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Enigmata: Scaling Logical Reasoning in Large Language Models with Synthetic Verifiable Puzzles

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A 32B model trained on auto-generated puzzles with rule-based verifiers surpasses o1 and o3-mini-high on puzzle benchmarks, and the puzzle data also transfers to math and STEM.

desk verdict A genuinely useful generator-verifier puzzle suite with a credible ARC-AGI signal, but the unverified verifiers and in-distribution eval make the headline margins softer than they look. read the letter →

arxiv 2505.19914 v2 pith:GARLY3AO submitted 2025-05-26 cs.CL cs.AI

Jiangjie Chen , Qianyu He , Siyu Yuan , Aili Chen , Zhicheng Cai , Weinan Dai , Hongli Yu , Qiying Yu
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Xuefeng Li Jiaze Chen Hao Zhou Mingxuan Wang
This is my paper · ORCID
classification cs.CLcs.AI
keywords logicalreasoningpuzzlereinforcementlearningwithverifiablerewardsrule-basedverifierssyntheticdatagenerationmulti-tasklargemodelsARC-AGI
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

Enigmata argues that puzzle reasoning—the logic problems humans can solve without specialist knowledge—can be trained into large language models at scale using the same verifiable-reward reinforcement learning that powers strong math and coding models. The paper builds 36 puzzle tasks across seven categories, each paired with an automatic generator that controls difficulty and a rule-based verifier that grades answers instantly. From this data, after rejection fine-tuning and multi-task reinforcement learning, it produces Qwen2.5-32B-Enigmata, which it reports surpasses o1 and o3-mini-high on its Enigmata-Eval benchmark and on ARC-AGI, while generalizing to out-of-domain puzzles and retaining math ability. When the same synthetic puzzle data is added to the training of a much larger mixture-of-experts reasoning model, the paper reports further gains on AIME, BeyondAIME, and GPQA Diamond. The intended upshot is that knowledge-free puzzle data is a cheap and scalable ingredient for strengthening general reasoning.

What carries the argument

The load-bearing object is the generator–verifier pair: for 30 of the 36 tasks a program generates unlimited puzzle instances while varying explicit difficulty parameters such as grid size, mask rate, or step count, and for all 36 tasks a rule-based verifier instantly judges a response correct or incorrect, producing the reward for PPO-style reinforcement learning. This is coupled to a two-stage training recipe: rejection fine-tuning, in which eight candidate solutions per puzzle are generated and the correct ones are used for supervised fine-tuning alongside math problems in a one-to-one ratio, followed by multi-task reinforcement learning with the VC-PPO variant. A second design choice, multi-stage RL, trains first on ARC-AGI and AIME problems and then introduces Enigmata data, which the paper shows is decisive for high-complexity tasks such as Light Up.

What would settle it

Have an independent solver or a human auditor re-judge a random sample of the 4,758 Enigmata-Eval instances and compare their verdicts against the paper's verifiers; any measurable disagreement would shift the reported accuracies. A complementary check is to re-run the full training recipe using only the released verifier and generator code and see whether the resulting model still reaches 32.8% on ARC-AGI 1 and 62.6% on Enigmata-Eval.

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Extended reading notes

Core claim

The paper's central claim is that a diverse collection of generator–verifier pairs, each producing unlimited puzzles with controlled difficulty and instant rule-based grading, is enough to make a 32B model a stronger puzzle reasoner than much larger frontier models. Concretely, it reports that Qwen2.5-32B-Enigmata reaches 32.8% on ARC-AGI 1 and 0.6% on ARC-AGI 2, against 25.8% and 0.4% for o3-mini-high and 29.0% and 0.4% for o1, and scores 62.6% on Enigmata-Eval; on the out-of-domain KOR-Bench it improves from 54.7% to 65.0% over its base model. The paper further claims that adding 20K Enigmata puzzle items to the reinforcement learning data of Seed1.5-Thinking, a 20B-activated-parameter mixture-of-experts model, lifts AIME 2024 by 0.8 points, AIME 2025 by 1.9 points, BeyondAIME by 0.4 points, and GPQA Diamond by 0.8 points, making puzzle data an almost free complement to math and STEM training.

Load-bearing premise

The load-bearing premise is that all 36 rule-based verifiers judge answers correctly and each generated puzzle has exactly one intended solution; if any verifier accepts wrong answers or rejects correct ones, both the reinforcement learning reward and the Enigmata-Eval scores become unreliable.

Editorial extensions

If this is right

  • An open 32B model can outperform much larger proprietary reasoning models on puzzle benchmarks, making puzzle-solving a comparatively cheap capability to train.
  • Puzzle training transfers out of domain: the trained model improves on KOR-Bench over its base and keeps the math ability gained during rejection fine-tuning, so puzzle data complements rather than replaces math data.
  • On a frontier 200B-parameter reasoning model, adding synthetic puzzle data improves AIME, BeyondAIME, and GPQA Diamond, so knowledge-free puzzles can strengthen knowledge-heavy reasoning.
  • Built-in difficulty tags enable curriculum experiments; the paper's balanced easy:medium:hard data mixing yields more stable generalization than a difficulty-skewed mix.
  • Rejection fine-tuning is the deciding factor for high-complexity puzzles: without relevant SFT data, mix-training RL fails on tasks such as Light Up.

Reading between the lines

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

  • Beyond the paper, the same generator–verifier architecture could extend to other rule-governed domains, such as theorem proving or constraint satisfaction, wherever a cheap checker and difficulty-tagged sampling exist.
  • The 0.6% score on ARC-AGI 2 is a small absolute number; whether it marks a genuine step toward open-ended abstract reasoning or only narrow gains on recurring patterns is a question the paper's reported numbers leave open.
  • The observation that in-prompt code writing hurts puzzle accuracy suggests that code only pays off with an execution loop; a testable extension is to give the model tool-use access and see whether the gap reverses.
  • Because the verifiers are the sole source of reward, an independent audit of their correctness on a sample of generated puzzles would directly test the trustworthiness of the benchmark scores.
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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 / 4 minor

Summary. The paper introduces Enigmata, a suite of 36 puzzle tasks across seven categories, each paired with an instance generator and a rule-based verifier, together with Enigmata-Eval (a benchmark sampled from the same generators) and a two-stage training recipe (rejection fine-tuning followed by VC-PPO multi-task RL). The authors train Qwen2.5-32B-Enigmata and report large gains over the base model on Enigmata-Eval and ARC-AGI 1, mixed results on KOR-Bench, and small gains on ARC-AGI 2. They also report that adding 20K Enigmata examples to a larger Seed1.5-Thinking RL run gives small improvements on AIME, BeyondAIME, and GPQA Diamond. The manuscript positions Enigmata as the first comprehensive, scalable, verifiable puzzle resource for RLVR training.

Significance. If the resource and results are sound, this is a useful contribution: it offers a diverse set of generator-verifier pairs, controllable difficulty, and a concrete RLVR recipe for puzzle reasoning. The external ARC-AGI 1 gain (6.0% to 32.8%) and KOR-Bench gain (54.7% to 65.0%) are credible signals that the training data transfers beyond the in-domain benchmark, and the public-release intent is commendable. However, the manuscript's central claims are currently weakened by three load-bearing issues: the verifiers are not auditable from the paper, Enigmata-Eval is in-distribution with respect to the training generators, and some headline superiority claims are contradicted or statistically unsupported by the paper's own tables.

major comments (4)
  1. [§3.2, §4.2, Appendix E.3] The entire RLVR loop and the Enigmata-Eval scores depend on the correctness of the 36 rule-based verifiers, but the manuscript provides no verifier code, test suite, or specification of how each verifier decides correctness. Section 3.2 states only that verifiers 'undergo manual validation,' and Section 4.2 defines vi as scoring a response as correct or incorrect without stating whether this is exact-match against a stored reference or semantic validation. Appendix E.3 shows multiple tasks that explicitly permit multiple valid outputs (Skyscraper: 'return any one'; Sudoku: 'provide one'; Hamiltonian Path: any valid path; Maze: any path; Game24: any valid expression), and Crypto KPA requires inferring a cipher from a single plaintext-ciphertext pair, which is generally underdetermined. Because the same verifiers generate both the PPO reward and the Enigmata-Eval scores in Table 3, a small systematic error rate could move the 62.6 vs 59.9 margin over o3-mini-high and would corrupt the training signal itself. The authors should release verifier code, test suites (including invalid and multiple-valid cases), and a precise statement of the matching policy.
  2. [Abstract and §5.2, Table 3] The abstract claims that Qwen2.5-32B-Enigmata 'consistently surpasses o3-mini-high and o1 on the puzzle reasoning benchmarks,' but Table 3 shows KOR-Bench, described in §5.1 as a knowledge-orthogonal reasoning benchmark containing puzzles from five categories, where the model scores 65.0 versus o3-mini-high at 69.6 and o1 at 69.9. The claim is therefore not consistent across the puzzle benchmarks reported in the paper. The authors should either restrict the claim to the specific benchmarks where the model wins (Enigmata-Eval, ARC-AGI 1, ARC-AGI 2) or discuss why the model does not surpass these baselines on KOR-Bench.
  3. [§3.4, §5.2] Enigmata-Eval is built by systematically sampling from the broader Enigmata dataset, and the training set is generated by the same generators described in §4.2. While the paper states that strict separation prevents data leakage, the in-domain column of Table 3 partly measures fit to the training distribution and is not an independent measure of puzzle reasoning. The external ARC-AGI and KOR-Bench results are more informative, but the manuscript should explicitly caveat the Enigmata-Eval comparisons and should provide per-task evidence of non-overlap (for example, generation-seed separation or hash-based decontamination) beyond the assertion of strict separation.
  4. [§5.1, §5.2, Table 3, §5.3, Table 5] The reported differences on ARC-AGI 2 are 0.2 to 0.4 percentage points (o1 0.4, o3-mini-high 0.4, Qwen2.5-32B-Enigmata 0.6), and the evaluation protocol in §5.1 averages only 4 samples for non-AIME benchmarks. These differences are within plausible sampling noise, so the abstract's mention of 'ARC-AGI 2 (0.6%)' as a point of superiority is not statistically supported. Similarly, Table 5 reports improvements of +0.4 on BeyondAIME and +0.8 on GPQA Diamond without confidence intervals or significance tests, which is too thin a basis for the 'free lunch' framing. The authors should add error bars, increase evaluation samples for these small-margin benchmarks, or temper the claims.
minor comments (4)
  1. [Appendix E.3, Listing 14] The Car Painting output format section incorrectly instructs the model to output a sequence of rotation steps, which appears to be copy-pasted from the Twiddle listing; it should describe the car ID array format used in the example.
  2. [Appendix E.3, Listing 22] The Kakurasu example states 'Board size: 4 X 4' and 'Row sums: [0, 5, 10, 5]', but the displayed grid is 5x5 with five rows and five column-sum entries; the inconsistency should be corrected.
  3. [Appendix E.3, Listing 29] The Countdown example says the expression should 'result in 24' and uses the failure message 'cannot form 85', while the task target is 85; this copy-paste error should be fixed.
  4. [§5.1 and Table 3] The evaluation protocol says AIME is sampled 32 times and other benchmarks 4 times, but no standard errors are reported anywhere; adding confidence intervals would make the comparisons across models more interpretable, especially where margins are small.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: in-domain evaluation is explicitly labeled, and the headline generalization claims rest on external public benchmarks, not on the paper's own construction.

full rationale

No quoted step exhibits a reduction of a claimed result to its own inputs. The paper's central claims are empirical training-and-evaluation results, not derivations from first principles. The one structurally in-domain element is Enigmata-Eval: Section 3.4 says it was built by 'systematically sampling from our broader dataset,' and Section 4.2 generates the Enigmata training split from the same task generators and verifiers. However, Table 3 explicitly labels Enigmata-Eval as 'In-Domain,' and the paper does not present it as evidence of out-of-distribution generalization. The headline puzzle claims are additionally supported by independent, externally defined benchmarks: ARC-AGI 1, ARC-AGI 2, KOR-Bench, AIME, BeyondAIME, and GPQA Diamond. These are measured on public held-out benchmarks and cannot reduce to the paper's data-construction choices by definition. Self-citations to Seed1.5-Thinking and to the DAPO/VC-PPO line provide the base model and RL algorithm, but the comparison in Table 5 is a controlled experiment (same base model, same RL data, with versus without 20K Enigmata examples), so the cited prior work is not doing the argumentative work. The verifier-correctness limitation noted in Section 3.2 is a reproducibility and validity risk, not a circularity: a flawed verifier would corrupt both the RL reward and the evaluation scores, but the paper's results are not true by construction solely because of how the verifiers are defined. No self-definitional equation, fitted-parameter-renamed-as-prediction, or imported-uniqueness step was found, so the appropriate finding is no significant circularity.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claims rest on unverified tooling assumptions, including generator and verifier correctness, no leakage, RFT solution quality, difficulty tier validity, and the fairness of the Seed1.5-Thinking comparison. They also depend on hand-set training choices such as mixing ratios, sample counts, upsampling factors, and step counts. No formal verification or released code is provided.

free parameters (7)
  • Puzzle-to-math ratio = 1:1
    Chosen by hand for both RFT and RL; no ablation isolates this ratio, and the generalization claim depends on it (Sections 4.1 and 4.3).
  • Enigmata samples per task in RL = 400
    Chosen by hand; Figure 2 shows performance changes with data size, so the central result depends on this quantity (Appendix B.2).
  • Difficulty mix in RL stage 2 = 1:1:1 easy:medium:hard
    Selected after comparing with 2:6:2 and HRV; no optimum is proven, and Table 6 shows different OOD trade-offs (Section 5.4).
  • ARC-AGI upsampling factor = 8x
    Hand-set for Mix-Training and Multi-Stage stage 1; no sweep is reported (Section 4.3 and Table 9).
  • AIME upsampling factor = 2x Mix-Training, 1x Multi-Stage stage 1
    Hand-set to balance math and puzzle data; no ablation is reported (Appendix B.2 and Table 9).
  • PPO training steps = 370 reported for the main model vs 425 in Appendix C vs 200+225 in ablations
    The main Qwen2.5-32B-Enigmata result is described as 370 steps in Section 5.1, while Appendix C says 425 and ablation tables use 200 plus 225 steps; this inconsistency affects reproduction.
  • RFT samples per task = 1,000 puzzle instances per task
    Chosen by hand to build the RFT dataset; no scaling study is reported (Appendix B.1).
assumptions (6)
  • domain assumption The 36 rule-based verifiers are correct and the generated puzzles are well-posed.
    The entire RLVR reward signal depends on verifiers that are said to undergo manual validation, but no verifier code or test suite is shipped (Section 3.2).
  • domain assumption No data leakage exists between Enigmata training and Enigmata-Eval.
    Section 3.4 asserts strict separation protocols but provides no leakage testing details, and eval instances come from the same generators as training.
  • domain assumption DeepSeek-R1 generated correct solutions for rejection fine-tuning.
    RFT data quality depends on selecting correct solutions from eight DeepSeek-R1 samples per instance, with correctness determined by the same verifiers (Section 4.1).
  • domain assumption Using the public ARC-AGI training sets during RL training does not contaminate the ARC-AGI evaluation results.
    The recipe trains on ARC-AGI 1 and 2 public training data and then evaluates on ARC-AGI benchmarks; the paper assumes the eval sets remain uncontaminated (Section 4.3).
  • domain assumption Difficulty tiers based on GPT-4o pass@k are meaningful for training.
    Difficulty levels are assigned by GPT-4o pass@k scores, which may not correspond to the difficulty experienced by the target model or by other models (Section 3.2 and Table 13).
  • domain assumption The Seed1.5-Thinking comparison is a controlled same-compute ablation.
    Section 5.3 says the same base model and data are used except for 20K Enigmata instances, but Table 5 reports the published Seed1.5-Thinking numbers, leaving ambiguity about whether a same-compute control was actually retrained.

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Cite this review

Pith. "Pith review of Enigmata: Scaling Logical Reasoning in Large Language Models with Synthetic Verifiable Puzzles." pith.science (2026). https://pith.science/paper/GARLY3AO

@misc{pith2026250519914,
  author       = {Pith},
  title        = {Pith review of: Enigmata: Scaling Logical Reasoning in Large Language Models with Synthetic Verifiable Puzzles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GARLY3AO}},
  note         = {Machine review of arXiv:2505.19914}
}
read the original abstract

Large Language Models (LLMs), such as OpenAI's o1 and DeepSeek's R1, excel at advanced reasoning tasks like math and coding via Reinforcement Learning with Verifiable Rewards (RLVR), but still struggle with puzzles solvable by humans without domain knowledge. We introduce Enigmata, the first comprehensive suite tailored for improving LLMs with puzzle reasoning skills. It includes 36 tasks across seven categories, each with 1) a generator that produces unlimited examples with controllable difficulty and 2) a rule-based verifier for automatic evaluation. This generator-verifier design supports scalable, multi-task RL training, fine-grained analysis, and seamless RLVR integration. We further propose Enigmata-Eval, a rigorous benchmark, and develop optimized multi-task RLVR strategies. Our trained model, Qwen2.5-32B-Enigmata, consistently surpasses o3-mini-high and o1 on the puzzle reasoning benchmarks like Enigmata-Eval, ARC-AGI (32.8%), and ARC-AGI 2 (0.6%). It also generalizes well to out-of-domain puzzle benchmarks and mathematical reasoning, with little multi-tasking trade-off. When trained on larger models like Seed1.5-Thinking (20B activated parameters and 200B total parameters), puzzle data from Enigmata further boosts SoTA performance on advanced math and STEM reasoning tasks such as AIME (2024-2025), BeyondAIME and GPQA (Diamond), showing nice generalization benefits of Enigmata. This work offers a unified, controllable framework for advancing logical reasoning in LLMs. Resources of this work can be found at https://seed-enigmata.github.io.

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Forward citations

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    Game Board : It t y p i c a l l y c on sis ts of an n x n grid matrix . Each cell r e p r e s e n t s a building , and the bu il di ng height is r e p r e s e n t e d by a number ranging from 1 to n , where n is the size of the matrix

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    Example :

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    Clue Mapping Logic : (1) Across Clues : - Rows without ’* ’ c h a r a c t e r s r e p r e s e n t across words . These words c o r r e s p o n d to across clues in order from top to bottom . - Rows c o n t a i n i n g ’* ’ do not c o r r e s p o n d to any across word or clue ...

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    Mat ch in g Letters at I n t e r s e c t i o n s : - Letters at the i n t e r s e c t i o n s of across and down words must match , en su ri ng valid words are formed both h o r i z o n t a l l y and v e r t i c a l l y . Output Format : Please output your answer within a code...

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    The goal is to restore the puzzle to its initial state ( as shown below ) : 1 2 3 4 5 6 7 8 9 Task : Please provide the steps to restore the puzzle to its initial state . Output Format : - Please output your answer within a code block ( ‘ ‘ ‘) as follows : ‘‘‘ < result > ‘‘‘ -...

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    Your goal is to mi ni mi ze the number of color sw it che s by o p t i m a l l y a r r a n g i n g the cars . Task : Find a r e a r r a n g e d s eq ue nce of cars that m i n i m i z e s the number of color s wi tc hes . You must provide a list of car IDs in their new order ( ...

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    Push (1)

    Pop : Remove el em ent s from the top of the stack and add them to the output se que nc e . Example : Suppose the input s eq ue nce is [1 , 2 , 3]. Here are some p os si bl e valid stack p e r m u t a t i o n s : - [1 , 2 , 3]: D ir ec tly push all e le men ts into the stack a...

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    - The puzzle usually starts from a s c r a m b l e d state

    Initial State : - The initial state of the puzzle is 8 n um be re d tiles r an do ml y d i s t r i b u t e d in a 3 x3 grid , with the blank space located a ny whe re . - The puzzle usually starts from a s c r a m b l e d state

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    ‘‘‘ Puzzle : ‘‘‘ 4 1 5 2 6 8 3 7 0 ‘‘‘ Listing 18 Case of Fifteen Puzzle Task : The Fifteen puzzle is a classic sliding puzzle game

    Goal : - The u lt im at e goal is to arrange the tiles in order from left to right , top to bottom , as follows : 1 2 3 4 5 6 7 8 0 Output Format : - Please output your answer within a code block ( ‘ ‘ ‘) as follows : ‘‘‘ < result > ‘‘‘ - If there is an answer , the is the se ...

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    - The puzzle usually starts from a s c r a m b l e d state

    Initial State : - The initial state of the puzzle is 15 nu mbe re d tiles ra ndo ml y d i s t r i b u t e d in a 4*4 grid , with the blank space located a ny whe re . - The puzzle usually starts from a s c r a m b l e d state

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    - Tiles can only move in the four d i r e c t i o n s : up ( U ) , down ( D ) , left ( L ) , and right ( R )

    Mov em en t : - The player can move a tile ad jac en t to the blank space into the blank space . - Tiles can only move in the four d i r e c t i o n s : up ( U ) , down ( D ) , left ( L ) , and right ( R ) . - Only one tile can be moved at a time

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    ‘‘‘ Puzzle : ‘‘‘ 4 9 2 1 12 3 11 5 7 8 14 0 13 10 6 15 ‘‘‘ Listing 19 Case of Nine Puzzle Task : The nine puzzle is a classic sliding number puzzle

    Goal : - The u lt im at e goal is to arrange the tiles in order from left to right , top to bottom , as follows : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 Output Format : 34 - Please output your answer within a code block ( ‘ ‘ ‘) as follows : ‘‘‘ < result > ‘‘‘ - If there is an ...

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    Initial State : - The initial state of puzzle co ns is ts of 9 number tiles r an dom ly arr an ge d on a 3 x3 grid - The puzzle t y p i c a l l y starts from a s c r a m b l e d state

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    For example : 1 2 3 , sh if ti ng by 1 step results in 2 3 1 , s hi fti ng by 2 steps results in 3 1 2

    Mov em en t : - Players can choose to move an entire row or column , s hi ft in g it by 1 to 2 steps in a ci rcu la r manner . For example : 1 2 3 , sh if ti ng by 1 step results in 2 3 1 , s hi fti ng by 2 steps results in 3 1 2. - We r e p r e s e n t row moves as RAB , wher...

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    R11 " ,

    Goal : - The u lt im at e goal is to arrange the tiles in order from left to right , top to bottom as follows : 1 2 3 4 5 6 7 8 9 Output Format : - If a so lu tio n exists , output the s eq uen ce of moves within a code block ( ‘ ‘ ‘) , for example : ‘‘‘ [" R11 " , " C23 " , "...

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    - The puzzle t y p i c a l l y starts from a s c r a m b l e d state

    Initial State : - The initial state of the puzzle co ns is ts of 16 number tiles ran do ml y ar ra ng ed on a 4 x4 grid . - The puzzle t y p i c a l l y starts from a s c r a m b l e d state

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    For example : 1 2 3 4 , shi ft in g by 1 step results in 2 3 4 1 , s hi ft in g by 2 steps results in 3 4 1 2 , and s hif ti ng by 3 steps results in 4 1 2 3

    Mov em en t : 35 - Players can choose to move an entire row or column , s hi ft in g it by 1 to 3 steps in a ci rcu la r manner . For example : 1 2 3 4 , shi ft in g by 1 step results in 2 3 4 1 , s hi ft in g by 2 steps results in 3 4 1 2 , and s hif ti ng by 3 steps results ...

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    R11 " ,

    Goal : - The u lt im at e goal is to arrange the tiles in order from left to right , top to bottom as follows : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Output Format : - If a so lu tio n exists , output the s eq uen ce of moves within a code block ( ‘ ‘ ‘) , for example : ‘‘‘ [...

  124. [134]

    The puzzle is played on an NxN grid ( where N is an even number ) , with each cell c o n t a i n i n g a number

  125. [135]

    black out

    Your goal is to " black out " certain cells , f o l l o w i n g these rules : - In each row and column , the same number cannot appear more than once . To e l i m i n a t e repetitions , you must black out some of the cells . - Black cells cannot be adjacent , either h o r i z...

  126. [136]

    The puzzle is played on a r e c t a n g u l a r grid ( with a r b i t r a r y row and column sizes )

  127. [137]

    black out

    Your goal is to " black out " certain cells , f o l l o w i n g these rules : - The black cells in each row must sum up to the target number for that row . - The black cells in each column must sum up to the target number for that column . - To c a l c u l a t e the row sum : ...

  128. [138]

    No valid so lu ti on exists for the given K aku ra su puzzle

    C o o r d i n a t e s are 1 - based . For example , the first row is row 1 , and the first column is column 1. Puzzle : Solve the f o l l o w i n g K ak ura su puzzle by b la ck ing out the cells where needed . Board size : 4 X 4 Row sums : [0 , 5 , 10 , 5] Column sums : [5 , ...

  129. [139]

    The puzzle is played on a r e c t a n g u l a r grid ( the number of rows and columns is not fixed )

  130. [140]

    1" means it must have exactly 1 light bulb around it , a

    The goal is to place light bulbs ( r e p r e s e n t e d by L ) on the empty squares of the grid , f o l l o w i n g these rules : - Each num be re d black square ( r e p r e s e n t e d by numbers 1 -4) must have the s p e c i f i e d number of light bulbs around it . For exa...

  131. [141]

    - N um be red cell : This cell di sp la ys the number of mines ad ja cen t to it

    Grid and Mines The game grid con si st s of several cells , each of which may be : - Mine : If the player clicks on a mine cell , the game ends . - N um be red cell : This cell di sp la ys the number of mines ad ja cen t to it . The number i n d i c a t e s how many of the eig...

  132. [142]

    Unable to d e t e r m i n e any mine l o c a t i o n s

    Current Grid State R e p r e s e n t a t i o n The grid state is r e p r e s e n t e d as : -2: I n d i c a t e s the cell is unknown ( not r ev ea le d ) . 0 -8: Re ve al ed non - mine cells , where the number i n d i c a t e s how many mines are ad ja ce nt to that cell . Fo...

  133. [143]

    The number ranges from 0 to 4 , r e p r e s e n t i n g the number of i n t e r s e c t i n g di ag on al lines

    Grid Numbers : - Each cell in the grid may contain a number , i n d i c a t i n g how many dia go na l lines meet at that i n t e r s e c t i o n . The number ranges from 0 to 4 , r e p r e s e n t i n g the number of i n t e r s e c t i n g di ag on al lines

  134. [144]

    /" ( forward slash , r e p r e s e n t i n g top - left to bottom - right ) or a

    Dia go na l Line Rules : - Each cell must contain one d ia go na l line , either a "/" ( forward slash , r e p r e s e n t i n g top - left to bottom - right ) or a "\" ( backslash , r e p r e s e n t i n g top - right to bottom - left )

  135. [145]

    For example : - Number 1: I n d i c a t e s 1 d ia gon al line i n t e r s e c t s at that point

    I n t e r s e c t i o n Numbers : - The number i n d i c a t e s how many d ia gon al lines meet at that i n t e r s e c t i o n . For example : - Number 1: I n d i c a t e s 1 d ia gon al line i n t e r s e c t s at that point . - Number 2: I n d i c a t e s 2 d ia gon al lin...

  136. [146]

    /" ( forward slash , top - left to bottom - right ) - -1 r e p r e s e n t s

    No Loops : - The d ia go nal lines must not form loops . All d iag on al lines must connect , and no closed cycle can be formed . Puzzle : Solve the f o l l o w i n g slant puzzle : 1 0 1 . 0 2 0 1 3 1 1 4 0 2 1 1 4 2 0 . 0 1 . 2 2 3 1 1 0 4 0 2 3 2 1 2 1 3 2 0 3 1 0 . . 1 2 0...

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    Listing 27 Case of Tic Tac Toe You are tasked with solving a Tic Tac Toe puzzle

    Bh6 + Kh8 24. Listing 27 Case of Tic Tac Toe You are tasked with solving a Tic Tac Toe puzzle . Task & Rules :

  138. [148]

    The board c on sis ts of 3 x3 cells

  139. [149]

    O " or

    Players take turns placing their mark on an empty cell , one move per turn . The two players use " O " or " X "

  140. [150]

    A player wins by placing three of their marks c o n s e c u t i v e l y in a row , column , or d ia go na l

  141. [151]

    X " " O

    If the board is c o m p l e t e l y filled without a winner , the game is a draw . You are playing tic - tac - toe as X . Puzzle : Current board : O | X | - - - - - - - - - | | O - - - - - - - - - X | X | O Qu es ti on : What is the best next move ? Please provide only your mo...

  142. [152]

    - The total number of l a n d m a r k s in the puzzle will range from 7 to 10

    L a n d m a r k s D e f i n i t i o n : - I de nti fy a set of l a n d m a r k s which include : store , bank , house , cinema , garden , and school . - The total number of l a n d m a r k s in the puzzle will range from 7 to 10

  143. [153]

    - The root node of this tree r e p r e s e n t s the sta rt in g point for n a v i g a t i o n

    S t r u c t u r e : - The l a n d m a r k s are o r g a n i z e d in a binary tree s t r u c t u r e . - The root node of this tree r e p r e s e n t s the sta rt in g point for n a v i g a t i o n

  144. [154]

    O b j e c t i v e : - Your goal is to find the s hor te st path from the s ta rt ing point to the nearest s p e c i f i e d type of la nd ma rk

  145. [155]

    ([ A -Z , ]+)

    Puzzle Input : - You will receive a q ue st ion . - Use the i n f o r m a t i o n pro vi de d in the qu es ti on to d e t e r m i n e the path . Output Format : - Please output your answer "([ A -Z , ]+) " within a code block ( ‘ ‘ ‘) , c o n t a i n i n g only the path letter...

  146. [156]

    You can only move up , down , left , or right , not d i a g o n a l l y

  147. [157]

    You cannot pass through o b s t a c l e s ( B )

  148. [158]

    You can move freely on open spaces (.)

  149. [159]

    Please find a valid path from the start point ( S ) to the end point ( E )

    The goal is to find a path from the start point ( S ) to the end point ( E ) . Please find a valid path from the start point ( S ) to the end point ( E ) . If there are mu lti pl e paths , provide any one of them . If no valid path exists , state that it is i m p o s s i b l e...

  150. [160]

    Knights always tell the truth

  151. [161]

    E n t a i l m e n t

    Use logical r e a s o n i n g to d e t e r m i n e the truth value of the s t a t e m e n t . Output Format : - Please output your answer within a code block ( ‘ ‘ ‘) as follows : ‘‘‘ < result > ‘‘‘ Options : - " E n t a i l m e n t ": Use this if the s t a t e m e n t is l o ...

  152. [162]

    These pr em ise s are the f o u n d a t i o n a l truths or a s s u m p t i o n s for the puzzle

    Pre mi se s : You will be p ro vi de d with a set of s t a t e m e n t s or p re mi se s . These pr em ise s are the f o u n d a t i o n a l truths or a s s u m p t i o n s for the puzzle

  153. [163]

    Your task is to e va lua te this c o n c l u s i o n in the context of the given p re mi se s

    C o n c l u s i o n : A s t a t e m e n t will be p r e s e n t e d as the c o n c l u s i o n . Your task is to e va lua te this c o n c l u s i o n in the context of the given p re mi se s

  154. [164]

    True " ,

    E v a l u a t i o n C ri te ri a : - True : The c o n c l u s i o n l o g i c a l l y follows from the pr em is es . - False : The c o n c l u s i o n c o n t r a d i c t s the p re mis es . - Unknown : The c o n c l u s i o n cannot be d e t e r m i n e d from the pr emi se s...

  155. [165]

    This may include context or sp ec if ic c o n s t r a i n t s that apply to the puzzle

    B a c k g r o u n d I n f o r m a t i o n : C a r e f u l l y read any i n t r o d u c t o r y i n f o r m a t i o n p ro vi de d with the puzzle . This may include context or sp ec if ic c o n s t r a i n t s that apply to the puzzle

  156. [166]

    These clues will guide you in d e t e r m i n i n g the r e l a t i o n s h i p s between d i f f e r e n t e le me nt s in the grid

    Clues : Analyze each clue given . These clues will guide you in d e t e r m i n i n g the r e l a t i o n s h i p s between d i f f e r e n t e le me nt s in the grid

  157. [167]

    Co ns id er all p os si bl e options and e l i m i n a t e those that c o n t r a d i c t the clues

    Logical D e d u c t i o n : Use logical r e a s o n i n g to deduce the correct p l a c e m e n t of each element in the grid . Co ns id er all p os si bl e options and e l i m i n a t e those that c o n t r a d i c t the clues

  158. [168]

    Your re sp on se should include a so lu tio n f oll ow ed by the final answer in a m ar kd ow n table format

    C o n s i s t e n c y Check : Ensure that your s olu ti on is c o n s i s t e n t with all the clues and b a c k g r o u n d i n f o r m a t i o n p rov id ed . Your re sp on se should include a so lu tio n f oll ow ed by the final answer in a m ar kd ow n table format . Use t...

    1984

  159. [169]

    N a t i o n a l i t y : can ad ia n is on the left of Job : writer

  160. [170]

    Hobby : card - games is on the right of Food : apricot Fill the f o l l o w i n g table to show your final answer . | Food | correct answer | correct answer | | Hobby | correct answer | correct answer | | Job | correct answer | correct answer | | N a t i o n a l i t y | correc...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.