. Qxe5+ Kh4 77. Qh2+ Kg4

  170. [182]

    Qh2+ Kg5 80

    Qg2+ Kh4 79. Qh2+ Kg5 80. Qg3+ Kf5 81. Bxg8 Rxa7+ 82. Kb1 Ra8 83. d7 Ke4 84. Qg4+ Ke3 85. Qd4+ Kf3 86. Bd5+ Ke2 87. Bxa8 Kf1

  171. [184]

    Nge2 a5 7

    Bd3 Nf6 6. Nge2 a5 7. O-O Bd6 8. c5 Bc7 9. f3 Nbd7

  172. [185]

    fxe4 fxe4 12

    e4 dxe4 11. fxe4 fxe4 12. Nxe4 Nxe4 13. Bxe4 Qh4 14. Nf4 O-O 15. g3 Qe7 16. Bc2 Nf6 8 rZbZ0skZ 7 Zpa0l0op 6 0ZpZpm0Z 5 o0O0Z0Z0 4 0Z0O0M0Z 3 Z0Z0Z0O0 2 POBZ0Z0O 1 S0AQZRJ0 a b c d e f g h

  173. [186]

    Nh5 A strong move that secures the 46 Mastering Board Games by External and Internal Planning with Language Models initiative. e5 18. Bg5 a4 19. Qd3 e4 20. Qe3 Bh3 21. Nxf6+ gxf6 22. Bxf6 Rxf6 23. Qg5+ Qg7 24. Qxf6 Re8 25. Qxg7+ Kxg7 8 0Z0ZrZ0Z 7 Zpa0Z0jp 6 0ZpZ0Z0Z 5 Z0O0Z0Z0...

  174. [187]

    Rae1 An unintuitive exchange sacrifice. Bxf1 27. Kxf1 Bd8 8 0Z0arZ0Z 7 ZpZ0Z0jp 6 0ZpZ0Z0Z 5 Z0O0Z0Z0 4 pZ0OpZ0Z 3 Z0Z0Z0O0 2 POBZ0Z0O 1 Z0Z0SKZ0 a b c d e f g h

  175. [188]

    Rxe4 MAV–MCTS decides to enter the opposite color bishops endgame which are notorious for their high margin of a draw. Rxe4 29. Bxe4 Bf6 8 0Z0Z0Z0Z 7 ZpZ0Z0jp 6 0ZpZ0a0Z 5 Z0O0Z0Z0 4 pZ0OBZ0Z 3 Z0Z0Z0O0 2 PO0Z0Z0O 1 Z0Z0ZKZ0 a b c d e f g h

  176. [189]

    d5 The pinnacle behind MAV–MCTS’s play, and the only winning move! When sacrificing the exchange on move 26, MAV– MCTS correctly evaluated that the resulting semi-forcing sequence will result in the winning opposite color bishops endgame. Bxb2 31. dxc6 bxc6 32. Bxc6 Ba3 33. Kg2 Bxc5

  177. [190]

    MAV–MCTS proceeds to display the winning technique.Kf6 35

    Bxa4It is known that the opposite-colored bishops endgames are winning if the distance between the passed pawns is four or more squares. MAV–MCTS proceeds to display the winning technique.Kf6 35. Bc2 h6 36. Kf3 Bg1

  178. [191]

    Kg4 Bg1 39

    a4 Bb6 38. Kg4 Bg1 39. h3 Bb6 40. Kh5 Kg7

  179. [192]

    g4 Be1 43

    Be4 Bf2 42. g4 Be1 43. Bg2 Bg3 44. a5 Be1

  180. [193]

    Be4 Bd4 47

    a6 Bf2 46. Be4 Bd4 47. Bd3 Bc5 48. Bc4 Be3

  181. [194]

    Bd3 Bg1 51

    h4 Bc5 50. Bd3 Bg1 51. Bc2 Ba7 52. Be4 Bf2

  182. [195]

    Bc2 Bc5 55

    Bb1 Bg1 54. Bc2 Bc5 55. Bb1 Bb6 56. Bc2 Bf2 57. Bb1 Bd4 58. Bc2 Bb6 59. Ba4 Bg1 60. Bd7 Be3 61. Ba4 Kf6 62. Bd1 Kf7 63. Ba4 Kf6

  183. [196]

    g5 Bxg5 66

    Bd1 Ke6 65. g5 Bxg5 66. hxg5 Kd6 67. Ba4 hxg5 68. Kxg5 Kc7 69. Kf5 Kb8 70. Ke4 Ka8 71. Ke5 Kb8 72. Bc2 Kc8 73. Kd6 Kb8 74. Bg6 Ka7

  184. [197]

    Bg6 Ka7 77

    Bd3 Kb8 76. Bg6 Ka7 77. Bd3 Kb6 78. Bf1 Ka5 79. a7 Kb4 80. a8=Q Kc3 81. Qd5 Kb2 82. Qa5 Kb1 83. Qd2 Ka1 84. Bg2 Kb1 85. Bd5 Ka1

  185. [198]

    Qa2# 1 – 0 Game 21 A menace knight Date: 2024-12-01 White: Stockfish–L19 Black: MAV–MCTS(𝑀 = 2000) Result: 0 – 1 Online PGN: https://lichess.org/wuIGT2Q5 47 Mastering Board Games by External and Internal Planning with Language Models

    2024

  186. [199]

    d4 d5 2. c4 c6 3. Nc3 e6 4. e3 f5

  187. [200]

    Bd3 a5 7

    Qc2 Nf6 6. Bd3 a5 7. b3 Ne4 8. Bb2 Nd7 9. Nh3 Bd6

  188. [201]

    c5 Bxh2+ 12

    O-O Nxc3 11. c5 Bxh2+ 12. Kxh2 Ne4 13. Rfe1 Ndf6 14. Kg1 g5 15. f3 g4 16. Nf4 gxf3 17. gxf3 Rg8+ 18. Kf1 Ng3+ 19. Kg1 Ngh5+ 20. Kf1 Ng3+ 21. Kg1 Ngh5+ 22. Kf1 Nxf4 23. exf4 Kf7 24. Qh2 b6 25. Ke2 Ba6 26. Bxa6 Rxa6 27. a4 Ra8 28. Rg1 Rb8 29. Rxg8 Qxg8 30. Rg1 Qf8

  189. [202]

    Kd3 h5 33

    Bc3 b5 32. Kd3 h5 33. Qg2 Qh6 34. axb5 Rxb5 35. Kc2 Rb8 36. Qg5 Qh7 37. Qh4 Rg8

  190. [203]

    Kb2 a4 40

    Rg5 Ra8 39. Kb2 a4 40. b4 Qh6 41. Ka2 Rh8 42. Ka3 Qf8 43. b5 Qa8 44. Rg1 Qb7 45. Rb1 cxb5 46. Be1 Qc8 47. Rb2 Qa8 48. Bc3 Qa7

  191. [204]

    Rxb8 Qxb8 51

    Rxb5 Rb8 50. Rxb8 Qxb8 51. Kxa4 Qb1 52. Qh2 Qd3 53. Qd2Even some extremely strong engines do not immediately realize that this is a losing move. 8 0Z0Z0Z0Z 7 Z0Z0ZkZ0 6 0Z0Zpm0Z 5 Z0OpZpZp 4 KZ0O0O0Z 3 Z0AqZPZ0 2 0Z0L0Z0Z 1 Z0Z0Z0Z0 a b c d e f g h 53...Qxd2 MAV–MCTS correctly...

  192. [205]

    Bh2 Kb4 69

    Kd3 Kb3 68. Bh2 Kb4 69. Bg3 Kb3 70. Bh2 Nb4+ 71. Ke3 Kc3 72. Ke2 Nc2 73. Kf2 Nxd4

  193. [206]

    Kf2 d4 76

    Ke3 Nc2+ 75. Kf2 d4 76. Kg3 d3 77. Kxh3 d2 78. Bg1 Nd4 79. Kh4 d1=Q 80. Be3 Qe1+

  194. [207]

    Kh6 Qc1 83

    Kh5 Qxe3 82. Kh6 Qc1 83. Kg5 Kc2 84. Kf6 Qa3 85. Kg7 Qe7+ 86. Kh6 Kd2 87. Kh5 Qg7 88. Kh4 Ke1 89. Kh5 Kf2 90. Kh4 Qh6# 0 – 1 Game 22 Dominating the queen Date: 2024-12-01 White: MAV–MCTS(𝑀 = 2000) Black: Stockfish–L18 Result: 1 – 0 Online PGN: https://lichess.org/UMurXJiP

    2024

  195. [208]

    d4 Nf6 2. c4 e6 3. Nf3 c5 4. d5 d6 5. Nc3 exd5 6. cxd5 g6 7. h3 Bg7 8. e4 O-O 9. Bd3 Re8

  196. [209]

    Bc2 Nc5 13

    Bf4 c4 12. Bc2 Nc5 13. Nd2 Qd7 14. Nxc4 Ncxe4 15. Nxe4 Nxe4 16. f3 Nf6 17. Nxd6 Rd8 18. Bg3 Nh5 19. Bh2 Qc7 20. Ne4 Qb6+ 21. Kh1 f5 22. Ng3 Nf6 23. Bb3 Be6 8 rZ0s0ZkZ 7 opZ0Z0ap 6 0l0ZbmpZ 5 Z0ZPZpZ0 4 0Z0Z0Z0Z 3 ZBZ0ZPMP 2 PO0Z0ZPA 1 S0ZQZRZK a b c d e f g h

  197. [210]

    Raxd1 Re8 26

    dxe6MAV–MCTS recognizes that the only way to fight for the advantage is by sacrificing the queen.Rxd1 25. Raxd1 Re8 26. e7+ Kh8

  198. [211]

    Nf1 f4 29

    Rfe1 Qb4 28. Nf1 f4 29. Bg1 h6 30. Re5 Rxe7 8 0Z0Z0Z0j 7 opZ0s0a0 6 0Z0Z0mpo 5 Z0Z0S0Z0 4 0l0Z0o0Z 3 ZBZ0ZPZP 2 PO0Z0ZPZ 1 Z0ZRZNAK a b c d e f g h

  199. [212]

    Rd4MAV–MCTS plays a zwischenzug that traps the opponent’s queen in the middle of the board!Qxb3 48 Mastering Board Games by External and Internal Planning with Language Models 8 0Z0Z0Z0j 7 opZ0s0a0 6 0Z0Z0mpo 5 Z0Z0S0Z0 4 0Z0S0o0Z 3 ZqZ0ZPZP 2 PO0Z0ZPZ 1 Z0Z0ZNAK a b c d e f g h

  200. [213]

    Instead, by taking the rook, MAV–MCTS pieces continue to dominate Black’s queen until the rest of the game.Qg8 33

    Rxe7 MAV–MCTS correctly evaluates that taking the queen only leads to equality. Instead, by taking the rook, MAV–MCTS pieces continue to dominate Black’s queen until the rest of the game.Qg8 33. Nd2 g5 34. Ne4 Nd5

  201. [214]

    hxg4 Qd8 37

    Rxb7 g4 36. hxg4 Qd8 37. Rd1 Qh4+ 38. Bh2 a6 39. Rc1 Ne7 40. b4 Bb2 41. Rd1 Bg7 42. b5 axb5 43. Rxb5 Ng8 44. Rh5 Qe7 45. Bxf4 Kh7 46. Rd6 Bb2 47. Bxh6 Nxh6 48. Rdxh6+ Kg8 49. Rg6+ Kf8 50. Rf5+ Ke8 51. Nd6+ Kd8

  202. [215]

    Rf7The queen is gone.Kxd6

    Rg8+ Kc7 53. Rf7The queen is gone.Kxd6

  203. [216]

    c4 c5 2. g3 g6 3. Bg2 Bg7 4. Nc3 Nc6 5. a3 e6

  204. [217]

    axb4 cxb4 8

    b4 Nxb4 7. axb4 cxb4 8. Nb5 Bxa1 9. Nf3 Qa5 10. O-O 8 rZbZkZns 7 opZpZpZp 6 0Z0ZpZpZ 5 lNZ0Z0Z0 4 0oPZ0Z0Z 3 Z0Z0ZNO0 2 0Z0OPOBO 1 a0AQZRJ0 a b c d e f g h 10...a6 MAV–MCTS finds the only move which securesasizableadvantageinaveryirrational position. 11. Nd6+ Ke7 12. c5 Qxc5 1...

  205. [218]

    dxc5 Qxc5 26

    Qb2 Rxc5 25. dxc5 Qxc5 26. Qxb3 Kc8 27. Qb2 Qb5 28. Qa1 d4 29. Bf1 Qc5 30. Qa2 Rd7

  206. [219]

    g4 Nf6 33

    Bh3 f5 32. g4 Nf6 33. gxf5 Kb8 34. fxg6 Rg7

  207. [220]

    Kf1 Ne4 37

    Qb3 Rxg6+ 36. Kf1 Ne4 37. Bf4+ Ka8 38. Be6 d3 39. Be3 Qxe3 40. fxe3 Nd2+ 41. Kf2 Nxb3 42. Bxb3 Rb6 43. Bc4 Rb2+ 44. Ke1 d2+

  208. [221]

    Bd3 a4 47

    Ke2 a5 46. Bd3 a4 47. Bxh7 a3 48. Bg8 a2

  209. [222]

    Kxd1 Rxa2 51

    Bxa2 d1=Q+ 50. Kxd1 Rxa2 51. Kc1 Rxh2

  210. [223]

    Kc1 b5 54

    Kb1 Ka7 53. Kc1 b5 54. Kd1 Rg2 55. Ke1 Ra2 56. e4 Kb7 57. e5 Kc7 58. Kf1 Kd7 59. e6+ Kxe6 60. Kg1 b4 61. Kf1 Rc2 62. Ke1 Kd5 63. Kd1 b3 64. Ke1 Kc6 65. Kd1 Kd5 66. Ke1 b2 67. Kf1 8 0Z0Z0Z0Z 7 Z0Z0Z0Z0 6 0Z0Z0Z0Z 5 Z0ZkZ0Z0 4 0Z0Z0Z0Z 3 Z0Z0Z0Z0 2 0orZ0Z0Z 1 Z0Z0ZKZ0 a b c d e ...

    2024

  211. [224]

    c4 dxc4 3

    d4 d5 2. c4 dxc4 3. Rxc4 Nc6 4. Nb3 e5 5. dxe5 Nxe5 6. Rc1 c5 7. f4 8 nZrablks 7 opZ0Zpop 6 0Z0Z0Z0Z 5 Z0o0m0Z0 4 0Z0Z0O0Z 3 ZNZ0Z0Z0 2 PO0ZPZPO 1 ZNSBAQJR a b c d e f g h 7...c4 Instead of retreating, Black counterat- tacks MAV–MCTS’s knight and the craziness starts. 8. fxe5 ...

  212. [225]

    Kxf2 Qa5 8 nZRZBlks 7 opZ0Zpop 6 0Z0Z0Z0Z 5 l0Z0O0Z0 4 0Z0Z0Z0Z 3 Z0Z0Z0Z0 2 0O0ZPJPO 1 ZNZ0ZQZR a b c d e f g h Despite being a queen up, Black is not better!

    Bxe8 Bxf2+ 13. Kxf2 Qa5 8 nZRZBlks 7 opZ0Zpop 6 0Z0Z0Z0Z 5 l0Z0O0Z0 4 0Z0Z0Z0Z 3 Z0Z0Z0Z0 2 0O0ZPJPO 1 ZNZ0ZQZR a b c d e f g h Despite being a queen up, Black is not better!

  213. [226]

    Qc7 Nxc8 16

    Qc1 Nb6 15. Qc7 Nxc8 16. Qxa5 Qxe8 17. Rd1 b6 18. Qb5 O-O 19. Qxe8 Rxe8 20. Rd7 a5

  214. [227]

    Nc4 Re7 23

    Na3 Kf8 22. Nc4 Re7 23. Rd8+ Re8 24. Rd7 Re6 25. Rc7 Ne7 26. Rb7 Ng6 27. Rb8+ Ke7

  215. [228]

    Rxf7 Nxe5 30

    Rb7+ Kd8 29. Rxf7 Nxe5 30. Nxe5 Rxe5 31. Rxg7 h5 32. h4 Rf5+ 33. Ke3 Re5+ 34. Kf3 Ke8

  216. [229]

    hxg5 Kf7 37

    Rg5 Rxg5 36. hxg5 Kf7 37. Ke4 b5 38. Kd5 b4 39. Kc4 Kg6 40. Kb5 b3 41. Kxa5 Kxg5 42. Ka4 h4 43. Kxb3 8 0Z0Z0Z0Z 7 Z0Z0Z0Z0 6 0Z0Z0Z0Z 5 Z0Z0Z0j0 4 0Z0Z0Z0o 3 ZKZ0Z0Z0 2 0O0ZPZPZ 1 Z0Z0Z0Z0 a b c d e f g h MAV–MCTS must settle for a draw despite being two pawns up in the pawn e...

  217. [230]

    Qb7+ Kg1 53

    Qb1+ Kg2 52. Qb7+ Kg1 53. Qb1+ Kg2

  218. [231]

    Qb7+ (threefold repetition).After all the crazinessthegameendspeacefully. 1/2 – 1/2 Game 25 Accelerated Benko gambit 51 Mastering Board Games by External and Internal Planning with Language Models Date: 2024-11-14 White: Stockfish–L20 Black: MAV–MCTS(𝑀 = 2000) Result: 1/2 – 1/...

    2024

  219. [232]

    d4 h5 2. c4 8 qabmnsks 7 opopopo0 6 0Z0Z0Z0Z 5 Z0Z0Z0Zp 4 0ZPO0Z0Z 3 Z0Z0Z0Z0 2 PO0ZPOPO 1 LBANMRJR a b c d e f g h 2...b5Even Stockfish gives the same evaluation as for the traditional Benko gambit.3. b3 Nf6

  220. [233]

    Nf3 c6 6

    h4 Qb7 5. Nf3 c6 6. e4 d6 7. Ne3 a6 8. Re1 Ng4 9. e5 dxe5 10. Nxe5 Nxe5 11. dxe5 g6 12. O-O Ba7 13. f4 bxc4 14. Kh1 cxb3 15. axb3 Ne6

  221. [234]

    8 0ZbZ0sks 7 aqZ0opZ0 6 pZpZnZpZ 5 Z0Z0OPZp 4 0Z0Z0Z0O 3 ZPZ0M0Z0 2 0Z0Z0ZPZ 1 LBA0SRZK a b c d e f g h 16...Qb4 But, MAV–MCTS finds astonishing resources to keep the game alive!17

    f5The position seems dire for MAV–MCTS. 8 0ZbZ0sks 7 aqZ0opZ0 6 pZpZnZpZ 5 Z0Z0OPZp 4 0Z0Z0Z0O 3 ZPZ0M0Z0 2 0Z0Z0ZPZ 1 LBA0SRZK a b c d e f g h 16...Qb4 But, MAV–MCTS finds astonishing resources to keep the game alive!17. Qa4 Qxa4 18. bxa4 Bxe3 19. Rxe3 gxf5 20. Bxf5 Rd8 21. R...

  222. [235]

    Bb3 Rb1 37

    Bd6 Ra1 36. Bb3 Rb1 37. Bg8 Ra1 38. Bb3 Rb1 39. Bg8 Ra1 40. Bb3 (threefold repetition). After surviving some terrifying moments in the middlegame, MAV–MCTS lives to fight another day. 1/2 – 1/2 Game 26 Delayed Benko gambit Date: 2024-11-14 White: Stockfish–L19 Black: MAV–MCTS(...

    2024

  223. [236]

    b3 c5 2. e4 h5 3. Ne3 b5 4. c4 8 bansqmks 7 o0Zpopo0 6 0Z0Z0Z0Z 5 Zpo0Z0Zp 4 0ZPZPZ0Z 3 ZPZ0M0Z0 2 PZ0O0OPO 1 ABMRL0JR a b c d e f g h 4...h4 Similar to Game 2, MAV–MCTS pushes b and h pawns.5. O-O e5 6. Nd5 Nb6 7. Bd3 Ne6

  224. [237]

    Ne2 Re8 10

    Qe3 Qf8 9. Ne2 Re8 10. Rde1 Rh6 11. Qf3 Nd4 12. Qh3 Bb7 13. Ne3 bxc4 14. bxc4 Bc8 15. Nxd4 exd4 16. Ng4 Rh8 17. f4 d6 18. f5 Ba6 19. f6 Bc8 20. e5 Rxe5 21. Qf3 g6 22. Bb2 Rg5 23. h3 Bxg4 24. hxg4 Nd7 25. Qb7 Ne5 26. Be2 h3

  225. [238]

    d3 Nxg4 29

    g3 d5 28. d3 Nxg4 29. Bxg4 Rxg4 30. Qd7 Rxg3+ 31. Kh1 52 Mastering Board Games by External and Internal Planning with Language Models 8 0a0Z0lks 7 o0ZQZpZ0 6 0Z0Z0OpZ 5 Z0opZ0Z0 4 0ZPo0Z0Z 3 Z0ZPZ0sp 2 PA0Z0Z0Z 1 Z0Z0SRZK a b c d e f g h 31...Rg2 MAV–MCTS again finds a saving ...

  226. [239]

    Kh1 Rh2+ 35

    Kg1 Rg2+ 34. Kh1 Rh2+ 35. Kg1 Rg2+ 36. Kh1 (threefold repetition). 1/2 – 1/2 Game 27 Not fearing ghosts Date: 2024-11-14 White: MAV–MCTS(𝑀 = 1000) Black: Stockfish–L19 Result: 1 – 0 Online PGN: https://lichess.org/4EMehaRm MBSKAQMR

    2024

  227. [240]

    c4 f6 2. f4 c6 3. Nb3 O-O-O 4. d4 Bh5 5. Bb4 g5 6. f5 Qg7 7. Bd3 Bf4 8. Rc3 Nh6 9. g3 Ng4

  228. [241]

    Kd2 e5 12

    Ke1 Bb8 11. Kd2 e5 12. d5 e4 13. Bxe4 Rhe8

  229. [242]

    Rc2 Ne3 8 nZksrZ0Z 7 opZpZ0lp 6 0ZpZ0o0Z 5 Z0ZPaPob 4 0APZ0Z0Z 3 ZNZ0mBO0 2 PORJPZ0O 1 Z0Z0ZQMR a b c d e f g h

    Bf3 Be5 15. Rc2 Ne3 8 nZksrZ0Z 7 opZpZ0lp 6 0ZpZ0o0Z 5 Z0ZPaPob 4 0APZ0Z0Z 3 ZNZ0mBO0 2 PORJPZ0O 1 Z0Z0ZQMR a b c d e f g h

  230. [243]

    Kxe3MAV–MCTS correctly evaluates that it is safe to capture the knight and to expose its king to a various discovered attacks in the center of the board. Bd6+ 17. Kf2 Bxf3 18. Nxf3 Bxb4 19. c5 Nc7 20. a3 Qe7 21. Qc1 Nxd5

  231. [244]

    Rd1 Qe4 24

    axb4 Nxb4 23. Rd1 Qe4 24. Nbd4 Nxc2 25. Qxc2 Qxc2 26. Nxc2 Re4 27. Nd2 Ra4 28. b3 Ra5 29. Ne4 d5 30. Nd6+ Kc7 31. Nd4 Rxd6

  232. [245]

    g4 Ra6 34

    cxd6+ Kd7 33. g4 Ra6 34. h4 gxh4 35. g5 fxg5 36. f6 Ke8 37. Ne6 b5 38. Nc7+ Kd7 39. f7 Kxd6 40. Nxd5 c5 41. f8=Q+ Ke5 42. Qe8+ Re6 43. Qh8+ Kf5 44. Qxh7+ Ke5 45. Qg7+ Ke4 46. Qh7+ Ke5 47. Qg7+ Kf5 48. e4+ Kg4

  233. [246]

    Rg1+ Kh5 50. Rxg5# Once MAV–MCTS obtained the advantage in the middlegame, it never let it slip.1 – 0 Game 28 Finishing games in style Date: 2024-11-14 White: Stockfish–L18 Black: MAV–MCTS(𝑀 = 100) Result: 0 – 1 Online PGN: https://lichess.org/kaZI3lB6 ARJRMBMQ

    2024

  234. [247]

    e4 g5 2. g3 53 Mastering Board Games by External and Internal Planning with Language Models 8 bsks0anl 7 opopopZp 6 0Z0m0Z0Z 5 Z0Z0Z0o0 4 0Z0ZPZ0Z 3 Z0Z0Z0O0 2 POPO0O0O 1 ARJRMBMQ a b c d e f g h 2...Nd6 A move that breaks a traditional wisdom not to block central pawns with p...

  235. [248]

    a4 Nf6 12

    Nh5 Bd4 11. a4 Nf6 12. Nhf4 e5 13. Ne2 d6

  236. [249]

    Rxh1 b6 16

    a5 Qxh1 15. Rxh1 b6 16. Bxa8 Rxa8 17. c3 e4 18. cxd4 exd3 19. Nf4 Kd7 20. b4 Rh8 21. Re1 Rae8 22. Rxe8 Rxe8 23. axb6 Re1+ 24. Kb2 Rxb1+ 25. Kxb1 cxb6 26. Nxd3 Kc6 27. Kc2 Ne4

  237. [250]

    d5 Kc4 30

    Kd1 Kb5 29. d5 Kc4 30. Ke2 Ng5 31. Bc3 Ne4 32. Bb2 Ng5 33. Ne1 Kxb4 34. Kd3 Kc5 35. Bf6 Nh7 36. Bc3 Kxd5 37. Bh8 Ng5 38. Ng2 Ne6

  238. [251]

    gxh4 Nf4+ 41

    Nh4 Nxh4 40. gxh4 Nf4+ 41. Kc2 Ke4 42. Bf6 b5 43. Kb3 a5 44. Be7 d5 45. Kc2 a4 46. Bd6 d4 47. d3+ Nxd3 48. h5 Ne1+ 49. Kd2 Nf3+ 50. Kc2 Kf5 51. h6 Kg6 52. h7 Kxh7 53. Kd3 Kg8 54. Bb4 Ne5+ 55. Kxd4 Nc6+ 56. Kc3 Nxb4 57. Kxb4 Kg7 58. Kc3 Kf6 59. Kb4 Kg5 60. Kc3 a3 61. Kc2 b4 62....

  239. [252]

    Kc4 Qe5 70

    Kd3 g3 69. Kc4 Qe5 70. Kb4 Qg7 71. Kc4 b2

  240. [253]

    Kc4 b1=Q 74

    Kb4 g2 73. Kc4 b1=Q 74. Kc5 Qd7 75. Kc4 8 0Z0Z0Z0Z 7 Z0ZqZ0Z0 6 0Z0Z0Z0Z 5 Z0Z0Z0Z0 4 0ZKZ0Z0Z 3 Z0Z0Z0Z0 2 0Z0Z0jpZ 1 ZqZ0Z0Z0 a b c d e f g h 75...g1=R Like in the traditional chess game 23, MAV–MCTS once again shows its prefer- ence for rook underpromotions.76. Kc3 Rc1# 0 –...

    2024

  241. [254]

    d4 d5 2. h4 b6 3. h5 Nd7 4. c4 e6 5. cxd5 exd5 6. h6 g6 8 bZrmqaks 7 o0onZpZp 6 0o0Z0ZpO 5 Z0ZpZ0Z0 4 0Z0O0Z0Z 3 Z0Z0Z0Z0 2 PO0ZPOPZ 1 ANSNLBJR a b c d e f g h

  242. [255]

    Qxe4 dxe4 9

    e4MAV–MCTS pushes a pawn on the most 54 Mastering Board Games by External and Internal Planning with Language Models protected square on the board to secure the piece activity.Qxe4 8. Qxe4 dxe4 9. Nbc3 Bd6

  243. [256]

    Nb5 Bf4 12

    Ne3 Bb7 11. Nb5 Bf4 12. g3 Bxe3 13. fxe3 Ne6 14. b3 O-O 15. a4 a6 8 0ZrZ0skZ 7 ZbonZpZp 6 po0ZnZpO 5 ZNZ0Z0Z0 4 PZ0OpZ0Z 3 ZPZ0O0O0 2 0Z0Z0Z0Z 1 A0S0ZBJR a b c d e f g h

  244. [257]

    Na7Not your everyday knight route.Rce8

  245. [258]

    Ne5 Rd8 19

    Nc6 Nf6 18. Ne5 Rd8 19. Bc4 Ng5 20. Kf2 c6

  246. [259]

    Be2 Nd5 23

    Bb2 b5 22. Be2 Nd5 23. Nxc6 Rd6 24. Ne5 Rf6+ 25. Ke1 Rd8 26. axb5 axb5 27. Rc5 Nxe3

  247. [260]

    Kd2 Nf5 30

    Rxb5 Ba8 29. Kd2 Nf5 30. Ng4 Rfd6 31. Rc1 e3+ 32. Ke1 Bf3 33. d5 Bxg4 34. Bxg4 Nxh6 35. Be2 Ne4 36. Be5 R6d7 37. Rb4 Nd6 38. Bf6 Re8

  248. [261]

    Rb6 Ng8 41

    g4 Kf8 40. Rb6 Ng8 41. Bb2 Red8 42. Ba3 Kg7 43. Bf3 Ne7 44. Bb2+ Kf8 45. Ba3 Kg8 46. Rc3 h5 47. gxh5 g5 48. Rxe3 Nxd5 49. Bxd5 Nf5

  249. [262]

    Bxd7 Nc2+ 52

    Bc6 Nxe3 51. Bxd7 Nc2+ 52. Kd2 Nxa3 53. Rd6 Rb8 54. Ba4 Nb1+ 55. Kc2 Na3+ 56. Kc1 Kh8 8 0s0Z0Z0j 7 Z0Z0ZpZ0 6 0Z0S0Z0Z 5 Z0Z0Z0oP 4 BZ0Z0Z0Z 3 mPZ0Z0Z0 2 0Z0Z0Z0Z 1 Z0J0Z0Z0 a b c d e f g h

  250. [263]

    Rxf5 g4 59

    Rd5The Black’s knight is trapped on the edge of the board!f5 58. Rxf5 g4 59. Rg5 Rg8

  251. [264]

    Kd2 Kf8 62

    Rxg8+ Kxg8 61. Kd2 Kf8 62. Ke3 Kg7 63. Kf4 Nc2 64. Kxg4 Kf6 65. Be8 Ne3+ 66. Kf4 Nd5+ 67. Ke4 Nc7 68. Bg6 Na6 69. Kd4 Kg5 70. Kc4 Kh6 71. Kb5 Nc7+ 72. Kc6 Na6 73. Kb5 Nc7+ 74. Kc5 Kg7 75. Bd3 Kh6 76. Kd6 Ne8+

  252. [265]

    Bg6 Nxh5 79

    Ke7 Ng7 78. Bg6 Nxh5 79. Bxh5 Kg5 80. Bd1 Kh4 81. Ke6 Kg5 82. Ke5 Kg6 83. b4 Kg5

  253. [266]

    Kf6 Kh7 86

    b5 Kh6 85. Kf6 Kh7 86. b6 Kg8 87. b7 Kh7

  254. [267]

    Rh8# 1 – 0 Game 30 Complications never end Date: 2024-11-14 White: MAV–MCTS(𝑀 = 100) Black: Stockfish–L19 Result: 1/2 – 1/2 Online PGN: https://lichess.org/iSEAtCyD SKLBSNAN

    b8=R Kh6 89. Rh8# 1 – 0 Game 30 Complications never end Date: 2024-11-14 White: MAV–MCTS(𝑀 = 100) Black: Stockfish–L19 Result: 1/2 – 1/2 Online PGN: https://lichess.org/iSEAtCyD SKLBSNAN

    2024

  255. [268]

    e4 Nhg6 2. d4 c5 3. dxc5 Qxc5 4. Nhg3 Qa5 5. c3 Bc7 6. Ne3 Nf4 7. Bc2 e6 8. a4 O-O-O 9. f3 Qa6 10. Rd1 f5 11. a5 d5 12. exd5 g6 13. Ba4 Re7 14. Nc2 Nxd5 15. Na3 e5 16. Nb5 Ne6 17. Nxa7+ Kb8 18. Bb5 8 0j0s0ZbZ 7 Mpa0s0Zp 6 qZ0ZnZpZ 5 OBZnopZ0 4 0Z0Z0Z0Z 3 Z0O0ZPM0 2 0O0Z0ZPO 1 ...

  256. [269]

    Nd4MAV–MCTS adds fuel to the fire.Rc7

  257. [270]

    Qg5 Bb6 28

    cxd5 Bxd5 27. Qg5 Bb6 28. Be3 Kb7 29. Bf4 Qe7 30. Qxe7 Rxe7 31. Bg5 Red7 32. Bxd8 Rxd8 33. b4 Nd3 34. Nc2 Nf2 35. Ba6+ Kc7

  258. [271]

    Nxe3 Be4+ 38

    Rd2 e3 37. Nxe3 Be4+ 38. Kc1 Nd3+ 39. Bxd3 Bxe3 40. Kc2 Bxd2 41. Bxe4The dust has finally settled. Bxb4 42. Ra7+ Kb6 43. Rxh7 g5 44. Rb7+ Ka5 45. Bc6 Rd2+ 46. Kb3 Rd3+

  259. [272]

    Kd5 Ba3 49

    Kc4 Rc3+ 48. Kd5 Ba3 49. Rb5+ Ka6 50. Rb1 g4 51. Bd7 Rc2 52. Ra1 Rd2+ 53. Ke5 Kb6

  260. [273]

    Rb3+ Kc7 56

    Rxa3 Rxg2 55. Rb3+ Kc7 56. Ba4 Rxh2 57. Rc3+ Kb8 58. Kd6 Rd2+ 59. Kc6 Rd8 60. Rb3+ Kc8 61. Bb5 Rd1 62. Re3 Rc1+ 63. Kb6 Rd1

  261. [274]

    Re3 Rc1 66

    Rc3+ Kb8 65. Re3 Rc1 66. Re7 Rc2 67. Re1 Rc3 68. Re4 Rc1 69. Bc4 Rb1+ 70. Bb5 Rc1 71. Bc4 Rb1+ 72. Bb5 Rc1 (threefold repetition). MAV–MCTS decides to call it a day.1/2 – 1/2 Game 31 Only active pieces count Date: 2024-11-14 White: Stockfish–L20 Black: MAV–MCTS(𝑀 = 100) Result...

    2024

  262. [275]

    e4 e5 2. b3 b6 3. d3 d6 4. f4 exf4 5. Ne2 Ne6 8 bsksqZna 7 o0o0Zpop 6 0o0onZ0Z 5 Z0Z0Z0Z0 4 0Z0ZPo0Z 3 ZPZPZ0Z0 2 PZPZNZPO 1 ARJRLNZB a b c d e f g h

  263. [276]

    g3MAV–MCTS sacrifices a pawn for a rapid development. fxg3 7. Qxg3 Ne7 8. Ne3 d5 9. exd5 Nxd5 10. Bxd5 Bxd5 11. Rf1 Bb7 12. Kd2 Rd7 13. Rbe1 g6 14. Bxh8 Qxh8 15. Nc3 O-O-O

  264. [277]

    Nf6 Rd4 18

    Ng4 Qg7 17. Nf6 Rd4 18. Nb5 Rb4 8 0Zks0Z0Z 7 obo0Zplp 6 0o0ZnMpZ 5 ZNZ0Z0Z0 4 0s0Z0Z0Z 3 ZPZPZ0L0 2 PZPJ0Z0O 1 Z0Z0SRZ0 a b c d e f g h

  265. [278]

    Re7 Qh6+ 21

    Nxc7 A stunning piece sacrifice that is hard to grasp.Nxc7 20. Re7 Qh6+ 21. Kd1 Ne6 56 Mastering Board Games by External and Internal Planning with Language Models 8 0Zks0Z0Z 7 obZ0SpZp 6 0o0ZnMpl 5 Z0Z0Z0Z0 4 0s0Z0Z0Z 3 ZPZPZ0L0 2 PZPZ0Z0O 1 Z0ZKZRZ0 a b c d e f g h

  266. [279]

    Qxg5 Nxg5 24

    c4 MAV–MCTS point is finally revealed – the Black’s rook is isolated from the rest of the board!Qg5 23. Qxg5 Nxg5 24. Kc2 Nf3 25. Kc3 a5 26. Rxb7 Nxh2 27. Rf2 Kxb7 28. Rxh2 b5 29. a3 8 0Z0s0Z0Z 7 ZkZ0ZpZp 6 0Z0Z0MpZ 5 opZ0Z0Z0 4 0sPZ0Z0Z 3 ZPJPZ0Z0 2 PZ0Z0Z0S 1 Z0Z0Z0Z0 a b c ...

  267. [280]

    Rg4 Ke6 41

    Rg8 Ke5 40. Rg4 Ke6 41. Rh4 a4 42. Rh2 Rg1 43. Ne2 Rb1 44. Rh5 a3 45. Ra5 Ra1 46. Nd4+ Kd6 47. Rd5+ Ke7 48. Kb3 a2 49. Nc6+ Ke6 50. Rh5 Rg1 51. Kxa2 Rg2+ 52. Ka3 Kf6 53. Rh1 Rg3+ 54. Ka4 Kf5 55. Ka5 Ra3+ 56. Kb6 Ra8 57. Rh5+ Ke4 58. Rh6 Ra1 59. Rh8 Rb1+

  268. [281]

    Na7 Ra2+ 62

    Ka6 Rb2 61. Na7 Ra2+ 62. Kb7 Rb2+ 63. Kc8 Rb1 64. Rh6 Kf4 65. Ra6 Rh1 66. Kd8 Ke3

  269. [282]

    Ra1 Kd3 69

    Ra5 Rh7 68. Ra1 Kd3 69. Nc8 Kc3 70. Rg1 Rh5 71. Kd7 Rh7+ 72. Ke6 Kc2 73. Rg8 Kd2 74. Rg5 Rh6+ 75. Kd7 Rh7+ 76. Kd6 Rh4 77. Rg8 Ke3 78. Na7 Kf4 79. Kc6 Rh6+ 80. Kb5 Rh1 81. Kc6 Rh6+ 82. Kb5 Ke4 83. Kb4 Kd5 84. Nc8 Rh1 85. Rg5+ Kd4 86. Kb5 Rh8 87. Na7 Ke3

  270. [283]

    Rg8 Ra1 90

    Kb6 Rh1 89. Rg8 Ra1 90. Rg4 Rc1 91. Rg6 Rb1+ 92. Kc6 Ra1 93. Rg7 Ke2 94. Rg2+ Kf1

  271. [284]

    Rg4 Rc1+ 97

    Rg7 Rb1 96. Rg4 Rc1+ 97. Kb7 Rb1+ 98. Ka6 Ra1+ 99. Kb5 Rxa7 100. Rg6 Ra8 101. Kc5 Kf2 102. Rb6 Ke3 103. Rb1 Ra5+ 104. Kd6 Ke4

  272. [285]

    Rb6 Ra1 107

    Rb4+ Kf5 106. Rb6 Ra1 107. Kd5 Rd1+

  273. [286]

    Kd4 Ra4+ 110

    Kc4 Ra1 109. Kd4 Ra4+ 110. Kd5 Ra5+

  274. [287]

    Rh6 Ra4+ 113

    Kd4 Ra1 112. Rh6 Ra4+ 113. Kd3 Kg5

  275. [288]

    Ke4 Kg6 116

    Rh7 Ra1 115. Ke4 Kg6 116. Rh2 Re1+ 117. Kd3 Ra1 118. Ke4 Kf7 119. Rh7+ Kg6 120. Rh2 (threefold repetition).An extraordinary game despite the outcome!1/2 – 1/2 Game 32 Returning knight Date: 2024-11-14 White: Stockfish–L19 Black: MAV–MCTS(𝑀 = 500) Result: 1/2 – 1/2 Online PGN: ...

    2024

  276. [289]

    f4 exf4 3

    e4 e5 2. f4 exf4 3. Qxf4 d5 4. O-O Ne6 5. Qh4 d4 6. c3 c5 7. b4 dxc3 8. dxc3 Nb6 9. Ne3 cxb4

  277. [290]

    Nb3 h5 12

    cxb4 Qxb4 11. Nb3 h5 12. Qf2 f6 13. e5 Bxe5 14. Nf5 Nc4 15. Bd3 Bd7 16. Bxc4 Qxc4

  278. [291]

    Nd6 Qg4 19

    Rxe5 fxe5 18. Nd6 Qg4 19. Qf7+ Kh7 8 0Z0ZrZ0s 7 opZbZQok 6 0Z0MnZ0Z 5 Z0Z0o0Zp 4 0Z0Z0ZqZ 3 ZNZ0Z0Z0 2 PZ0Z0ZPO 1 Z0A0ZRJ0 a b c d e f g h

  279. [292]

    Ne4White finds a cute resource to force the draw. Qxe4 21. Qxh5+ Kg8 22. Qf7+ 57 Mastering Board Games by External and Internal Planning with Language Models Kh7 23. Qh5+ Kg8 24. Qf7+ Kh7 25. Qh5+ (threefold repetition). 1/2 – 1/2 Game 33 Harrys’ deadlock Date: 2024-11-14 Whit...

    2024

  280. [293]

    h4 h5 8 blranmks 7 opopopo0 6 0Z0Z0Z0Z 5 Z0Z0Z0Zp 4 0Z0Z0Z0O 3 Z0Z0Z0Z0 2 POPOPOPZ 1 AQSBMNJR a b c d e f g h Pushing h-pawns is one of the best ways to start the game with this piece configuration according to strong engines.2. b4 b5 3. d3 a5

  281. [294]

    Nd2 Ng6 6

    c3 e6 5. Nd2 Ng6 6. e3 axb4 7. cxb4 Bxh4 8. Rh3 Qb6 9. g3 Bg5 10. Bd4 Qd6 11. Rxh5 e5 12. Ba1 Rxh5 13. Bxh5 Qe6 14. Bf3 Bxf3 15. Nexf3 Be7 16. Qb2 Ra8 17. Rc2 Qd5 18. Qb3 Qxb3 19. Nxb3 f6 20. Bc3 Ra4 21. d4 Bxb4 22. dxe5 Bxc3

  282. [295]

    Rc5 Nxe5 25

    Rxc3 Rxa2 24. Rc5 Nxe5 25. Nxe5 8 0Z0ZnZkZ 7 Z0opZ0o0 6 0Z0Z0o0Z 5 ZpS0M0Z0 4 0Z0Z0Z0Z 3 ZNZ0O0O0 2 rZ0Z0O0Z 1 Z0Z0Z0J0 a b c d e f g h 25...d6 This zwischenzug, instead of immedi- ately recapturing the knight, is the only way to secure the advantage.26. Rxb5 fxe5 27. f4 exf4 ...

  283. [296]

    Kd7 Rg1 51

    Ke7 Nf5+ 50. Kd7 Rg1 51. Kc7 Ke5 52. Kc6 Ne7+ 53. Kc5 Rc1+ 54. Kb4 Kd4 55. Kb3 Re1

  284. [297]

    Kb3 Ra1 58

    Kc2 Nc8 57. Kb3 Ra1 58. Kc2 Nb6 59. Kb2 Rg1 60. Ka3 Kc3 61. Ka2 Nc8 62. Ka3 Ra1# 0 – 1 Game 34 Fianchettoing the rook Date: 2024-11-14 White: Stockfish–L20 Black: MAV–MCTS(𝑀 = 2000) Result: 1/2 – 1/2 Online PGN: https://lichess.org/S9V6YA1i LBMRAKSN 58 Mastering Board Games by...

    2024

  285. [298]

    O-OOnce in a lifetime opportunities should be taken.Ng6 2. c4 c5 3. Ng3 Nb6 4. d4 cxd4 5. Ba5 Ne5 6. c5 Nc6 7. b4 Nxa5 8. bxa5 Nc4 9. a6 d5 10. cxd6 Nxd6 11. Rxd4 Bc7 12. Rh4 h6 13. a4 bxa6 14. Nd3 Qd5 15. Ba2 Qa5 16. Rc1 Bb6

  286. [299]

    Rf4 8 0Z0sbjrZ 7 o0Z0opZ0 6 pa0m0Zpo 5 l0Z0Z0Z0 4 PZ0Z0S0Z 3 ZBZNZ0M0 2 0Z0ZPOPO 1 L0S0Z0J0 a b c d e f g h 18...Rg7 MAV–MCTS decides to fianchetto the rook

    Bb3 g6 18. Rf4 8 0Z0sbjrZ 7 o0Z0opZ0 6 pa0m0Zpo 5 l0Z0Z0Z0 4 PZ0Z0S0Z 3 ZBZNZ0M0 2 0Z0ZPOPO 1 L0S0Z0J0 a b c d e f g h 18...Rg7 MAV–MCTS decides to fianchetto the rook. 19. Ne5 Kg8 20. e3 Kh7 21. h4 h5 22. Nc6 Bxc6 23. Rxc6 e6 24. Bc2 Kg8 25. Ne4 Nxe4 26. Rxe4 Qd2 27. Qf6 Kh7 ...

  287. [300]

    Rc4 Rc8 32

    Qf6 Qd7 31. Rc4 Rc8 32. Ree4 Kh7 33. g3 Rc7 34. Rxc7 Qxc7 35. Bd3 Qd7 36. Bb5 Qd5

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    URLhttps://arxiv.org/abs/2203. 14465. E. Zelikman, G. Harik, Y. Shao, V. Jayasiri, N. Haber, and N. D. Goodman. Quiet-star: Lan- guage models can teach themselves to think before speaking, 2024. URLhttps://arxiv. org/abs/2403.09629. Z. Zeng, Y. Liu, Y. Wan, J. Li, P. Chen, J. ...

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    com/GPTsChessEloRatingLegalMoves/

    URL https://blog.mathieuacher. com/GPTsChessEloRatingLegalMoves/. R. Agarwal, A. Singh, L. M. Zhang, B. Bohnet, S. Chan, A. Anand, Z. Abbas, A. Nova, J. D. Co- Reyes,E.Chu,etal. Many-shotin-contextlearn- ing. arXiv preprint arXiv:2404.11018, 2024. R. Aksitov, S. Miryoosefi, Z....

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    URLhttps://arxiv.org/abs/2402. 11291. O. Gramopadhye, S. S. Nachane, P. Chanda, G. Ramakrishnan, K. S. Jadhav, Y. Nandwani, D. Raghu, and S. Joshi. Few shot chain-of- thought driven reasoning to prompt llms for open ended medical question answering.arXiv preprint arXiv:2403.04...

    2024 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.