Enigmata:  Scaling Logical Reasoning in Large Language Models with Synthetic Verifiable Puzzles
Authors: Jiangjie Chen
Qianyu He
Siyu Yuan
Aili Chen
Zhicheng Cai
Weinan Dai
Hongli Yu
Jiaze Chen
Xuefeng Li
Qiying Yu
Hao Zhou
Mingxuan Wang
Venue: NeurIPS 2025 spotlight ()
Keywords: Reasoning Model
Large Language Model
Reinforcement Learning
Logical Reasoning
Primary area: deep_learning

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 7 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. Project page: https://seed-enigmata.github.io.

ENIGMATA: Scaling Logical Reasoning in Large Language Models with Synthetic Verifiable Puzzles
Jiangjie Chen ,, Qianyu He H2. Siyu Yuan 1.2.+.f, Aili Chen
Zhicheng Cai
Weinan Dai 335.t Hongli Yu 33,5.t, Qiying Yu 13.5.,, Xuefeng Li!.4.t
Jiaze Chen Hao Zhou 3.5 Mingxuan Wang'.5
slaanniec@byytdannee com
zhouhao@air.tsinge edu. wangmingxuan.
*ByteDance Seed
FFuda University
Institute for Al Industry Research (AIR), Tsinghua University
*Shanghai Jiao Tong University
SIA of Tsinghua AIR and ByteDance Seed
Abstract
Large Language Models (LLMs), such as OpenAl's ol and DeepSeek's RI, 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. includes 36 tasks across 7 categories, each with: 1) generator that produces unlimited examples with controllable difficulty, and rule-based verifier for automatic evaluation. This generator-veriin design supports scalable, multi-task RL training, fine- ine-grained analysis, and seamless RLVR integration. We further propose ENIGMATA- rigorous benchmark, and develop optimized multi-task RLVR strategies. Our trained model, (0wen2.5-32B-ENIICE
consistently surpasses 03-mini-high and ol on the puzzle reasoning benchmarks like ENIGMATA- ARC-AGI (32.8%), and ARC-AGI (0.6%). It also generalizes well to out-of-domain puzzle bench- marks and mathematical reasoning. with little multi-tasking trade-off. When trained on larger models like Seed1.5-Thee Thinking (20B activated parameters and 200B total parameters), puzzle data from ENIGMATA further boosts SoTA performance ad- vanced math and STEM reasoning tasks such as AIME (2024-2025), BeyondAIME and GPQA (Diamond), showing nice generalization benefits of ENIGMATA. This work offers unified, controllable framework for advancing logical reasoning LLMs. Project page: https: pttps://seeee /ess:://seedeeee en eni--ataaaaaaaaaa gmata.. gi thub.
1 Introduction
Large Reasoning Models (LRMs) such as ol and RI, trained from Large Language Models (LLMs) with Reinforcement Learning (RL), have demonstrated excellent performance in complex reasoning tasks [1; 2; 3; 4; 5], such as mathematics, STEM, and coding. The success LRMss highlights the effectiveness of the Reinforcement Learning with Verifiable Rewards (RLVR) paradigm, which stresses the great importance of obtaining high-quality verifiable prompts [6; 7: 8). However, existing LRMs still struggle to complete various puzzle tasks that require purely logical reasoning skills rather than professional knowledge, which are easy and even obvious for human [9; 10; 11; 12]. Most
Equal Contribution.
Work done during internship ByteDance Seed.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Overview of the ENIGMATA dataset: 36 puzzle tasks across seven categories, designed to enhance and evaluate diverse reasoning capabilities in large language models.
Figure 1:
existing works targeting at puzzles mainly focus on designing benchmarks for evaluation [13; 10; 12]. lacking the training methods and resources for modern LLMs to tackle this challenge. Existing puzzle datasets often lack diversity and scalability, covering limited puzzle types and offering little control over generation or difficulty [12; 10; 11]. Few works target solving puzzles, yet with either designing a prompting workflow and relying on a code interpreter [14], or training LLMs upon one or a few puzzles [15; 16], which is difficult to generalize.
Based on the success of the "LLM+RLVR" paradigm, has become crucial obtain a large, diverse and challenging set of verifiable puzzle prompts for training agents to master logical reasoning, To meet this need, we introduce ENIGMATA -a comprehensive suite that pairs scalable, controllable puzzle dataset with a training recipe that equips LLMs with strong muzzle-solving abilities. ENIG- MATA delivers fully synthesizable data across multiple categories, difficulty levels, and scales with automatic evaluation, thus offering effortless integration into RL pipelines.
The whole ENIGMATA suite comprises of ENIGMATA-Data, ENIGMATA-Evva and ENIGMATA-Modee First, ENIGMATA-Dade comprises 36 task types spanning seven broad categories, each probing a distinct facet of logical reasoning. Every task is backed by an automated generator-verifie pair: the generator can vary both the quantity and the difficulty of the puzzles it produces, while the verifier instantly checks the correctness. This design yields three key benefits: 1) can generate an unlimited supply of self-verifying puzzle prompts, which plug seamlessly into the RLVR framework and support long chain-of-thoon of-thought training. 2) Programmatic difficulty control allows researchers to mix puzzles in desired difficulty ratios and to conduct fine fine-grained experiments on how curriculum design influences reinforcement learning. 3) Generators can emit arbitrary sample counts per task, enabling studies of task balancing and cross-task generalization.
Based on ENIGMATA- puzzles that challenges even state-of-the-art LRMss such as 04-mini-high, providing a comprehensive assessment of logical reasoning capabilities. We further introduce the ENIGMATA-Moded recipe, systematic approach for training nigh-performance LLMs on puzzle tasks. This recipe incorporates extensive research on multi-task training strategies, examining various training paradigms and data mixing techniques regarding data amount and difficulty control. Built upon (wen-22.5322) [17], our resulting models surpass current state-of-thh-an LRMs on both the ENIGMATA Eval benchmark and the challenging ARC-AGI benchmark, while exhibiting remarkable improvements on out-of-domanl (OOD) puzzle tasks and maintaining robust generalization in mathematical reasoning. When scaling the base model from Qwen2.5 (32B dense model) to Seed1.5 -ee....iiinning [5] (20B/200B Mixture-of- Expert (MoF) model), we observe a nice generalization from Knowledge-orthogonal puzzle data from ENIGMATA to tasks that require advanced math and STEM domain knowledge and reasoning
we present the ENIGMATA--EN -Eal benchmark, diverse collection of

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skills. Such improvement over a SoTA model seems like a nice "free lunch", since the puzzle data are mostly synthetic.
Overall, our key contributions are: 1) We introduce ENIGMATA, the first suite for enabling LLMs with advanced and comprehensive logical reasoning abilities for solving puzzles. 2) The ENIGMATA suite consists of ENIGMATA-Date featuring 36 distinct tasks across 7 categories with controllable difficulty, scalable generation, and automatic verification, which seamlessly fit the RLVR training paradigm. 3) We establish ENIGMATA-Eval, a benchmark that rigorously and comprehensively evaluates puzzle reasoning abilitiess, and propose the ENIGMATA-Modes recipe that trains models with superior performance on in- in-oomiin and OOD puzzle reasoning tasks.
2 Related Work
Reinforcement Learning with Verifiable Rewards. Reinforcement Learning (RL) has become a key method for improving models' reasoning capabilities. Unlike Reinforcement Learning with Human Feedback (RLHF), Reinforcement Learning with Verifiable Rewards (RLVR) removes the need for a reward model by directly assigning rewards based on objectively verifiable answers [2; 5; 18]. which has shown strong performance in mathematics [19; 20; 21]. STEM [2; 22]. and coding [18; 5]. For example, mathematical models are rewarded when their answers match standard solutions [20; 22], while coding models receive rewards when their code passes unit tests [23]. This method is appealing due to its automation and resistance to reward hacking. Puzzles are particularly well-suited for RLVR. They can be generated automatically without expert annotations [7; 15], follow clear rules with algorithmically verifiable solutions [11], and support precise control over difficulty [15; 10J. However, most prior RLVR research has focused on other domains, overlooking puzzles' potential for delivering effective reward signals.
Puzzle Reasoning of LLMs. Unlike Knowledge-intensive tasks, puzzles purely test a model's reasoning abilities rather than its knowledge memory capacity [9; 11; 12]. Researchers have proposed various benchmarks to assess different types of reasoning. including abstract [9; 12], deductive [10], and compositional reasoning [13]. Some benchmarks support scalable generation and difficulty control but lack puzzle diversity [12: 10), while others offer diverse formats without controllable difficulty [11]. A comprehensive benchmark that balances diversity, scalability, and controllability is still missing. Efforts to improve LLMs' puzzle-solving abilities mainly fall into two categories: tool integration and RLVR. Tool-based methods [14] incorporate external resources such as code or symbolic solvers but do not directly enhance the model's 's internal reasoning. Recent RLVR approaches leverage the scalable generation and verification properties of puzzles [16; 15), but often focus on a single task type, such as countdown [16] or zebra logic [15], limiting generalization. ACES [24] generates code synthesis puzzles, while ENIGMATA provides RL-ready generator-verifier pairs for broader logical reasoning.
3 ENIGMATA-Data: The Puzzle Dataset
In this section, introduce ENIGMATA-Data. comprehensive dataset designed to enhance and evaluate complex reasoning capabilities of LLMs in RLVR training.
3.1 Puzzle Categories
As shown in Figure 1, the ENIGMATA- is composed of 36 puzzle tasks 7 primary categories, including:
Crypto Puzzle is designed to evaluate models' understanding of cryptography and pattern recog- nition. Tasks such as Crypto KPA require models to decode encrypted messages or solve crypto- graphic challenges, testing their ability to work with hidden encoded information.
Arithmetic Puzzle challenges models to solve problems that require numerical reasoning and basic arithmetic operations. Puzzles like G4me24 test a model's ss ability to perform arithmetic calculations under constraints.
3

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Logic Puzzle assesses deductive reasoning and the ability to infer conclusions based on premises. Puzzles like Knights and Knaves test model's logical thinking through challenging scenarios that require applying logical rules to solve problems.
Grid Puzzle includes tasks that challenge models to solve problems involving structured grids. Puzzles in this category, such as Sudoku, require models to reason about numbers, patterns, and placements in grid format, testing both logical and spatial reasoning abilities.
Graph Puzzle involves tasks where models must reason about nodes, edges, and paths within graph structures. Challenges such as Hami comiltone tonian Path test a model's ability to understand and traverse graphs, evaluating its capacity for aath-finding and network navigation.
Search Puzzle includes tasks that require models to explore state space efficiently find correct solution under specific rules and constraints. Puzzles in this category, such as Minesweeper, challenge models simulate or search through potential action sequences, evaluate game or puzzle states, and make optimal decisions.
Sequential Puzzle focuses on tasks that involve understanding and predicting sequences of steps. Examples include 8 Puzzle, which test models' abilities to manipulate objects sequence or follow a series of logical steps to reach a solution.
3.2 Data Construction
We outline our three-phase data construction pipeline for ENIGMATA-Data.
Table 1: Task statistics in ENIGMATA-Data.
Phase I: Tasks Collection and Design. First, we curate and design 36 logic puzzle tasks that demand complex reasoning capabilities. Among these, 30 tasks scalable with custom generators for creating additional puzzle instances, while the remaining 6 tasks draw puzzle instances from existing datasets. Most tasks feature multi-step reasoning that inte- grates various complex reasoning skills.
Data Type
Task Number
All Puzzles
36
Auto nuto-Generated Puzzles
30
Crypto Puzzle
Arithmetic Puzzle
Logic Puzzle
Grid Puzzle
10
Graph Puzzle
4
Phase II: Auto-Generator and Verifier Develop- ment. We equipped 30 tasks with custom auto- generators for scalable data creation, and all 36 tasks with manually validated auto-verifiers that evaluate solution correctness or provide reward scores for rea- soning chains.
Search Puzzle
Sequential Puzzle
Phase III: Sliding Difficulty Control. For each puzzle task, we identify key variables that control difficulty, such grid size and blank cell count in Binario. These variables serve parameters in our auto-generator to create puzzle instances across difficulty levels. We evaluate model performance
on these puzzles using the pass k= Eroollems
metric (n 200; A: 10, 100) follow-
ing [25]. By analyzing performance trends across different parameter settings, we establish three difficulty levels (Easy, Medium, Hard) for each task.
3.3 Task Statistics
Table 2 shows that ENIGMATA stands out the only dataset that encompasses multiple task categories, offers scalability, provides automatic verification, and is publicly available. Additionally, uniquely employs the RLVR approach fundamentally enhance models' puzzle reasoning capabilities. Table presents the distribution tasks across the ENIGMATA.
3.4 ENIGMATA-Eval
We develop ENIGMATA -vval by systematically sampling from our broader dataset. each task, we aimed to extract 50 instances per difficulty level (Fasy, Medium, Hard). However, due inherent constraints in some tasks, we collected total of 4,758 puzzle instances rather than the theoretical maximum of 5,400. This discrepancy arises because some tasks generate fewer than 50 instances per

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Table 2: Comparison of different puzzle reasoning benchmarks.
Resources
Categories
Tasks
Scalable
Auto-Verifier
Trainable
Solution
KOR-Bench [11]
5
125
X
N/A N/A
NPR [12]
1
1
X
oogra [10]
1
/
N/A
SearchBench [13]
11
FCoReBench [14]
5
40
/
X
Prompt-based
Logic-RL [15]
1
1
J
RLVR
ENIGMATA
7
36
RLVR
difficulty level, while others rely manually collected and annotated data rather than nuto-generation. Importantly, we ensured no data leakage between training and evaluation sets by implementing strict separation protocols during the sampling process.
4 ENIGMATA-Model: The Training Recipe
Developing advanced logical reasoning language models requires a carefully structured training approach that develops diverse reasoning skills while avoiding overfitting to specific problem types. Our training methodology follows a two-stage process designed to systematically build reasoning abilities: (1) rejection fine-tuning to establish foundational reasoning patterns, and (2) multi-task RL to develop general reasoning skills that transfer across diverse problem domains.
4.1 Rejection Fine-tuning
Directly applying reinforcement learning to a base model often results in training instability and may not unlock the model's full performance potential [2]. To address this, we begin with rejection fine-tuning (RFT), i.e., leveraging high-quality solutions during supervised fine-tuning (SFT) to establish solid foundational reasoning patterns. We strategically combine math problems with puzzles in our training data, as mathematics elicits diverse reasoning patterns and contribute to the model generalization [26].
For puzzles, we uniformly sample tasks and difficulty levels from the ENIGMATA dataset to ensure comprehensive coverage and balanced distribution of reasoning patterns. We also include the training data of the ARC-AGI puzzle [9; 27] in the RFT data, since is too difficult to learn without RFT as coad-start. For each puzzle instance, we utilize DeepSeek-RI [2] generate candidate solutions, from which we select the correct solution for RFT. The mathematical component consists of carefully curated examples from high-quality R Rddistileed mathematical dataset [28]. Throughout RFT, we maintain balanced 1:1 ratio between puzzles and mathematical problems to ensure comprehensive reasoning development across domains. Detailed implementation and dataset specifications are provided in Appendices B and C.
4.2 RL with Verifiable Puzzles
We VC-PPO [29], PPO variant [30], train our models. Each of the 36 tasks in ENIGMATA has an automated verifier v that instantly scores a response as correct or incorrect. For 30 tasks we also have generator gi that can create examples any difficulty; the other tasks draw from fixed pools F, For each task i and difficulty level d E Di, we choose how many examples Ni,d d to use. Then:
(N
i i 30,
dE
8;
Sample (.d (N (F))),
i >>> 30.
dED,
The full training set is S = U; with total size |S] = -, |sa|. By changing the Ni.d- we can easily adjust: How many examples come from each task, 2) The mix of easy vs. hard items, 3) Overall dataset size. During training, each generated example is fed to its verifier which returns a reward that VC-PPO uses to update the policy. This loop provides fully automatic RI. pipeline for puzzle reasoning.
5

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4.3 Multi-task Training
Developing general logical reasoning is hard because different puzzles require different thinking skills. To build strong, transferable gobblem-solving abilities, explore two multi-task training methods: Mix-training RL and Multi-stage RL, since single-task training often leads to narrow expertise and poor transfer to new puzzles [15].
rich diversity of tasks can significantly enhance generalization and actively prevent overspecializa- tion [2]. Therefore, we employ Mix-training RL to integrate multiple puzzle types simultaneously during the training process. Our methodology involves a meticulously constructed dataset that integrates three critical components: a) The training split of ENIGMATA, featuring balanced task and difficulty distributions; b) The public training set of ARC-AGI 1 and 2, which improve the generalization of existing reasoning abilities to unseen tasks; and c) AIME mathematical problems (1983-2023), which are difficult enough to elicit diverse reasoning patterns and enhance generaliza- tion. A strategic 1:1 puzzle-to mathematics ratio is maintained throughout this training process to foster the development of complementary reasoning systems within the model.
Mix-training RL offers broad exposure to different puzzle types, but the varied reasoning skills required can cause conflicts between tasks. To address this, we adopt Multi-stage RL., a curriculum- based approach that builds core skills before introducing new challenges. For difficult tasks like ARC-AGI, we use a two-phase strategy: 1) train intensively on ARC-AGI 1, 2, and AIME until the model generalizes well and performance stabilizes; 2) gradually introduce ENIGMATA-DaiE while retaining earlier data to avoid forgetting. This step-by-step method helps the model learn complex reasoning more effectively and maintain strong performance carlier tasks. More implementation details are provided in Appendix B.
5
Experiments
5.1 Experiment Setup
We adopt several challenging reasoning benchmarks for evaluation: ENIGMATA- and abstract reasoning challenges ARC-AGI 1 [9] and ARC-AGI 2 [27] known for their extreme difficulty for LLMs. We also include the knowledge-orthogon 2024 monitor OOD behaviors. We evaluate each AIME problem 32 times and others 4 times, reporting the average performance. Verformance.Baselines are described in Appendix E. We train our models from Qwen2.5-33BBIInttrus [17], solid starting point for training strong reasoning models [31; 2]. After acquiring the RFT model Dewen2.5-32B-R. -FF),, we leverage the Mix- Bix-Trrining approach with Beeen2.5-32B-ENEEEE
reasoning benchmark KOR-Benchl [11] and AIME
showing superior overall performance in our experiments, Note that each PPO step performs multiple gradient updates by iterating over 4 mini-batches derived from the training batch. Details are in Appendix C.
370 training steps and get the RL model
5.2 Results
According to Table 3, our model outperforms most of the public models on ENIGMATA- with 32B parameters, demonstrating the effectiveness of our dataset and training recipe. Besides, our model stands out on the challenging ARC-AGI benchmark, surpassing strong reasoning models such as Gemini 2.5 Pro, 03-mini, and ol. Additionally, both RFT and multi-task RL training strategies yield significant performance gains on OOD benchmarks. This indicates that our training recipe effectively enhances the model's general logic reasoning abilities and can generalize to unseen tasks. Moreover, after RL training, our models still maintain comparable math reasoning abilities gained from rejection fine-tuning, showing that our training strategy preserves general reasoning capabilities while enhancing logic-specific skills.
Next, let's dive into detailed analysis across reasoning categories in ENIGMATA-Evar In Table 4, Qwen2.5-32B. 3-2-GGGAAAATA demonstrates exceptional performance in structured reasoning categories, particularly excelling in Crypto, Arithmetic, and Logic tasks. This suggests our training approach effectively develops capabilities in rule-based reasoning with clear constraints and patterns. Besides, our model shows competitive performance in search tasks, outperforming most baseline models. Search problems require strategic exploration of solution spaces and planning capabilities. The strong performance suggests our approach effectively develops these higher-order reasoning skills. Notably,

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Table 3: Performance of reasoning, generic, and our trained LLMs reasoning benchmarks.
Puzzle
Math
Model
In-Domain
Out-of-Doman
AIME 24
ARC-AGI1
ARC-AG12
ENIGMATA-Eval
KORBench
04-mini-high 03-mini-high
54.7
2.6
65.1
72.7
92.3
25.8
0.4
59.9
69.6
79.3
ol
29.0
0.4
54.9
78.0
DeepSeek-R1
17.8
0.2
49.2
Gemini 2.5 Pro
22.7
1.4
50.6
68.2
90.3
Claude-3.7-Sone S.0.....-Snnee-Thiiiin
37,6
1.4
53.2
67.8
60.3
DS-RI Distilled Qwen 32B
7.9
0.0
31.1
63.5
72.0
QwQ-32B
7.0
0.0
43,8
61.8
69.2
Grok-2-1212
7.5
0.0
13.6
54.9
16.7
GPT-4o-1122
7.3
0.0
14.2
57.9
10,0
Claude 3.7 Sonnet
12.9
0.0
22.7
56.9
23.0
Qwen2.5-32B Swen2.5-32B-Instrucc
6.0
0.0
12.6
54.7
16.6
Owen2.5-3222 --32---TRR
8.4
0.0
46.6
61.0
62.0
32.8
0.6
62.6
65.0
60.6
Table 4: Performance of reasoning LLMs, generic LLMs, and our trained LLMs on ENIGMATA-Eval.
Model
Crypto
Arithmetic
Logic
Grid
Graph
Search
Sequential
Overall
o4-mini-high
97.0
93.3
82.2
71.0
62.4
64.3
34.0
65.1
03-mini-high
88.1
74.3
65.5
63.1
61.5
29.6
$9.9
o1
97.8
80.0
60.1
63.3
56.2
50.9
23.6
54.9
DeepSeck-R1
82.7
77.1
71.4
51.1
62.6
38.4
19.5
49,2
Gemini 2.5 Pro
75.2
95.4
71.5
58.9
37.3
49.4
17.5
50.6
Saaue=3.7-Sooneee ihinking
81.5
75.4
76.6
$7.7
50.5
49.4
26.4
53.2
DS-RI Distilled Qwen 32B
62.9
36.4
18.8
31.1
QwQ-32B
59.0
74.4
47.3
28.2
24.4
43.8
Grok-2-1211
50.0
17.6
6.4
13.6
GPT-4o-1120
26.2
34.5
17.8
19.3
3.9
14.2
Claude 3.7 Sonnet
38.1
16.7
60.0
22.9
22.4
7.8
15.0
22.7
Qwen2.5-322 --32uu-nstruct
4,0
10.3
46.4
15.3
7.3
2.5
8.2
12.6
Dlwen2.5-32B-R-
62.0
71.7
71.6
51.3
55.1
39.3
18.4
46.6
Qwen2.5-32B. 322-ENIGMAATA
96.0
93.7
90.2
62.6
54.0
70.4
29.7
62.6
we observe consistent performance hierarchy across categories for most models. Tasks in Crypto and Arithmetic tend to yield the highest accuracy, while spatial tasks and sequential tasks remain more difficull. These challenges point to promising directions for future work.
5.3 Generalization with Scaling: Free Lunch from ENIGMATA
Can logical reasoning data, such as puzzles that do not require domain knowledge, benefit general reasoning capabilities? We do not observe such a generalization for Qwen2.5-32B. Therefore, we scale the experiments to much larger model. We follow Seed1.5-Thinde [5] and train from the same Mexturee-of-xppeeee
(MoE) model the RL stage, which features 20B activated and 200B total parameters. To make fair comparison between SeedI.5-Thinkin (20B/200B) and the same RL training data except for 20K ENIGMATA-Date and train the models for comparable PPO steps.
adopt the same base model
Surprisingly, the results in Table show that our dataset enhances general capabilities like math and STEM problen.solving. When compared to Seed1.5- --hhhiiking,, a leading reasoning model, additional training with ENIGMATA, i.e.. Seedd..-Thinking --hinninnnENIGGMM -NII- ATA, generally improves performance on AIME 2024 and 2025, BeyondAIME) [5] (an expert-curated, more challenging evaluation dataset), and GPQA Diamond [32]. Given the difficulty of further improving SoTA models like Seed1. 5--Thinking, simply incorporating ENIGMATA'S 's synthetic puzzle data during the RL training stage appears almost
7

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Table Results on benchmarks for general reasoning capabilities. This demonstrates that additional RL training on ENIGMATA-Dale generalizes well on larger models, showing the benefits of puzzle data for math and STEM reasoning.
Model
AIME 2024
AIME 2025
BeyondAIME
GPQA Diamond
04-mini-high
93.4
92.7
55.7
81.4
03-mini-high ol DeepSeek-RI
87.3
86.5
63.6
79.7
74.3
79.2
50.0
78.0
79.8
65.0
42.4
71.5
Seed B.cdl.5-TTe Thinking
86.7
74.0
48.0
77.3
Seed1.5- Seenkig- Thinking-ENIGMATA ENIGMATA
87.5(-0.8)
(5.9(+1.9)
48. 48.4(++0.)
78. (8.1(4).8)
Enigmata-En
ARC-AGI1
ARC-AGI2
AIME 2024
KOR-Bench
40
60.0
Performance
0.6
57.5
64
Data Points
0.4
Stage1
35
55.0
0.2
50
52.5
0.0
20
20
20
20 Training Data(K)
Training Data(K)
Training Data(K)
Training Data(K)
Training Data(K)
Figure 2: Impact of training data size in the second stage of Multi-stage Training model perfor- mance across different benchmarks. The blue dashed line represents model performance after the first training stage, while the red solid line shows performance after the second stage.
like a "free lunch" for expanding the capability spectrum of reasoning models, even leading to generalization improvements in general advanced reasoning."
5.4 Ablation Studies
Training Data Size. We study the impact of varying sizes of ENIGMATA-Train data during the sec- ond stage of Multi-stage Training model performance. To ensure fair comparison, all checkpoints were evaluated using models obtained at step 150 from the sample stage-1 checkpoint, with equal sampling across all difficulty levels. As shown in Figure 2, first, a small amount of ENIGMATA- -rraa data in the second stage significantly improves ENIGMATA-Eval performance while better preserving first-stage knowledge and OOD performance. Second, increasing ENIGMATA-Tran data progres- sively enhances in-domain ENIGMATA Eea performance. Third, excessive ENIGMATA -rain data leads to catastrophic forgetting and slightly degraded OOD performance.
Data Difficulty Control. We study how the distribution of data difficulty affects performance. As described in $ 4.2, we the data size at different difficulty levels N With N; 400 per task, compare two ratios in Mtlti-stage Training's second stage: balanced (Nuuay _Nrmed =1:hadd1:1111)
versus coddimm-focusee (Nv.myy _ -Nhaad 2:6:2). The latter setting quantifies how extreme samples can undermine RL training [20]. We also compare with historical reward variation (HRV) [33] baseline data selection strategy, using the same stage-1 checkpoint and 1,50 second-stage steps. As shown in Table 6, the balanced difficulty ratio (1:1:1) enables the model to demonstrate more robust complex reasoning performance. Also, effortless difficulty control method based on difficulty tags in ENIGMATA data performs comparably to HRV on ENIGMATA- Evall while delivering superior results on OOD benchmarks.
Multi-task Training. For the two training paradigms in & 4.3, we evaluate the impact of SFT/RFT using two variants: 1) SFT which excludes three tasks (Countdown, Minesweeper, Light Up) to test transfer ability; and 2) SFT ALL. which includes all tasks as a full baseline. To ensure fair comparison, all checkpoints were trained for identical total steps: (I) Multi-Stage training consisting of 200 steps in stage and 225 steps in stage 2; and (2) Mix-Training using the combined dataset
NNot that, we primarily experiment upon Qwen2.5-32B for further analysis due to the prohibitive resources required to train 20B/200B model.

---

Table 6: Comparison between different data mixing strategies in the stage 2 of Multi-stage RL.
Puzzle
Math
Mixing Method
In- Hi-Domain
Out-of-Doman
AIME 24
ARC-AGI1
ARC-AG12
ENIGMATA-Eval
KORBench
IRRV
34.7
0.1
57.7
60.1
50.1
Easy:Medium: Hard = 1:1:1
34.5
0.6
58,5
64.0
56.6
Easy:Medium:Hard 2:6:2
35.1
58,9
63.8
48.0
Table 7: Comparison between different training strategies.
Puzzle
Math
Model
In-Domain
Out-of-Domanl
AIME 24
ARC-AGI1
ARC-AG12
ENIGMATA-Eval
KORBench
Mix Training + SFFT-Part
22.7
0.0
46.7
56.7
51.8
Multi-Stage SFT-Part
27.5
0.0
56.1
51.6
45.9
Mix Training SFT-AIl
0.1
58.8
Multi-Stage SFT-AIl
33.4
0.4
61.1
60.2
55.6
across all 425 steps. As shown in Table 7, first, Multi-Stage and Mix- approaches show complementary strengths. Multi-Stage builds deeper task-specific reasoning, while dix-nrrin improves generalization. Second, With limited pre-training data, Multi-Stage transfers better to unseen tasks, especially complex ones like ENIGMATA and ARC-AGI, reflecting curriculum learning benefits. Third, Mix- fix-nirannng generalizes better OOD tasks, suggesting that diverse training helps models learn broader reasoning strategies beyond specific tasks.
Moreover, Figure 4 shows the training dynamics of Mix- Trainiri RL and Multi-Stage RL. We observe a positive correlation between rewards and response length. Both approaches achieve similar final rewards, showing the signs of test-time scaling effect [34; 35] where models learn to explore more tokens to find better solutions. However, the more volatile response length under State RL suggests instability in generation. In contrast, Mix-Training RL yields more consistent outputs with smoother length curves, indicating greater training stability.
5.5
Analysis
How does SFT affect RI training? We further analyze the influence of SFT RL training explore the reason behind the performance gap in Table 4.3. We represent the reward curves across different training steps for different training approaches in Figure 3. For simple tasks like Countdown, all methods achieve similar improvements, suggesting SFT is not essential for these tasks. For medium-come ceeiuui-cmpplxxtty tasks like Minesweeper, Mix-Training struggles without SFT, while Multi-stage RL still learns effectively. When the task is included in SFT AT. both approaches start from higher baseline and quickly optimize to near- rear-perfect accuracy. This pattern typical for tasks with fixed solution patterns or shortcuts once the model grasps the solving approach, performance improves dramatically. For high-complexity tasks like Light Up. the combination of comprehensive SFT and Multi-stage RL dramatically outperforms all other approaches, particularly for difficult variants of the task. Interestingly, Mix-Training RI. without relevant SFT (c) fails entirely, highlighting that complex reasoning tasks require both strong foundational knowledge (from relevant SFT) and structured learning approaches (from Multi-stage RL) to achieve optimal results.
Response Length Analysis. Figure 5 presents the distribution of reasoning token lengths for both cor- rect and incorrect responses in ARC-AGI 1. Tasks where the model produces more concise reasoning showed significantly higher accuracy compared to those requiring longer reasoning chains. This anal- ysis identifies potential inefficiencies in the model's reasoning process and highlights opportunities for reasoning optimization.
Table 8: Impact of code utilization accuracy
Condition
ARC-AGI1
ENIGMATA
Overall
34.5
62.6
With code
12.5
41.8
Without code
35.7
63.6

---

Hard
Medium
Easy
PFT + Multi-stage RL
SFTAi
Multi-stage RL
SFT Mix Mix-Training RL
SFTAy Mix-Training RL
1.00
Countdown Minesweeper
0.75
0.50
0.25
Average Reward
0.00
1.00
0.75
0.50
0.25
0.00
1.00
0.75
Light Up
0.50
0.25
0.00
75
150
225
75 Training Steps (b)
150
225
150
300
425
150
300
425
Training Steps (a)
Training Steps (c)
Training Steps (d)
Figure 3: Learning curves across training approaches for representative puzzle tasks. Each row represents different task, and each column represents different training approach. The curves show how average reward changes with training steps for different difficulty levels.
20k
Mix-Training RL
Token Length Analysis
Token Length Success Rate
Incorrect
Success Rate
1.0
15k
200
Correct
100
Fitted Curve
Response Length
Awerage Reward
Accuracy
0.0
5k-
10000
20000
30000
10k
2ök
30k
Multi-Stage RL
B22 ZUueno Selgmmaee
20k
Rossonse Longth
Avcrage Rowaard
Incorrect
Success Rate
1.0
Fitted Curve
15k
Count
50
Correct
Accuracy
10k
0.0
5k
10000
20000
30000
10k
30k
200
400
Thinking Taken Length (a)
Thinking Taken Length
Training Steps
(b)
Figure 4: The response length and reward curves during Mix- Training RL and Multi-Stage RL training.
Figure 5: Performance analysis of Qwen2.5-32BB 32--EEIGMMAAA
ARC-AGI I: (a) distribution of reasoning token lengths for correct and incorrect responses, (b) and success rate by reasoning token length.
Code Utilization in Puzzle Reasoning. We investi-
gate whether code utilization enhances model performance in reasoning tasks. Using pattern matching, we identified code elements model outputs and classified responses accordingly. According to Table 8, code utilization actually hindered the performance on puzzle tasks. This suggests that current models fail to effectively use the code without executing for complex reasoning.
Conclusion
In this paper, we present ENIGMATA, suite for equipping LLMs with advanced puzzle reason- Angg.ENIGMATA-DEE features 36 tasks across seven reasoning categories, with its scalable generation, automated verification, and adjustable difficulty. We also introduce the ENIGMATA-EVv benchmark assessing puzzle reasoning abilities and guiding research on generalizable reasoning models. ENTGMATA-Modes trained with RLVR, demonstrates its superior performance and exhibits ro- bust generalization and reasoning skills. We hope ENIGMATA serve solid foundation for the community to push forth the research on reasoning models.
10

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A Reward Curves Across Individual Tasks in ENIGMATA
We conducted a detailed analysis of our model's learning dynamics by examining reward curves across all individual tasks in the ENIGMATA dataset. Figure 6 presents these learning trajectories, revealing several important patterns in how models acquire puzzle-solvee solving capabilities.
The reward curves across tasks reveal three distinct learning patterns:
1. Gradual Mastery Tasks: A number of tasks, such Light Up and Zebra Logic show smooth and consistent reward gains over the course of training. These tasks typically involve complex reasoning chains require integrating multiple constraints, making them suitable for learning. The steady upward trend suggests the agent is progressively refining its decision-making strategy through extended exploration.
Difficult--Stratiier Tasks: Tasks like Car Painting, Star Battle, and Hitori demonstrate clear separation between difficulty levels: easy instances are learned relatively early, while medium and hard variants require significantly more training to improve. This stratification indicates that the difficulty scaling mechanism is effective and yields meaningful distinctions in learning complexity.
Some tasks, including Big Bench Symbolic and Magic Square, show little to no improvement across all difficulty levels, particularly on the hard setting. This suggests that these tasks may suffer from challenges such as sparse rewards, long-term dependencies. or overly complex solution spaces that PPO struggles to handle without additional guidance.
3. Stagnant Low- Tasks:
These learning patterns offer actionable insights for currieulum design. Tasks that are mastered early can serve as warm-up phases to bootstrap the agent's core skills, while more challenging tasks can be introduced later push reasoning boundaries. Understanding which tasks exhibit positive transfer or potential interference is key to optimizing multi-task training regimes.
Table 9: Training data distribution across different strategies and stages.
Method
ENIGMATA-
ARC-AGI1
ARC-AGI 2
AIME
Training
11557
3160
5934
1738
Multi-Stage Stage 1
0 11557
3160
5934
869
Multi Multi-Stage Stage 2
395
989
869
B Training Dataset Details
Rejection Fine-tuning. For the puzzle part of the Rejection Fine-tuning (RFT) dataset, we sample 1,000 instances from each task in the ENIGMATA dataset. We also include synthetic ARC-AGI data'. We then use DeepSeek-RI R1 to generate 8 candidate solutions for each instance and select one correct solution per instance. The final puzzle dataset contains 12,041 high-quality puzzle samples. For the mathematical part of the RFT dataset, we collected mathematical problems from light-R1 [28] that answered by DeepSeck-RA R1. We included all data from stage2 of light-Rl and sampled 3,000 additional problems from stagel. resulting in a total of 12,533 mathematical samples.
Reinforcement with Verifiable Puzzles. For our reinforcement learning with verifiable puzzles, we implemented two training paradigms. The detailed data mixing ratio is shown in Table 9 and introduced below. As for the Mix-Training, the dataset for this approach consists of three components: (1) ENIGMATA-
400 samples per task with equal distribution across difficulty levels (2) ARC- AGI 1 and 2: Official datasets upsampled 8x address specific reasoning challenges (3) AIME problems from 1983-2023 upsampled 2x as the mathematical component We maintained a 1:1 puzzle- to-math ratio throughout training to ensure balanced exposure different reasoning types. As for the Multi-stage RL, this approach follows a curriculum strategy with two distinct stages: (1) Stage
14

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1: Training ARC-AGI 1 and 2 (upsampled 8x) along with AIME problems (upsampled 1x) (2) Stage Incorporating ENIGMATA- while maintaining the datasets from Stage (ARC-AGI ARC-AGI 2, and AIME) without additional upsampling prevent catastrophic forgetting
C Implementation Details
We fine-tune the model on this balanced dataset for epochs using maximum sequence length of 32768 tokens and a learning rate of le-5. establishing a strong reasoning foundation before proceeding to reinforcement learning.
We adopt variant of Proximal Policy Optimization (PPO) [30], i... VC-PPO [29], train our reasoning agent on verifiable puzzles. This variant introduces several modifications tailored to oog-chain-offthoughh generation, which improves both training stability and performance over long sequences.
Standard PPO optimizes the following clipped objective:
(0) B(g.a) Decemmoo
min r.(0)A,, -
where (0) 00(0))).00z) (orl the importance sampling ratio, and A, the estimated advantage. We compute A, using Generalized Advantage Estimation (GAE) [30]:
A =
(GAR(T)))
Σ (7))66441 where 6 = = 7VV(+())))) - VV(s).
To optimize long-CoT training, we follow the VC-PPO and adopt the following modifications the standard PPO algorithm:
Removing KL Divergence Constraint. In traditional RLHF [36], KL penalty is used to prevent the policy from diverging too far from the reference model. However, in long-CoTI settings, this constraint often limits exploration and learning capacity. Following VC-PPO, remove the KL term by setting kl_loss seessweeggt weight = 0.0, allowing the model to freely deviate from the initial policy distribution.
Value Prefraining. We observe that initializing the value model from the reward model causes unstable training due to their objective mismatch. To address this, we adopt value pretraining: we first sample responses from a fixed SFT policy Tan compute Monte Carlo returns, and train the value model until the loss and explained variance converge. The pretrained value model is then used to initialize the critic for PPO.
Decoupled GAE. To reduce reward decay and improve optimization over long token sequences, we use different À values for the policy and the value model. Specifically, we set Acritic 1.0 for unbiased value estimation, and Apolicy 0.95 to improve sample efficiency and learning speed.
Training Details. We the maximum prompt length 6, 144 tokens and the maximum response length to 26,624 tokens. PPO training is conducted for 425 steps with batch size of 4,096 and mini-batch size of 512. The actor and critic are optimized using Adam, with learning rates of 1 x 10-6 and 2 x 10-6, respectively, and linear warm-up schedule over 10 steps. Before PPO begins, we perform value pretraining [29] for steps by collecting Monte Carlo returns from fixed SFT policy and fitting the value model to these returns. Our implementation is based on the VeRL framework. We enable gradient checkpointing for both the actor and the critic to reduce memory consumption. Rollouts are generated using temperature sampling (r = 1.0), with enforced end-of-sequence tokens. We leverage vL.LM [37] for efficient batched decoding with 256 rollout slots and paged attention.
15

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Table 10: Cross-domain transfer when models are trained via SFT on single puzzle domain. Entries show absolute accuracy improvements (in percentage points) on held-out domains relative to the Qwen2.5-32B-RR baseline. Parentheses mark in-domain evaluation.
Train Domain
Crypto
Arithmetic
Logic
Graph
Search
Sequential
Avg.
Scarch
+6.67
+14.37
+14.71
+11.22
+25.97
(+26.38)
+8.96
+13.65
Scquential
+10.67
+18.37
+12.49
+6.33
+21.61
+3.88
(+9.16)
+12.23
Grid
+9.33
+10.70
+13.38
(+25.07)
+25.25
+1.37
+9.16
+11.53
Graph
+5.33
+15.03
+10.04
+5.07
(+37.79)
+4.38
+7.38
+7.87
Crypto
(+51.33)
+16.03
+10.93
+4.70
+16.15
+0.38
+7.77
+9.33
Arithmetic
+4.33
(458.70)
+8.49
-1.97
+11.25
-1.12
+4.90
+4.31
Logic
+0.33
+2.37
(+18.27)
-2.11
+6.70
-0.50
+3.21
+1.67
D
Cross-Domain Transfer Study
Table 10 details how puzzle skills transfer when supervised fine-tuning is restricted to single domain. The Search domain provides the strongest average gains (+13.65 points) across unseen domains, indicating that long-horizon planning and constraint satisfaction learned from Search puzzles readily generalize. Sequential and Grid training also transfer broadly, while Logic-only supervision yields limited improvements outside its home domain. These observations support constructing compact yet diverse curricula by prioritizing high-traud high-transfer domains such as Search.
E Baselines
We categorize our baseline models into the following groups for comparison:
General-purpose LLMs.
As reference benchmarks, we also included several top-tier general-
purpose models:
GPT-4-11220 1120 [38]: OpenAl's model, representing the current highest standard of general models.
Gemini 2.5 Pro [3]: Google's advanced model with excellent performance across various tasks.
Claude 3.7 Sonnet [4]: Anthropic's high-performance model known for its reliability and compre- hensiveness.
Grok-2-1212 [39]: xAI's open-weight model, demonstrating the potential of open-source models.
Qwen-2.5-32B Instruct [17]: Alibaba's moderate-scale language model with strong performance both Chinese and English tasks, which serves as our main backbone models for training.
Reasoning-specciaizee LLMs. selected the current state-of-the-art models specialized in reasoning:
ol [1], o3-mini [1], and o4-mini [1]: Representing OpenAl's advancements in reasoning capabilities, particularly excelling solving complex problems.
DeepSeek-RV R1 [2]: model optimized for mathematical and reasoning tasks, with outstanding performance in multi-step reasoning.
--S-onnnt--hhniinn [4]: Employing specialized chain-of-thought design to enhance capabilities in solving complex reasoning tasks.
[2]: Transferring DeepSeek-RI ''s reasoning capabilities to smaller model through knowledge distillation.
QwQ-32B [31]: Alibaba's reasoning model trained from Qwen 2.5-3-.....BB..
This categorized comparison allows us comprehensively evaluate model's performance in both specialized reasoning capabilities and general abilities, while comparing against different types of state-of-the-art models. For evaluation, temperature sampling 1.0).
inttps//giituu..cm/ooeennggn
16

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Limitations
Due the limited time and resources, we did not train on other backbone models with ENIG- MATA-Data, nor with other RL algorithms other than VC-PPO. But we believe the results from our experiments can generalize other back models and algorithms. Additionally, our data all in single-turn textual form, and we did not include multi-tun puzzles or visual puzzles into ENIGMATA, which we leave for future research. Due to the research purpose of this work, we only train LLMs with hundreds of competitive math problems along with ENIGMATA- easily include our training resources to an already comprehensive training set while maintaining (if not surpassing) the original performance of tasks in the original dataset.
In practice, one could
17

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Hard
Medium
Easy
Big Bench Symbolic
Binario
Campsite
Car Painting
1.00
0.75
0.50
0.25
0.00
Checkmate In One
Countdown
Crypto Kka
Crypto Kpa
1.00
0.75
0.50
0.25
0.00
Full Crosswords
Hamiltonian Cycle
Hamiltonian Path
Hitori
1.00
0.75
0.50
0.25
0.00
Kakurasu
Knights And Knaves
Light Up
Magic Square
Average Reward
1.00
0.75
0.50
0.25
0.00
Minesweeper
Skyscraper
Stack Permutation
1.00
0.75
0.50
0.25
0.00
Star
Sudoku
Sudoku2
Sum Skyscraper
1.00
0.75
0.25
Symbalic Hard
Tac Toe
Twiddle
Zebra Logic
1.00
0.50
0.00
150
300
425
0
150
300
425 0
150
300
425
150
300
425
Training Steps
Figure 6: Reward curves for D0wen2.5332-EB.. ENIGMATA across all individual tasks during training. Each subplot represents different puzzle task, with the x-axis showing training steps and the y-axis showing average reward. Colors indicate different difficulty levels: Easy (green), Medium (blue), and Hard (red).
18

---

ENIGMATA Details
This section provides details about the ENIGMATA dataset, including task specifications, difficulty estimation methodology, and example cases.
Table 11: Details of 36 tasks in ENIGMATA.
Task
Categories
Source
Difficulty Vriables
Rules
Binario
Grid
Auto
grid
Given partially filled binary grid, determine its unique completion such that each row and culumn contains an equal numher
mask rate
filled cells f
Is, no adjacent, and all rows and columas are pairwise distinct.
identical digits
Campsite
Grid
Auto
grid height b. grid widhh ut tent count
Given a grid with designated tree cells and empty cells, place empty cells such that cent exactly one tree, no rwn tents are adjacent (including diagonally), the number tents each row column matches the specified totals.
orthogonaaly adja-
Magic Square
Grid
Auto
grid size mask rate r
Completc the partially filled NxN magic square assign- ing distinct integers to empty cells columns, and both main diagonals sum to magic number, while preserving the given cutries and satisfying structural construints.
rows,
Skyscraper
Grid
Auto
grid size
Given NxN grid, assign each cell unique building height from N per now and column, such that number visible buildings from each edge matches the provided visi- bility constraints, with taller buildings obscuring shorter ones. Retum valid configuration report infensibility.
Sun apes
Grid
Auto
grid size
with numbers N without repetition in any row or column, such that from each edge, the sum visible building heights- where taller buildings block shorter ones behind-is is equal to the coresponding
Given X grid,
Battle
Grid
Auto
grid number of star
Given NxN grid composed of emply cells and blocked cells, place exactly one star in each row and each column that stars occupy only empty adjacent hori horizontally vertically, diagonally.
Sudoku2
Grid
Auto
mask rale
Given a partially filled 4x4 Sudoku grid with digits 1-4, fill remaining cells subgrid contains each digit exactly once.
mw, column,
Sudoku
Grid
Auto
mask vate
Given a partially filled 9x9 Sudoku grid, complete that subgrid contains the digits
zow, column, through exactly once.
Full Crosswords
Grid
[40]
grid size n
Given a fixed crussword grid with blank and blocked cells, blank cells that satisfy the provided across and down clues, ensuring consistency intersections.
valid English words
Symbolic Harnd
Grid
42]
passrate/pass@kk
Given a symbolic grid of numbers, learn transforma- lion rule across and apply to generate the consistent output pattern, preserving vertical segment structures and alterating Os specific column bands.
Crypto KKA
Crypto
Auto
plaintext length i, encryption method e. keyword length k.
Known Key Attack-style Decryption. Given encrypted ciphertext and the specilication of an encryption method (e.g Caesar, Vigenère, Hill, etc.), recover the original plaintest without heing given the encryption algorithm or decryption procedure. The requires understanding classical ci- phers and applying the correct decryption logic based on the cipher name and parameter settings.
shift in Caesar s, rails Rail Fence r.
w/b range in Affine r. matrix size Hill
Crypto KPA
Crypto
Auto
plaintext length 1,
Known Plaintext Attack-style Decryption. Given ciphertext and saintext-ciphertter example pair only, reverse-cugineer encryption transformation- wittbrrnattion-itttttt being given the algo- rithmie rule or key-and apply the inferred pattern to decrypt target cipheriext. The involves cryptanalytic gener- alization: learning transformation from single annotated example, and applying unseen input.
encryption method e. keyword length k. hint range hs. shift Caeuar rails Rail Fonce range in Affine r.
matris size in Hill
19

---

F1
Task Details
We present the detailed specifications all tasks in ENIGMATA in Tables 11, 12, and 13. These tables provide comprehensive information about each task, including: (1) Task category: ENIGMATA encompasses seven distinct reasoning categories. (2) Data source: Whether the task is automatically generated or sourced from existing datasets. (3) Difficulty control variables: The specific parameters used to adjust task difficulty. (4) Rule descriptions: Concise explanations of the rules governing each puzzle type. These details illustrate the diversity and controllability of the ENIGMATA dataset, highlighting how each task contributes different aspects of reasoning assessment.
F.2 Difficulty Estimation
In Section $ 3.2, we introduce our method for difficulty control in ENIGMATA: determining different difficulty levels (easy, medium, hard) for puzzles through model pass @k metrics. Specifically, we use GPT-4o's s pass@k (k=1,10,, (0=1000,000 to establish these difficulty tiers. Table 14 showcases specific examples of puzzles at different difficulty levels. As demonstrated, there are clear distinctions in model pass @k performance between easy, medium, and hard levels across various tasks.
20

---

Table 12: Details of 36 tasks in ENIGMATA.
Task
Categories
Source
Difficulty Vriables
Rules
Twiddle
Soquential
Auto
grid size number of rotations r
Given NXN grid distinct numbers from to row-major order using sequence of counterclockwise kok subgrid rotations, each specified by top-left coordinate.
it
Car Painting
Soquential
Auto
be painted, each number of color type L. cur may be moved up to K positions forward or backward; objective reorder the cars, within constraint, minimize adjacent color transitions (color switches).
number cars
Given fixed initial sequence of
shift runge k. skow ninge
Stack Permutation
Sequential
Auto
sequence length
Given an input sequence, determine whether a targel outpui sequence can produced using a order) and pop operations, ensuring last-in- ia--in--frttoon (LIFF) behavior.
only push
Big Symbol
Sequential
[4]; 42]
passrote/pasr@tt
Given sequence input-output list pairs, identify and apply the underlying symbolic transformation function to new inpat list to produce coresponding output.
Puzzle
Sequentia
Auto
number of inversion
Given 3x3 grid representing an puzzle state, output shortest sequence of moves (Up. Down, Left, Right) to reach the goal configuration [[1, 2, [4, [4. 6), [7. 0]). or indicate if solution exists.
15 Puzzle
Soquential
Auto
number of inversion
Given a 4x4 grid representing 15-puzzle state, output the shortest sequence of moves (U. D. L., R) to reach the goal configuration [[]. 4). [5. 6, 8). 9, 10, 11, (13, 14, 15. OJ], or report if no solution exists.
Nine Puzzle
Soquential
Auto
number of inversion
Given a 3x3 grid of numbers 1-9, the player can circularly shill any row column by or positions. Determine sequence of moves that results in the grid sorted in ascending order, report
unsolvable.
Sixtaen Puzzle
Sequential
Auto
number inversion
Given 4x4 grid containing tiles numbered I-16, players may circularly shift any row or column by to 3 positions the grid into ascending order; determine valid move sequence or prove it unsolvable.
Hi tori
Search
Anto
grid size n
Given an NxN grid of numbers, black out cells to ensure each and column contains duplicate numbers, two blacked cells are orthogonally adjacent, and all remaining white cells form single connected group.
Kakurasu
Search
Auto
grid black rate r
Given grid with and column sum constraints, select black cells such that the sum of their positions in cach row and column equals the respective largets, where each cell's value is its based index.
Light Up
Search
Auto
grid size u black. cell ratio va
Given a rectangular grid with black numbered and unnum- bered cells, place hulbs on empty cells illuminate all white cells without li lighting another hulb, ensuring each numbered black cell has exactly that many adjacent bulbs.
numbered natio
Menewwepper
Search
Auto
board size n. mine density d., reveal ratio
Given partially revealed Minesweeper grid, identify unrevealed cells that must contain mines, based solely on the numerical clues and adjacency constraints.
Search
Auto
Given a grid numeric constraints at intersections, assign dingonal slashes(two directions) to each cell such that inter- section counts are satisfied and no loops are formed.
board col d, hint ratio
Checknate in One
Search
[43]
passratelpass@k
configuration. find move that results immediate checkmate-io Le.. the opposing king is placed in check and has no legal way to escape.
Given a logal
Tic Tac: Toe
Search
Auto
board size n comparative potential center control e. fork
Given a partially filled NxN Tic Tac Toe board and active player, identify the optimal move that maximizes the player's winning chances or prevents immediate loss, according to standard game rules.
G4
Arithmetic
Auto
number integers
Given four to six inlegers, each exactly once with X, +. and parentheses to construct valid expression that evaluntes 24.
Countdown
Arithmetic
Auto
number integers range larget valae r
Given live integers and target value, form valid arithmetic expression using each number exactly once and the operatons
x, such that all intermediate results positive integers and the final result equals the target.
21

---

Table 13: Details of 36 tasks in ENIGMATA.
Task
Categories
Source
Difticulty Vriables
Rules
Monintonian Cycle
Gruph
number of nodex
Given undirected graph, determine whelber Hamiltonia
edge donsity d
cycle exists; if so, output one such cycle.
tonian Path
Gruph
Auto
number of nodes a.
Given an undirected graph, determinn whelher a Hamillonian path if output such poth.
edge density
Navigation
Graph
Auto
shortest path length
road network among city landmarks, identify the shortest path from designated start- ing point to the nearest landmark of specified type.
Given spatial description
Graph
obstucle percentage p
Given grid based tions, the objective is to determine valid path from start end. Movement restricted (up. down, left, right), and traversal through blocked or im- passable
with designated and end posi-
cardinal directions
not allowed.
Knights
Logic
[44]
ambiguity
individuals are either knights nntruthful) or knaves (lying), decide if specific conclusion is logically entailed, contradicted. or undetermined.
Given statements
number of inhabitants
FOLIO
Logic
[45]
number of primises
Given set of premises, determine whether conclusion logically follows from them.
Zebra Logic
Logic
Auto
logic rule type columns c.
Given fixed table structure and set of categorical items with positional equality constraints. deduce unique one- to-one assigmment of all elements that satisfies all logical conditions without transposing rows and columns.
minimal conditions
Table 14: pass @k cases of GPT-4o
Task Names
Easy
Medium
Hard
pass
pass @
@ass@ pass@1
pass pass@10
pass
pass
pass 10
pass
Sudoku
Zebra Logic
Campsite
Crypto KKA
Crypto KPA
Maze
Magic Square
22

---

F.3
Task Cases
The following listings present examples of each puzzle type the ENIGMATA dataset. Each task includes a rule description and few-shot examples demonstrating puzzle mechanics. Color-coding in prompts indicates different puzzle categories. These examples highlight our dataset's diversity, from grid-based challenges (Sudoku, Star Battle) to sequential puzzles (Fight Puzzle, Fifteen Puzzle). Each puzzle challenges distinct cognitive faculties, collectively forming comprehensive dataset that spans a wide spectrum of reasoning capabilities.
Listing 1: Case of Binario
You tasked with solving Binario puzzle.
Rules:
1. The Binario puzzle is played on a grid of size NxN, where N is an even number.
filled either 3. No more than half of the cells in any row or colunn can centain the same number. more two identical nunbers
Each in grid
adjacent horizontally or vertically. 5. The puzzle must have unique solution.
Task:
Solve the following Binario puzzle by filling in the nissing cells (denoted by "L") with 8s rules
according
Output Format:
Please output answer within code block (*)) and fornat the grid numbers, for example:
0118
6
If solution exists, output within the code block:
No valid solution exists for the given Binario puzzle.
Puzzle:
Listing 2: Case of Campsi te
You are tasked with solving campsite puzzle.
Rules:
1. Notations
represented by tents represented ".", and enpty spaces represented by "." You will given and indications for the number of tents in each rew and column. Your goal is to place tents on empty spaces.
trees and enpty spaces, total number tents,
2. Constraints
board associated with tent, which is always horizontally vertically adjacent to it.
Every
horizontally, verticully disgonally adjacent another tent.
(3) The number of tents in each row and column should natch the given indications.
(4) The nunber of tents on the board should natch the given total number tents.
(5) tent only be associated with one tree. but can be adjacent to more than
tree.
Output Fornat:
board. 2. You nust not change trees (*x*) on the board, but only place tents (***) on empty spaces
cutput
include
followed by
(".
3.
The final board should wrapped between "<begin_boaree board>" and beend_borrd>" tags.
23

---

Place tents the enpty according to the given grid and rules.
Final Board:
<begin_hoard>
[Final Board]
<end_board>
Puzzle:
Here is the puzzle:
total number of tents: tents each 1 tents each colunn:
x
Listing 3: Case of Magic Square
You provided with Magic Square puzzle. Sone cells filled with numbers, while blank cells are represented by dots.
to
puzzle based follewing
Rules:
Magic square is 3x3 partially filled matrix.
2. You need to fill in the blanks in the natrix so that the sun of the nunbers in each row, column,
the diagonals equal.
3. You can only fill the blanks with integers, the filled natrix only consists of integers. filled
duplicate already filled nunbers.
5. Make that the of the numbers in each each colunn, and the two diagonals is equal.
Task:
blank cells according given numbers and Find valid magic square solution for the given puzzle.
Output Fornat:
Please output your answer within code block (*"), fornatted as a grid of numbers, for example:
2
Puzzle:
...
35 10
15
29 38
Listing Case of Skyscraper
Skycraper is logic puzzle where the goal is to fill grid natrix based an the given clues. Here
rules:
Rules:
grid natrix. Each cell building. and the building height is represented by number ranging from 1 to n. where is natrix.
Board: typically consists
2.
Building Heights: Each row and column nust be filled with numbers that represent building heights. Each number can only appear once a row or column, similar to Sudoku constraints
3. Visibility Clues: The hint nunbers outside the matrix indicate how many buildings can be of
that direction. Taller buildings block Thus, a hint number represents how many buildings are visible from one end of row or column.
buildings behind
For example, if the clue for a column is "3", it means that from the or hotton of that colunn. buildings can be seen, and the building heights nust increase, as shorter buildings will blocked taller ones.
4. Objective: Fill the entire matrix based on the clues, ensuring that the heights the buildings distinct
follow visibility
edges.

---

Example:
[11 [2] [3]
[2]
[2]
[3]
[2]
[1]
[2]
[4]
[1]
[2] [3] [2] (1)
This is example of Skycraper puzzle:
the numbers at the top and botton of the columns indicate how many building can be seen from direction. that buildings can be seen from the top, indicating that at least one building is hidden by a taller one
clue
top the first column is neaning
The left and right clues for the rows work similerly, indicating how buildings visible from that rew 's left or right side.
Task:
Now. given Skycraper puzzle, your
reconstruct height of the buildings in each cell. If there are multiple solutions, return any one. If valid solution exists, state that no solution exists.
Output Fornat:
your within code block (*")), formatted of numbers, for example:
Please
.**
2 3
2
Puzzle:
The input clues are:
Top:
Left:
2 3
Right: 2 2
Bottom:
Listing 5:
Case of Sum Skyscraper
Skycraper logic puzzle gane clues and the height and nunber restrictions of rows and colunns. Here are the basic rules:
fill natrix
Rules:
an n grid matrix. Each cell represents a building, with its height represented number ranging from n, where is the length of the natrix side.
Gane Board: Typically,
representing heights of the buildings. Each number can only appear ance in rew column, sinilar to Sudoku canstraints.
Building Heights: Each
filled
3. Visibility Clues: The hint nunbers outside the matrix tell you how many buildings can be seen from that direction. Taller buildings will block shorter buildings behind then. Therefore, hint number indicates the total height of buildings visible from end of the row or column.
For example, if colunn's that column, the total height of the visible buildings is 11. Additionally. the building heights ones in front.
"I1," means looking
botton
increasing order, neaning shorter buildings blocked by taller
4. Objective: Fill the entire matrix according to the clues, ensuring that the heights of each colunn different, and that they comply with the visibility clues on the sides.
buindings in
Example:
[4] [5] [9]
[7]
[6]
[9]
[5]
[4]
[7]
[18]]
[4]
25

---

[9] [7]
the above is exanple of
Skycraper:
The numbers at the top and botton of the columns indicate how many buildings can be seen direction.
example, , the hint that the total height of the visible buildings fron to The hints on the left and right of the rows are similar to those for the colunns.
first column "7," meaning top botton is 7.
indicating the total height of buildings visible fron the left and right sides of that For example, if the hint is 18, it means the visible buildings could possibly be 1, 2, 3, 4.
Task:
Now, given Sun Skycraper scenario, please restore height each building scenario. If there are nultiple answers, output any of then. If there valid solution, then respend with "no valid solutien."
Output Fornat:
four lines: first line represents the height from top to botton, the second line from left to right, the third line from right to left, and the fourth 1ine fron botton to top. Using the previous example, the input data would be:
input data
7
8 10
5
within code block (**"), formatted grid of numbers storing the heights the buildings, for example:
Please output
2
4
1
If no solution exists, the result should be: "No valid solution."
Puzzle:
7 9
7
Please provide the answer according to the above requirenents.
Listing 6: Case of Star Battle
You are tasked with solving custonized version of the star-battle puzzle. The star-battle puzzle is lagic puzzle requires the placement stars grid. There is only one solution to this puzzle.
Your goal is to determine the location of each star, adhering to the following rules:
Rules:
The initial board consists of empty cells and blocked cells.
-Enpty cells are denoted by '.', blocked cells are denoted by 'x' stars are denotad 'x'
2. STAR MUST BE PLACED IN EMPTY CELL:
-Blacked cells cannot contain stars.
You only change
denoted by '.', must not change denated by 'x',
3. EXACTLY STAR IN EACH ROW AND COLUMN:
Each row and column nust contain EXACTLY one star.
column.
There shouldn' 't be rows or columns without stars.
COLUMN-WISE
No two stars can be adjacent to each other, even diagonally.
Rew-wise adjacency: two stars are them.
same row, and there is no empty between
Column- -wise adjacency: two stars are in the same colunn, and there is no empty cell be then.
Diagonal adjacency: two stars are in the same diagonel, and there is no empty cell between
Note that you are prone to nake nistakes in diagonal adjacency, so be extra careful. Here are examples of diagonal adjacency a partial grid:
26

---

The final board is revised version of the initial board way that need In both cases, the two stars are diagonally adjacent. You should avoid this situation.
5. CHECK FOR CONSTRAINTS AND BACKTRACK:
each step, you should check you find inconsistencies, you should backtrack and try a different placenent..
violates the canstraints 2., and
you find that you should backtrack and try different placenent. Since there is only one solution, there great chance your first placement incorrect, which neans night need to start over.
place
without violating the constraints,
Task:
Deternin the location of each star and place then into the grid, adhering to the rules.
Output Fornat:
Your output should include the final board.
place a star in an empty cell, replace the "." with a 'x".
Don change the blocked cells denoted by 'x'.
Your final board should be wrapped between <begin.boards and <end_board> tags. following structure:
Final Board:
<begin_board [Final Board] <end_board>
Puzzle:
The input clues
X
X
X
Please provide the answer according to the abave requirements.
Listing 7: Case of Sudoku2
Sudoku puzzle. Some cells are filled numbers, while empty cells are represented by dots.
You are provided
Your task is to find a valid solution for the puzzle based on the following rules:
Rules:
Structure:
grid, divided
snaller subgrids
regions)
2. Number Range: Each cell can only contain number between
Row Rule: Each
contain the nunbers through 4, with repeats.
4. Column Rule: Each column must contain the numbers through 4, with no repeats.
Subgrid Rule:
subgrid
through
repeats.
FFind valid Sudoku solution for the given puzzle.
If there are multiple solutions, provide one.
Output Fornat:
within block (*1) representing solved board, formatted as grid of numbers, far example:
Please output your
4
Please provide answer according to the above requirements.
27

---

Listing 8: Case of Sudoku
You are provided with 9x3 Sudoku puzzle. Some cells are filled with numbers, while empty cells represented by dots.
Your task is to find valid solution for the puzzle based on the following rules:
Rules:
Structure:
grid, divided
snaller subgrids
regions).
2. Number Range: Each cell can only contain number between and 9.
3. Row Rule: Each
contain the nunbers through 9, with repeats.
4. Column Rule: Each column must contain the numbers through 9, with no repeats. Subgrid
subgrid
through
repeats.
Finnd valid Sudoku solution for the given puzzle.
If there are muitiple solutions. provide one.
Output Fornat:
Please output your answer within Formatted as grid of nunbers, far example:
(*)) representing solved board,
7
4
5
8
5
9
6
7
9
2
8
9
5
6
Puzzle:
5
3
8
2
2
9
4
Please provide according the above requirements.
Listing Case of Full Crosswords
Task:
Your task is to complete the crossword puzzle grid based on the given clues. Ensure that all words neaningful and semantically valid. The grid consists of fillable blank spaces ') and blocked spaces ('+'), You must strictly follow the provided grid layout, with no modifications allowed.
Rules:
1. completing the Grid:
Fill each blank space ('_") with letter to form valid words according the given
clues
Blocked spaces ('*') nust remain unchanged and cannot contain any letters. of
natch provided exactly. -1ll blank spaces ('"") must be filled--no empty spaces are allowed.
2. clue Mapping Logic:
(1) Across Clues:
without 'a' characters represent words. These words correspond clues in order from top to botton.
containing
correspond any across
(2) Down Clues:
Columns without '+' characters represent down words. These words correspond down clues in order from left to right.
Columns containing '.' do correspond any word clue.
Matching Letters Intersections:
Letters at the intersections of across and down words must match, ensuring valid wards are forned both horizontally and vertically.
Output Fornat:
Please output your within code (*1)) follows:
28

---

across:
ANSMER1, ANSWER2, ANSWER_N
down: ANSWERT, ANSMER2 ANSWER_N
"across" contains list across words, ordered top botton.
"down" contains the list of down words, ordered from left right.
Puzzle:
Across clues:
"London farewell!" (2001)
2. "Fassil mollusk" (1972)
"Radial counterpart" (2813)
Down clues:
"Heasure electric charge" (1999)
2. "Is baokends?" (2014)
Law-abiding (2018)
The grid follows:
Listing 10: Case of Symbolic Hard
Task and Rules:
Figure the pattern the following examples and apply the test Your nust follow the fornat of the examples.
Output Fornat:
Please output your answer within code block (*)), formatted as grid of nunbers.
Puzzle:
# Examples
Example 1:
[[2. 2, 2,
8,
2,
2,
2,
8
2,
2,
2]
[2, e, 2,
8,
2,
2,
2,
8
[2, 2. 2,
8
2,
2,
2,
8,
2,
2,
2]
8.
2.
8,
[2. 2.
2.
8
2.
2,
2,
8
2,
2,
21
[2, 2. 2,
8,
2,
8,
21
[2, 2. 2,
8,
2,
2,
2.
8,
231
[[2, 2.
8,
[2. 2.
2,
8
2,
0
2,
8
2,
8
2]
[2. 2.
2.
8
2
2.
[2. 2.
2,
8,
2
8
2
8
2,
8
21
[2, 2. 2,
8
2,
2,
8
21
[2. 0. 2,
8,
2,
2,
2,
8
2]
[2. 2. 2.
8
2.
2.
2.
8,
2,
2]]
Exanple 2:
[L2. 2. 2,
8
[2, 2. 2,
8
2
8
2,
8
2
8
2
2,
21.
[2, 2. 2,
8
2,
2,
2
8
2
2,
21.
[2. 2. 2.
8
2
8
8
2
8
2].
[2. 2. 2
8
2
2
2
8
2.
8,
21.
8.
2,
8
2,
2].
[2. 2.
2.
8
2
2
2.
8
2.
2
2.
8,
2.
2.
2].
[2, 6. 2,
8
2,
21,
[2, 2. 2,
8,
2,
2,
2,
8,
2,
2, 213
[[2. 2.
2.
8,
2.
2
2.
8
2
2].
[2. 2.
2.
8
2.
2.
2.
8
2.
8
8
8.
21.
2.
2.
8
2.
2].
[2. 8.
2.
8
2
2
2.
8
2.
8
2
8
2.
8,
21.
[2, 2. 2,
8
2,
2,
21.
[2. 2.
2,
8,
7
8
2
8
2
8
2
2,
8,
21.
[2. 2.
2,
8
2,
2,
2,
8
2
2,
2].
[2. 2.
2.
8
2.
8
8
21
[2. 2. 2,
8
2,
2,
2,
8
2,
2,
2,
2]]
29

---

Example
[[2, 2, 2,
8,
2,
2,
2,
8
2,
2,
8
2.
2,
21.
[2,
21
[2. 2, 2,
8,
2,
2,
2,
8
2
2
2
8
2,
2,
21.
[2, 2, 2,
8,
2,
8,
2,
8
2,
8,
21.
[2. 2, 2,
8
2,
2,
2.
8
2,
2].
[2. 6. 2,
8
2.
2,
2.
8
2,
21.
[2. 2.
2,
8
2,
8
21.
[2, 2, 2,
8,
2,
9,
2,
8
2,
2,
8,
2,
8,
21.
2,
8,
[[2, 2, 2,
8,
2,
2,
2,
8,
2,
2,
2, 2,
21.
[2. 2. 2.
8
2.
2.
2.
8
2
8
2].
[2. 2. 2.
8
2.
2.
2.
8
2.
8
21.
[2, 2,
2,
8
2,
8
21.
[2, 2, 2,
8
2,
2,
2,
8
2,
2,
2,
8
2, 2, 21.
8
2.
8
2,
2]
[2, 2,
2,
8,
2,
2,
2,
8
2,
21.
[2. 2,
2,
8
2,
2,
2,
8
2,
8, 2].
[2. 2. 2.
8,
2.
2.
2.
8
2. 2. 21]
Test Problen:
[[2, 2,
2,
8,
2,
2,
2,
8,
2,
2,
2,
8,
2.
2,
2,
8,
2. 2,
21.
[2.
2.
2.
8
2,
8
2]
[2,
2,
2,
8
2,
2,
2
8
2
2
2,
2,
21.
[2. 2,
2,
8
2,
2.
2.
8
2.
2.
2.
2.
2]
[2. 2,
2.
8
2.
2
2.
8
2.
21.
[2, 6, 2,
8
2,
2,
2,
8
2,
8
2.
8
2.
8,
2,
2,
8
2].
[2,
21.
[2.
2,
2,
8
2,
8
2,
8
2,
8
2.
8
2.
0
2.
8.
2.
8,
2].
L2.
2,
2,
8
2.
2.
2.
2,
[2. 8
2,
8,
2,
2,
2,
8
2
2,
2.
8,
2,
8,
2,
8,
21.
[2. 2.
2,
8
2
2,
2.
8
2.
2.
2.
2.
2]]
[Your Answer Here]
Listing 11: Case of Crypto KKA
Task:
to apply provided decryption decipher given ciphertext.
Rules:
Review the decryption rules carefully understand the encryption method works.
2. Decrypt the provided ciphertext according to the rules, and derive the correct plaintext.
Output Fornat:
output your answer within
(*1))
<result>
<result> should decrypted plaintext corresponding given ciphertext,
example:
LOVE
Puzzle:
Ciphertext: DBUDIBUFO0
Encryption Method: (" ('nethod's 'Caesar Cipher', 'shife': 1)
What's the corresponding plaintext?
Listing 12: Case of Crypto KPA
Your task is to decrypt the given ciphertext and provide the corresponding plaintext.. You be given sample hint that illustrates pair ciphertext matching plaintext to guide in solving the puzzle.
Rules:
1. Analyze the provided ciphertext.
sample hint reference understand encryption pattern 3. Apply the deciphering technique to convert the ciphertext into plaintext.
Output Fornat:
Please output your answer within code block (*)) as follows:
30

---

<result>
<result> shauld be the decrypted plaintext corresponding to the given ciphertext, for example:
ABCIDEFG
Puzzle:
hint: Known plaintext and ciphertext pair:
Plaintext:
Ciphertext:
Use the example above to decode:
RONYYGBERT
Listing 13: Case of Twiddle
Given a 3x3 sliding puzzle where each cell contains a number (1 to 9), your goal is to the puzzle its original sorted order through
rotation operations.
You select 2x2 region within the 3x3 puzzle and rotate the positions of these cells counterclockwise.
The goal restore the puzzle its initial state shown below):
5 6
Task:
Please provide the steps the puzzle its initial state.
Output Fornat:
Please output your answer within code block (**) follows:
<result>
Replace '<result>" with a sequence of rotation steps, where each step is represented by coordinate (i, j) indicating selection 2x2 region with (i, i) the top-left corner for counterclockwise rotation. For example:
(0,0)->(1,1))200)
The problem guarantees that solution exists.
Puzzle:
3
9 2
Please provide the answer according to the above requirements.
Listing 14: Case Car Painting
You tasked with solving Painting Problen.
In the Car Painting Problem, you play the role of scheduler at car painting factory. Your job arrange painting sequence
cars nininize nunber color switches, reducing paint waste and production time.
Rules:
1. There are cars nunbered from to that need to be painted. Each predetermined (labeled A, B, c. etc.. for colors).
31

---

3. Cars enter the painting workshop in fixed order. but be rearranged within
4. Each car can be noved forward or backward by at nost pasitions from its original position.
5. A color switch occurs when two adjacent cars have different colors, adding to the cost. Your goal is to minimize the number of color
by optinally arranging cars.
Task:
Find rearranged sequence of that mininizes the number of color switches.
You provide list of 10s their order (rearranged minimize color switches
)
Puzzle:
Given the following information:
Nunber Cars (N): 14
Nunber Colors (H): 2
Maximun Novement Range (K):
Initial car sequence: [1, 2, 3, 4, 5, 6, 7, 8, 9,
18
11,
12, 13,
14]
Corresponding color: ['B',
8 8
18 18",
Please find rearranged sequence that mininizes the nunber of colar switches. Remenber: Each car can only be noved at most positions forward or backward from its original position.
Output Fornat:
Please output your within code block (*)),, formatted representing the new order of car IDs, for example:
integers
[2, 5.
10]
Replace "cresult>" with a sequence rotation steps, where each step is represented by 20 coordinate (i, indicating the selection the 2x2 region with (i, j) the left corner far counterclockwise rotation. For exanple:
(0.0)-2(1,1)>0660
The problem guarantees that solution exists.
Listing 15:
Case of Stack Permutation
Stock permutation is a permutation problen related to stacks (a last-in, first-out data structure). Given input sequence, the goal generate different output using stack operations (push and pop). sequence is called a valid stack permutation if it can
through
operations.
Task and Rules: Given
of natural numbers, such [I, 4J, output sequence can be obtained through stack operations. The stack operations include:
to know if particular
1. Push: Add numbers from the input sequence to the stack in order. top
Pop: Remove
output sequence.
Example:
Suppose the input sequence is [1, 2, 3]. Here are sone possible valid stack permutations: Directly push elements into the
then pop then order. [2, 1, 3]: Push and 2 into the stack. pop 2. then pop 1, and finally push and pop 3. pop
[3, 1]:
stack,
reverse order.
Output Fornat:
Please your within code block (**"), for
("Push(1)", "Fop()", "P)sh "P)sh "Pop()". "Pop ()
output sequence not within the code black:
permutation input sequence, autput
"The output
valid stack pernutation."
Puzzle:
Input sequence: [5, 2, 1, Output sequence: [5, 3, 1, 2]
Please provide the solution according requirenents below.
32

---

Listing 16: Case of Big Bench Symbolic
Task and Rules:
Apply function the final input list to generate the list. Use any preceding inputs and outputs as examples to find what is the function used. All exanple outputs have generated using
function.
Output
Please output answer within code block (*"),, formatted list of numbers, for example:
[e, 2, 3]
Puzzle:
Examples
Example 1:
17. 7. 9. 21. 7. 4, 4, 91, 81 17. 9, 21. 7. 4, 4, 91. 91
Example 2:
[7. 78, 78,
L78, 78,
Example 3:
[9, 7, 72, 44,
44]
(9. 72, 44, 443
Example 4:
17. 8, 7. 7] >> 18, 7, 7J
* Test Problem:
[5. 37, 97, 48, 7, 11
Listing 17: Case of Puzzle
Task:
The puzzle is classic sliding puzzle game. consists of containing numbered tiles from 1 to and one blank space (represented by 8) The player moves these tiles with the ultimate goal arranging in order rules:
Below the detailed
Rules:
1. Initial State:
initial of puzzle numbered tiles randomly distributed
wwit the blank space located anywhere.
The puzzle usually starts from scrambled state.
2. Movement:
The player can nove a tile adjacent to the blank space into the blank space.
-iles only
the four directions: up (U), down (D), left (L), and right (R). Only one tile can be moved at time.
The ultimate goal is to arrange the tiles in order from left to right, top to bottom.
as follows:
1 2
7 8 8
Output Fornat:
Please output your answer within a code block (*1) as follows:
<result>
answer, the is sequence noves, example:
LRURDL
If there is no answer, the is:
No feasible nove path exists.
Puzzle:
5
3
Listing 18: Case of 15 Puzzle
33

---

Task:
The 15 puzzle classic sliding puzzle gane. It consists a 4+4 grid conteining 15 numbered tiles fron to 15 and blank space (represented by 8). The player moves these tiles with the ultimate goal arranging then in order from to 15. Below are the detailed rules:
Rules:
1. Initial State:
The initial state of the puzzle is 15 numbered tiles randonly distributed 4*4 grid, with the blank space located anywhere.
The puzzle usually starts from a scrambled state.
Movement:
The player can nove a tile adjacent to the blank space into the blank space.
directions: up (U), down (D), left (L), right (R). Only one tile can be noved at time.
only
3. Geal:
The ultimate is to the tiles order from left to right, to botton,
as fallaws:
1 2
18 11 12
13
14
15
Output Fornat:
Please output your within block (*1)) follows:
<result>
If there is the is the sequence far example:
LRURDL
answer, the
No feasible nove path exists.
Puzzle:
3
11
5
8
14
8
Listing 19: Case of Nine Puzzle
Task:
The nine puzzle is classic sliding number puzzle. consists of containing 9 tiles numbered from 1 to 9, Players can choose to move an entire row or column in circular The ultimate goal arrange tiles numerical 9. The detailed rules are as follows:
fashion
Rules:
1. Initial State:
of puzzle consists number tiles randonly arranged The puzzle typically starts from scrambled state.
The initial
grid
Movement:
Players can choose to move an entire row ar column, shifting it by to 2 steps in example: steps results in 3 2.
circular manner.
3, shifting by step results
1, shifting by
represent row moves as RAB, where A is the row number and is the nunber steps. represented CAB, where
Similarly, column
colunn number and 8
the number of steps.
The ultinate goal is to arrange the tiles in order from left to right, top to botton as follows:
1
Output Fornat:
If solution exists, output the sequence of moves within code black (**"), for example:
["RI1", "C23", "R32", "C12", "R23", "C31"]
solution output within code block:
34

---

"No valid sequence noves exists."
4
Listing 20: Case of Sixteen Puzzle
Task:
The sixteen puzzle is classic sliding number puzzle. consists of 4x4 containing tiles numbered from 1 to 16. Players can choose to nave an entire row or column in a circular fashion each time. The ultimate goal
the tiles in numerical order
from to 16. The detailed rules are as follows:
Rules:
1. Initial State:
The initial state of the puzzle consists of 16 number tiles randenly arranged 4x4 grid.
The puzzle typically
from scrambled
2. Movenent:
row column, shifting it by circular nanner. For example: 2 4, shifting
steps step results in 3 1, shifting by 2 steps results in 3 2, and shifting by steps results in 2. RAB, where A the number and the nunber of steps. similarly, column moves are represented as CAB, where is the column number and 8 is number steps.
Players can
an
3. Goal:
ultimate goal arrange tiles
to right, top
follows:
1 2
9 10 11 12
Output
If solution exists, output the of within code block (*)), for example:
["R11", *C23", "R32", "C12", "R23", "c31"]
output within
valid sequence noves exists."
Puzzle:
12
5 16
Listing 21: Case of Hi tori
You are tasked with solving litari puzzle.
Rules:
1. The puzzle is played on an NxN grid (where N is an even number). with each cell
containing number.
2. Your geal is to "black out" certain cells, following these rules:
colunn,
number cannet appear
once.
repetitions,
must black out sone af the cells.
-Baakk cells cannot be adjacent, either horizontally or vertically.
-All white cells (cells that not blacked out) must be connected, neaning travel between any two white cells through harizontel or vertical moves.
Task:
puzzle by blacking out cells You must pravide the coordinates of the blacked-out cells.
Salve fallewing
Output Fornat:
Please output your answer within a code block (**),, formatted as list of coordinates, for example:
35

---

[[e, 8), (1, 3), (3, 2)1
If no solution exists, output within the code black:
"No valid solution exists for the given Hitori puzzle."
Puzzle:
[[5,
3,
1, 3, 51.
[1,
4, 3, 5,
31
[1.
3.
2, 4,
31.
[3.
5.
1,
1
21.
C3.
1,
4,
2,
211
Listing 22: Case of Kakurasu
You are tasked with solving Kakurasu puzzle.
Task Rules:
puzzle played on rectangular grid (with arbitrary rew
sizes).
2. Your goal is to "black out" certain cells, following these rules: The black cells in each row must sum up to the target number for that row.
The black cells in each colunn must
the target nunber for that column.
To calculate the row suft:
position deternined by For example, the first black cell in row has a value of 1, the second black cell has on.
value
cells
row.
value of 2,
To calculate the column sum:
The value the black cells in each column is deternined by their position in the column. For example, the first black cell in column has value of the second black cell has a value of 2, and so on. Coordinates column 1.
For example, first row 1, first column
Puzzle:
Solve the following Kekurasu puzzle by blacking out the cells where needed.
Board size: 4
Row sums: [o, 5, 10, 5]
Colunn suns:
You must provide the coordinates of the blacked-out cells.
Output Fornat:
Please output your answer within a code block (*)), formatted as list of coordinates, far example:
[(1 1), (2, 4), (4, 3)1
If no solution exists, output within the code block:
"No valid solution exists for the given Kakurasu puzzle."
Puzzle:
[[5.
3.
1,
3,
5]
[l.
4,
3,
5,
[1
3,
2,
4,
3]
1,
1,
[3.
1.
4,
2,
2]]
Listing 23: Case of Light Up
You
Light puzzle.
1. The puzzle is played rectangular grid (the number of rows and colunns is not fixed).
2. The gosl is to place light bulbs (represented by L) on the enpty squares of the grid, following these rules:
Each nunbered black square (represented by nunbers 1-4) nust have the specified number around
light
example, black square
"1" means
have exactly
1 light bulb around it, "2a neans exactly light bulbs, and so
Light bulbs can only placed on empty squares (represented by .) nust only light they cannot light other light bulbs.
enpty
Black squares (represented by U) cannot be illuninated and block the light the light bulbs.
36

---

Puzzle:
Solve the following Light puzzle by placing the required light bulbs (L)
d.
...l..
You need to provide the coordinates of the light bulbs in the following format:
Output Fornat:
Please output your answer within a code block (*)), formatted as 1ist numbers. For example:
[[2, 4), 2), 5), (0, (4, 3). (1, 1), 9), 1), O, (3,
If no solution exists, output within the code block:
"No solution, unable to solve the given Light Up puzzle."
Listing 24: Case of Minesweeper
You are playing Minesweeper gane...
composed of different cells, the player's goal deduce cells that contain mines and avoid clicking on nines. in gane may either
adjacent
Rules:
Grid and Nines
The game grid consists of several cells, each of which may be: gane -Nubered cell: This cell displays the number of nines adjacent to it. The number indicates how many
-Mine:
player clicks on
eight neighboring 2. Current Grid State Representation
The grid state is represented as:
Indicates the cell unknown (not revealed).
Revealed non-mine cells, where the nunber indicates how many mines are adjacent to example, cell nine adjacent, and on.
0-8:
adjacent, means
means
Task:
Based on current grid state, infer positions of the mines that can be definitively deternined.
Puzzle:
The current grid state is:
[[-2, -2,
-2,
-2 -2,
-2, -2, 81,
[-2, -2, -2
8,
8,
-2,
-2 8].
[-2, 2, 2,
2,
2,
-2,
-2, -2 -2]
[-2,
-2,
-2
-2],
[-2, -2, 2,
2.
1,
=2 -2 -2]
[-2, -2,
-2,
-,
-2, -2], 2, -21
[1, -2,
1,
2,
22
22,
[2, -2,
-2,
-2,
-2
8,
-2 -21 8,
[-2, -2 -2
1,
22
1]]
Coordinate Explanation:
Coordinates start Fron (0, 8), where the first row, first column is (0. 0), the second row second
(1, 1),
on.
You need to provide the coordinates of the determinable nines in the following fornat:
Output Fornat:
Please output your answer within a code block (**),, formatted as list of coordinates (r for example:
[o. 1), (2, 4) (4, 3)]
If no solution exists, autput within the code black:
"Unable determine any nine locations."
37

---

Listing 25: Case of Slant
You need to solve a Slant puzzle.
Task Rules:
Grid Numbers:
Each cell in the grid contain number, indicating how many diagenal lines meet at 4, representing
intersection.
number ranges
intersecting
diagonal lines.
2. Diagonal Line Rules:
Each cell nust contain diagonal line, either "/" (forward slash, representing
-left to botton-right) or a "\" (backslash, representing top-right to botton-left).
Intersection Numbers:
The number indicates how many diagenal lines neet at that intersection. For example: Indicates diagenal line intersects that point.
Number 2: Indicates diagonal lines intersect that point.
Number 8: Indicates no diagonal lines intersect at that point.
Numbers 3 and Represent and intersecting diagonal lines, respectively.
4. No Loops:
diagonal cycle can be formed.
loops. diagonal
connect, clased
Puzzle:
Solve the Following slant puzzle:
6
I 8 2
2
2
Output Fornat:
Please output your answer within (***), formatted grid numbers, example:
represents (forward slash, top-left botton-right
-1 represents * (backslash, top-right to botton-left)
Listing 26: Case of Checkmate in One
Task:
You are tasked with finding move chess position resulting checkmate:
Output Fornat:
-Please output your within code block (*11) follows, for exanple:
Rg54
Puzzle:
Here is the chess position:
2. g3 e6
Ag2
4. exd5 exds
Nc3
6. Nf3 b6
7.
9. Ne5 Bd6
11. Qc2 Nc6
12. Nxd5 Nxd4
13. 4 Kha
14. Qxc4 Bxe5
Bxe5
15. Qxd4 Qe7
17.
18. Bxg7+ Kg8
19. 0xd5 RFd8
28. QeS Qb4
21. Rad1 Re8
Qe4
23. Bh6+ Kh8
24.
38

---

Listing 27: Case of Tic Tac Toe
You are tasked with solving Tic Tac Toe puzzle.
Task Rules:
consists 3x3 cells.
2. Players take turns placing their nark on an empty cell, one nove per turn. The two players use or "x",
3. A player wins by placing three of their marks consecutively in colunn, or diagonal
4. the board is completely filled without winner. the is draw.
playing tic-tac-toe
Puzzle:
Current board:
0 I x
0
x
Question: What is the best next move? Please provide only your nove and display the board.
Output Fornat:
Please output your answer within (**), example:
"x" on
"X" "0- "X"
Listing 28: Case of Game24
You given four or five six integers, ranging from to 13, provide arithmetic expression that results in 24.
Task Rules:
You nust all the
numbers,
exactly
The operators you can use include: addition (+), subtraction (-), multiplication (+). and division (/).
You use parentheses to change the order of operations.
Output Fornat:
Please output your answer within code block (*") as follows:
<result>
...
If there is solution, <result> is the sequence of numbers and operators that results in
24, for example:
(8 / (8 2)
If there is no solution, <result> is "annnot forn 24"
Puzzle:
Input: 7, 1, 7, 13
Please provide solution for the according the above rules and input.
Listing 29: Case of Countdown
You given integers and a target number. Your expression that results in exactly the target number.
create arithmetic
Task Rules:
You must use ALL the given numbers, each exactly once. include:
operators you division (/).
(+), subtraction (+), multiplication (*),
You can use parentheses change order operations.
All intermediate results nust be positive integers (no fractions or negative numbers
allowed).
Output Fornat:
Please output your answer within
(*") follows:
<result>
If there is a solution, <result> is the sequence of numbers and operators that results in for example:
39

---

(8 / 2) (8 2)
If there is no solution, <result> is "cannot form 85"
Puzzle:
Numbers: 10, 5, 15, 2,
Target: 85
Please provide a solution for the game accarding to the above rules and input.
Listing 30: Case of Hami I tonian Cycle
You are tasked with solving Hamiltonian Cycle Puzzle.
Task Rules:
A Haniltonian Cycle an undirected graph a cycle that visits every vertex exactly once and returns the starting vertex. The task daternine whether Hamiltonian Cycle exists in the given graph.
The graph is represented as follows:
The first 1ine contains a single integer 'N", the number of vertices the graph. The subsequent lines each describe edge in the graph. Each edge rearesented by two space-separated integers "u" and "v", which indicate that there is an rnditdcted edge "v",
between vertex and
The vertices are numbered from "0" to 'N-1".
Output Fornat:
Please output your answer within a code block (*1) as follows:
<result>
...
Hamiltonian Cycle exists, <result> should
vertex indices
cycle. where the last vertex is the same the first vertex to complete the cycle, for example:
[o. 2. 3, 0]
If no Hamiltonian Cycle exists, <result> should be "NO".
Puzzle:
18
Listing 31: Case of Hami tonian an Path
You are tasked with solving Haniltonian Path Puzzle.
Task Rules:
Haniltonian Path in an undirected graph is path that visits every vertex exactly once. determine whether Hamiltonian Path exists the graph.
The task
The graph is represented The first line contains a single integer 'N', the nunber of vertices in the graph. subsequent
each describe edge the graph. edge represented by two space-separated integers "u" and "v", which indicate that there is an undirected edge between vertex "u" and vertex "v", The
numbered from
Output Fornat:
Please output your answer within code block (**") follows:
<result>
<result> should list vertex indices that form the Hamiltonian Path exists, for example: ...
[o, 2, 3, 1, 41
If Hamiltonian Path exists, <result> should "NO".
Puzzle:
40

---

12
13
16
11
5 13
10
11
13
7 11
10
12
Listing 32: Case of NL Navigation
You are tasked with solving spatial reasoning puzzle involving navigation in city, Follow these guidelines to deternine the shortest
specific landnark:
1. Landwarks Definition: Iddntify
landnarks which include: store, bank, house, cinemo, garden,
school.
The total nunber of landmarks in the puzzle will range from to 10.
Structure:
The landnarks are organized in binary tree structure.
root
represents starting point navigation.
3. Objective:
Your goal
shortest path
starting point nearest specified
type of landnark.
4. Puzzle Input:
will receive question.
Use the information provided in the question to determine the path.
Output Fornat:
Please output your answer "([A-Z, 1+)* letters, for example:
block (*"),, containing only path
EFJ
Your answer should only include the uppercase letters representing landnarks
path.
If the path is direct with no internediate landmarks, provide an empty code black:
Puzzle:
Here 1s the puzzle:
Story: There is of roads and set of landnarks. The start point cinema H.
There is a road which is 128 neters long from cinena to house F.
which There is a road which is 288 neters long from store I to cinema A.
There
neters long from
F
to bank There road which is 198 neters long from bank to house B.
long
There is à road which is 200 neters long from cinema to house D.
There road which 188 neters
fram house D bank E.
There is a road which is 280 neters long from hause D cinema C.
Question: Fron the start point, haw to reach the nearest bank?
Please provide the solution according to the requirenents above.
41

---

Listing 33: Case of Maze
You are tasked with solving Maze Puzzle.
Puzzle: 5x5 maze nap, below:
Where:
represents start point (located top-left corner coordinates (1, 1)) E represents the end point (located in the botton-right corner coordinates (5, 5)) represents an obstacle (impassable) represents open
(passable)
Rules:
1. You can only nove up, down, left, right, not diagonally.
You cannot pass through obstacles (8).
freely 4. The goal is to find a path from the start point (S) to the end point (E).
You
spaces (.).
Please find a valid path from the start point (S) to the end point (E). If there are multiple paths, pravide any one
no valid exists,
impossible to reach the end point.
Output Fornat:
-Please output your answer within a code block (*)) as follows:
<result>
...
-If there is path, <result> is the sequence of coordinates in the path. For example:
(1.11==((13)--(33))
II exists, output diractly:
not path start to
Please provide the solution according to the requirements above.
Listing 34: Case of Knights and Knaves
In this puzzle, each person is either knight or knave. Knights always tell the truth, while knaves always 1ie. Your information provided.
presented with scenario involving inhabitants of an island where
determine the truth value of statement based
Task Rules:
Knights always tell the truth.
2. Knaves always lie.
3. Use logical reasoning to determine the truth value of the statement.
Output Fornat:
Please output your answer within
(***) follows:
<result>
Options:
"Entailment":
statement logically
information
provided.
"Contradictian";
the statement contradicts
logical deductions.
*Unknown" : Use this if the truth value of the statement cannot be determined with the given information.
Puzzle:
either
knight, knights always
truth while knaves always lie.
You neat four inhabitants: Alice, Bill,
and -sill tells you that Hel and Ted are not the same.
-Mel claims that it is false that Alice is knave.
knight and Hel Alice tells you that only a knave would say that Ted is knave.
Ted that Alice
knave.
Can deternine who is knight and who is knave?
Question: Is the knight?
42

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Listing 35: Case of FOLIO
Analyze the given premi ses determine the validity the conclusion. Your task whether the canclusion is "True," "False, or "ünknown" based the information provided.
Task Rules:
statements or premises. prenises are the foundational truths or assunptions for the puzzle.
Premises:
provided
2. Conclusion: A statenent will be presented as the conclusion. Your task evaluate this conclusion in the context of the given
3. Evaluation Criteria:
True: conclusion logically
prenises.
False: The conclusion contradicts the prenises.
Unknown: The conclusion cannot determined insufficient information.
premises due
Output Fornat:
Please output your answer within a cade block (*))) as follows:
<result>
...
Replace "<result>" with one of the following options: "True", "False", or "Unknown",
Puzzle:
premises:
Elephantopus genus perennial plants
daisy family.
Elephantopus is widespread over much of Africa, southern Asia, Australia, and the Americas
Several species Elephantepus are native to the southeastern United States. Elephentopus scaber is traditional medicine.
conclusion: Elephantopus is native to the southeastern United States.
Note: Ensure that your evalustion is based solely an the information provided in the premises introducing
knawledge assumptions.
Listing 36: Case of Zebra Logic
You
solving grid puzzle. type puzzle requires careful analysis the previded background information and clues to deduce the correct arrangenent of elements grid fornat. Follow steps the specified fornat.
puzzle present your
Task Rules:
1. Background Information: Carefully read any introductory information provided with the apply puzzle.
puzzle.
may include context er specific constraints
2. Clues: Analyze each clue given. These clues will guide in determining the
relationships between different elements in
grid.
3. Logical Deduction: Use logical reasoning to deduce the correct placement of each element the grid. Consider all possible options and eliminate those that contradict the clues. Check: Ensure that
solution
clues
background information provided.
respense should include salution followed by final answer format. Use the follawing structure:
Assume column is Year, column 2 Wine, column is Type.
Output Fornat:
Pleass output your
within
block (*11) follows:
1984
[Correct Wine]
[Correct Type]
1988
[Correct Wine]
[Correct Type]
1992
[Correct Wine] [Correct Type]
1996
[Correct Wine]
[Correct Type]
Puzzle:
apricot, Hobby: baking, card-games Job: bartender. writer Nationality: canadian. egyptian
Natiomality:comadian
Jobswriter is the right of Fapd:apricet
2.
Fill the following table to show final answer
Food
correct answer correct answer correct correct answer correct answer correct answer correct answer
Habby Job
correct
Nationality
You must stick given uncompleted table must not transpose table.
43

---

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49


Reviews:
[{"review_id": "tsOAMs1AZ7", "reviewer": "Reviewer_jHNx", "summary": "The paper introduces Enigmata, a comprehensive and scalable benchmark suite aimed at enhancing puzzle reasoning in LLMs. It features 36 tasks across 7 categories, each equipped with a deterministic rule-based verifier, enabling large-scale multi-task training and integration with Reinforcement Learning with Verifiable Rewards (RLVR). The authors also propose Enigmata-Eval as a rigorous evaluation benchmark and develop multi-task RLVR strategies. Their trained model, Qwen2.5-32B-Enigmata, outperforms leading models like o1 and o3-mini-high on several reasoning benchmarks about puzzles and maths.", "questions": "**Choice of VC-PPO**: The paper adopts VC-PPO, but it’s unclear why this particular variant was chosen over alternatives such as VC-GRPO or VC-DAPO. A more comprehensive comparison with other common RL algorithms would help clarify whether VC-PPO is especially well-suited for this setting or whether other methods might perform similarly or better.", "limitations": "See Weaknesses part.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The model demonstrates impressive performance across a comprehensive suite of puzzle and math benchmarks, even performing on par with the proprietary LLMs. The writing is clear and accessible, and the method and analysis is presented in a simple yet effective manner. However, the paper may lack key experiments as follows:\n\n**Transferability of Puzzle Training Data Alone**: It’s encouraging to see that combining puzzle and math tasks in the proposed training framework yields strong performance across both domains. However, it remains unclear how effective puzzle training data alone is at supporting generalization, especially given that many puzzles involve complex, long-horizon reasoning and mathematical computation. A deeper investigation into the standalone impact of puzzle data on reasoning capabilities would be valuable. Also, identifying which specific puzzle domains contribute most to transferable skills could guide future efforts in data selection for promoting general reasoning abilities.\n\n**Cross-Domain Transferability in Puzzle Games**: It would also be insightful to explore whether training on one puzzle domain facilitates generalization to others. Since many puzzles share underlying cognitive skills (e.g., constraint satisfaction, pattern recognition), cross-domain transfer is a promising area of inquiry. A data efficiency study, such as identifying the most representative subsets of puzzles for skill acquisition, could further illuminate how to build compact yet powerful training curricula.\n\n**Transfer Across Difficulty Levels**: The dataset spans tasks of varying difficulty: easy, medium, hard. An interesting question is whether models trained only on easy tasks can generalize to harder ones, which was observed in other domains such as SimpleRL (Zeng et al., 2025). Studying this easy-to-hard generalization would provide insight into how task difficulty impacts skill development and transfer.", "was_revised": true, "final_justification": "Thank you for the comprehensive evaluation results! I’m particularly impressed by the range of transferability experiments. It’s worth highlighting that puzzles can be highly effective in enhancing other skills as well. I’ll go ahead and raise my score to 5.", "rebuttal": "Thank you for the insightful feedback! We appreciate your recognition of the model's strong performance across both puzzle and math benchmarks, as well as the clarity of our presentation.\n\n## Here are the responses to your concerns\n> W1: Transferability of Puzzle Training Data Alone. It remains unclear how effective puzzle training data alone is at supporting generalization. A deeper investigation into the standalone impact of puzzle data on reasoning capabilities would be valuable.\n\nA1: Thanks for highlighting this. To better evaluate the standalone effect of puzzle data, we conducted an experiment where the model was trained only on the Enigmata dataset, without any additional math or external reasoning tasks. The results are shown in the table below.\nWe observe that training with Enigmata puzzle data alone not only substantially improves performance on Enigmata-Eval, but also yields strong generalization to out-of-domain puzzle benchmarks such as ARC-AGI 1, ARC-AGI 2, and KORBench during the RL phase.\n\n| **Model** | **ARC-AGI 1** | **ARC-AGI 2** | **Enigmata-Eval** | **KORBench** | **AIME 24** |\n|-------|-----------|-----------|---------------|----------|---------|\n| **Qwen2.5-32B-RFT** | 8.4 | 0.0 | 46.6 | 61.0 | 62.0 |\n| **Qwen2.5-32B w/ Enigmata** | *15.3* | *0.1* | *62.6* | *64.0* | 58.5 | \n| **Qwen2.5-32B w/ Enigmata+ARC-AGI+AIME** | 32.8 | 0.6 | 62.6 | 65.0 | 60.6 |\n\nFurthermore, we observe similar trends at a larger scale using a proprietary MoE model (20B activated parameters, 200B total parameters, anonymized during review). By adding Enigmata to the original full training corpus (**MoE-Model+Enigmata**), we observe consistent performance improvements across multiple reasoning benchmarks, including AIME and GPQA for math and STEM reasoning.\n\n| **Model** | **Enigmata-Eval** | **AIME 2024** | **AIME 2025** | **GPQA Diamond** |\n|:--------------------|:------------------|:--------------|:--------------|:-----------------|\n| **MoE-Model** | 52.9 | 86.7 | 74.0 | 77.3 |\n| **MoE-Model w/ Enigmata**| 64.6 (+11.7) | 87.5(+0.8) | 75.9(+1.9) | 78.1(+0.8) |\n\n\n> W2: Cross-Domain Transferability in Puzzle Games. It would be insightful to explore whether training on one puzzle domain facilitates generalization to others. A data efficiency study could help identify compact training curricula.\n\nA2: We appreciate your suggestion. To directly evaluate cross-domain transferability, we conducted a new experiment where models were trained via SFT on only one puzzle domain, and then evaluated on the remaining six. This setup allows us to measure how effectively reasoning skills learned in one domain transfer to others.\nThe results are summarized below:\n\n| Training Domain | Crypto | Arithmetic | Logic | Grid | Graph | Search | Sequential | **Avg. Improvement** |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| **Search** | +6.67 | +14.37 | +14.71 | +11.22 | +25.97 | (+26.38) | +8.96 | **+13.65** |\n| **Sequential** | +10.67 | +18.37 | +12.49 | +6.33 | +21.61 | +3.88 | (+9.16) | **+12.23** |\n| **Grid** | +9.33 | +10.70 | +13.38 | (+25.07) | +25.25 | +1.37 | +9.16 | **+11.53** |\n| **Graph** | +5.33 | +15.03 | +10.04 | +5.07 | (+37.79) | +4.38 | +7.38 | **+7.87** |\n| **Crypto** | (+51.33) | +16.03 | +10.93 | +4.70 | +16.15 | +0.38 | +7.77 | **+9.33** |\n| **Arithmetic** | +4.33 | (+58.70) | +8.49 | -1.97 | +11.25 | -1.12 | +4.90 | **+4.31** |\n| **Logic** | +0.33 | +2.37 | (+18.27) | -2.11 | +6.70 | -0.50 | +3.21 | **+1.67** |\n\nAs shown, training on the Search domain leads to the strongest average transfer (+13.65) across other domains. This suggests that Search puzzles—characterized by long-horizon, combinatorial reasoning—promote general cognitive skills such as constraint satisfaction, backtracking, and planning. These skills appear to transfer well to a variety of symbolic reasoning tasks, making Search a promising candidate for curriculum-efficient training.\nWe believe this analysis provides a concrete foundation for selecting representative domains to construct compact yet generalizable reasoning datasets.\n\n> W3: Transfer Across Difficulty Levels. An interesting question is whether models trained only on easy tasks can generalize to harder ones, which was observed in other domains such as SimpleRL (Zeng et al., 2025). Studying this easy-to-hard generalization would provide insight into how task difficulty impacts skill development and transfer.\n\nA3: We appreciate you for raising this insightful point. We conducted targeted experiments to explore easy-to-hard generalization, particularly within the ARC-AGI benchmark.\nWe trained models using ARC-AGI data with two different prompt length limits—4K (representing \"easy\" short-context training) and 6K (medium-length). We then evaluated performance on test samples grouped by their required context lengths. As shown below, models trained with shorter prompts struggled significantly on longer/harder inputs, demonstrating that generalization across difficulty levels remains limited:\n\n| Train Prompt Len | Model | Overall Performance | Test Prompt Len: (0, 4096] | Test Prompt Len: (4096, 6144] | Test Prompt Len: (6144, ∞) |\n|------------------|-------|-------------------|---------------------------|------------------------------|---------------------------|\n| 6k | arc-agi1 | 39.6 | 49.35% | 17.31% | 7.33% |\n| 6k | arc-agi2 | 0.3 | 0.64% | 0.00% | 0.00% |\n| 4k | arc-agi1 | 32.3 | 43.28% | 3.37% | 0.00% |\n| 4k | arc-agi2 | 0.1 | 0.32% | 0.00% | 0.00% |\n\n\nFurthermore, we conducted qualitative analysis of DeepSeek-R1 solving maze puzzles at different difficulty levels. In easy cases, models adopt a greedy, forward-only strategy with no need for revision or backtracking. In contrast, hard instances require global search, recursive planning, and backtracking, occasionally involving code-style planning logic. Below is an excerpted comparison:\n```\n// Easy\n  - Starting at (1,1). Possible moves: right is blocked (B at (1,2)), down is (2,1). So first move down to (2,1).\n  - From (2,1), possible moves: up is back to start, down to (3,1), right to (2,2). Let's go right to (2,2).\n  - At (2,2), possible moves: up is (1,2) which is B, down to (3,2), left to (2,1), right to (2,3) which is B. So move down to (3,2).\n  - At (3,2), possible moves: up to (2,2), down to (4,2), right to (3,3) which is B. Let's move down to (4,2).\n  - At (4,2), possible moves: up to (3,2), down to (5,2), right to (4,3) which is B. So move down to (5,2).\n  - Now at (5,2). Possible moves: up to (4,2), right to (5,3), left to (5,1). Let's go right to (5,3).\n  - At (5,3), can move right to (5,4), then to (5,5) which is E. So path would be:\n  - (1,1) -> (2,1) -> (2,2) -> (3,2) -> (4,2) -> (5,2) -> (5,3) -> (5,4) -> (5,5)\n```\n```\n// Hard\n  -  ...\n  - Neighbors:\n  - Up: (0,0) visited.\n  - Down: (2,0). maze[2][0] is '.', not visited. Add to queue. Path: [(1,1), (2,1), (3,1)]. Mark visited[2][0] = True.\n  - Right: (1,1). maze[1][1] is '.', not visited. Add to queue. Path: [(1,1), (2,1), (2,2)]. Mark visited[1][1] = True.\n  - Left: invalid.\n  - So queue now has (0,1), (2,0), (1,1).\n  - Next, process (0,1) (1,2 in 1-based).\n  - Check if end? No.\n  - ....\n  - Let me retrace.\n  - Possible path:\n  - Start at (1,1) → (2,1) → (3,1) → (3,2) → (3,3) → (3,4) → (4,4) → (4,5) → (5,5).\n  - Wait, (4,4) is array[3][3], which is B. So that's not possible. Hmm.\n  - Wait, the third row (array index 2) is ['.', '.', '.', '.', 'B']. So (3,4) is array[2][3], which is '.' (coordinates (3,4)). Then moving down to (4,4) (array[3][3] which is B) is blocked. So that's a dead end.\n  - Alternative path from (3,4) could be right to (3,5), but that's B. So can't go there.\n  - So maybe from (3,3), move right to (3,4), then up to (2,4) (but array[1][3] is B), so no.\n  - Wait, let's think of another approach. Maybe go from (1,1) right to (1,2), then down.\n```\n\n## Responses to your question\n\n> Q1: VC-PPO vs. Other RL Variants. The choice of VC-PPO over VC-GRPO or VC-DAPO is unclear. A broader comparison would help assess its suitability.\n\nA1: Great question. We do not assume any requirements for the adopted RL algorithms in our work. In early-stage experiments, we did run our experiments on GRPO. However, we ultimately adopted VC-PPO due to its faster convergence, better runtime efficiency, and smoother learning dynamics. Due to limited time and resources, we chose to run our experiments on VC-PPO as a standard algorithm to ablate over data and training strategies."}, {"review_id": "uqeG8VFOVT", "reviewer": "Reviewer_5LFd", "summary": "This paper studies the problems of reinforcement learning from verifiable feedback in the context of logical puzzles. The paper introduces a collection of logical puzzles, called ENIGMATA, for training and evaluating LLMs. ENIGMATA contains 36 logical puzzles across 7 categories in total, where each of the puzzles supports generation of problems with controlled difficulty and rule-based evaluators for reliable evaluation.\n\nUsing ENIGMATA, the paper trains Qwen-32B models which achieves strong performance on logical puzzles, and also sees some generalization to math domains. The paper also provides some analysis such as impacts of training data size and the impacts of SFT training.", "questions": "See weakness for questions.\n\nLastly I have one practical concern (not affecting the scores), the practical value of this  work maybe overshadowed by some more comprehensive collections of these kinds of tasks, such as reasoning gym (containing over 100 tasks). The paper might consider some analysis to set it apart from similar work.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\nThe collected data suite, ENIGMATA, contains over 30 puzzles from diverse types. The tasks have some nice properties of being able to generate new examples, configurable difficulty and reliable evaluation.  This collection of data suites can be useful for the research community in the field of RLVR for reasoning.\n\nThe authors also conduct experiments of running RL over the logical puzzles, which shows good in-domain results. The experiments contain multiple settings, which shows some insights on when SFT is necessary.\n\nThe paper is mostly well-written and easy to follow.\n\n## Weakness\n\nThe experiment only contains one model of one family and one scale. It is unsure how the results could generalize across different model families.\n\nThe paper claims AIME as OOD evaluation, though the paper uses some math data for training as well (ln 161). I feel to some extent, this is not completely OOD.\n\nThe paper presents a collection of gams and empirical study on using them for training model, I feel the paper could benefit from more analysis on 1) generalization across games and 2) generalization more realistic settings, like what kind of games could enable generalization on what types of OOD games.\n\nWhile the paper compiles a set of tasks, most of them can be found in existing resources. The paper does not add new things for testing LLMs.", "was_revised": true, "final_justification": "I thank authors for the updates and increased my score a bit. At the same time, this work is largely reusing existing resources and presents a series of empirical studies. There's no new techniques proposed and somewhat limited insights. Therefore I am giving a score slightly leaning the positive side to reflect the engineering efforts presented in this work.", "rebuttal": "Thank you for the detailed feedback and for recognizing the value of the Enigmata benchmark suite, our RL experiments, and the overall clarity of the paper.\n\n## Here are the responses to your concerns\n\n> W1: The experiment only contains one model of one family and one scale. It is unsure how the results could generalize across different model families.\n\nA1: Thank you for raising this. While our main experiments were conducted using Qwen2.5-32B, we have extended our evaluation to include models with different architectures and scales to assess the generality of our findings.\nIn particular, we conducted additional experiments using a proprietary mixture-of-experts (MoE) model (with 20B activated parameters and 200B total parameters, anonymized during review). This model differs substantially from Qwen in both size and structure.\nAs shown in the table below, adding Enigmata to the MoE model's training corpus leads to consistent and significant gains across both in-domain (Enigmata-Eval) and math and STEM reasoning benchmarks such as AIME and GPQA:\n\n| **Model** | **Enigmata-Eval** | **AIME 2024** | **AIME 2025** | **GPQA Diamond** |\n|:--------------------|:------------------|:--------------|:--------------|:-----------------|\n| **MoE-Model** | 52.9 | 86.7 | 74.0 | 77.3 |\n| **MoE-Model w/ Enigmata**| 64.6 (+11.7) | 87.5(+0.8) | 75.9(+1.9) | 78.1(+0.8) |\n\n\n> W2: The paper claims AIME as OOD evaluation, though some math data is also used in training, making this not completely OOD.\n\nWe would like to clarify that AIME is not presented as an out-of-distribution (OOD) evaluation in our paper. Our actual OOD evaluation is conducted on KORbench, which comprises symbolic reasoning tasks in different tasks and domains, with no overlap with our training data. We will make this clearer in the revision to avoid potential misunderstandings.\n\n> W3: Generalization Across Games and Realistic Settings. The paper would benefit from more analysis on (1) generalization across games, and (2) generalization in more realistic settings—e.g., what kinds of games help generalize to which types of OOD tasks.\n\nA3: Thank you for this valuable suggestion. We address both aspects of your question through two complementary experiments:\n\n(1) Cross-Game Generalization within Enigmata:\nWe conducted a domain-level transfer experiment, where models were trained using supervised fine-tuning on only one puzzle domain, and evaluated across the remaining domains. This allows us to quantify how reasoning skills acquired in one domain transfer to others. Among all domains, the Search category produces the strongest average transfer to other domains (+13.65). This suggests that puzzles involving combinatorial exploration, planning, and constraint satisfaction can cultivate general reasoning capabilities applicable across task types. These results also offer practical guidance for curriculum-efficient training, where a subset of domains can be selected to maximize generalization.\n\n| Training Domain | Crypto | Arithmetic | Logic | Grid | Graph | Search | Sequential | **Avg. Improvement** |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| **Search** | +6.67 | +14.37 | +14.71 | +11.22 | +25.97 | (+26.38) | +8.96 | **+13.65** |\n| **Sequential** | +10.67 | +18.37 | +12.49 | +6.33 | +21.61 | +3.88 | (+9.16) | **+12.23** |\n| **Grid** | +9.33 | +10.70 | +13.38 | (+25.07) | +25.25 | +1.37 | +9.16 | **+11.53** |\n| **Graph** | +5.33 | +15.03 | +10.04 | +5.07 | (+37.79) | +4.38 | +7.38 | **+7.87** |\n| **Crypto** | (+51.33) | +16.03 | +10.93 | +4.70 | +16.15 | +0.38 | +7.77 | **+9.33** |\n| **Arithmetic** | +4.33 | (+58.70) | +8.49 | -1.97 | +11.25 | -1.12 | +4.90 | **+4.31** |\n| **Logic** | +0.33 | +2.37 | (+18.27) | -2.11 | +6.70 | -0.50 | +3.21 | **+1.67** |\n\n-------------\n\n\n(2) Transfer to Realistic OOD Settings:\nTo examine transfer to distributionally different tasks, we trained a model only on the Enigmata puzzle data, without any exposure to ARC-AGI or math-specific data, and then evaluated it on ARC-AGI, KORBench, and AIME 24. The performance is summarized below:\n\n \n\n| **Model** | **ARC-AGI 1** | **ARC-AGI 2** | **Enigmata-Eval** | **KORBench** | **AIME 24** |\n|-------|-----------|-----------|---------------|----------|---------|\n| **Qwen2.5-32B-RFT** | 8.4 | 0.0 | 46.6 | 61.0 | 62.0 |\n| **Qwen2.5-32B w/ Enigmata** | *15.3* | *0.1* | *62.6* | *64.0* | 58.5 | \n| **Qwen2.5-32B w/ Enigmata+ARC-AGI+AIME** | 32.8 | 0.6 | 62.6 | 65.0 | 60.6 |\n\n\nThe Enigmata-only model shows non-trivial improvements over baseline on ARC-AGI and KORBench, despite never seeing these benchmarks during training. This indicates that puzzle training alone can yield partial generalization to unseen, structurally different tasks, even in out-of-domain scenarios. The gains observed here affirm Enigmata's value for reasoning models.\n\n> W4: Novelty of Enigmata as a Benchmark. While the paper compiles a set of tasks, most can be found in existing resources. The paper does not seem to add new capabilities for testing LLMs.\n\n\nA4: Thank you for the thoughtful feedback. While many Enigmata tasks are inspired by classic puzzles, our contribution lies in turning them into a unified, diverse, and RL-compatible benchmark. Most existing resources offer only small, static datasets—suitable for evaluation but insufficient for large-scale training or systematic analysis. Enigmata addresses this by supporting automatic generation, rule-based verification, and controllable difficulty across 36 tasks and 7 reasoning domains, enabling reinforcement learning, curriculum design, and generalization studies.\n\nSecond, Enigmata enables new ways to test LLMs by going beyond static evaluation. It supports active training, difficulty scaling, and complex reasoning, allowing us to systematically analyze how models learn and generalize across tasks. which together allow systematic analysis of how models learn, adapt, and generalize. Importantly, Enigmata focuses on symbolic puzzle reasoning—which, unlike knowledge-intensive tasks, targets core reasoning skills without relying on prior knowledge or memorization. This makes it a valuable resource for isolating and understanding the fundamental reasoning abilities of LLMs. We will clarify this distinction more explicitly in the revised version.\n\n## Here are the responses to your question\n\n> Q1: Comparison with Reasoning Gym. The practical value of this benchmark may be overshadowed by more comprehensive collections like Reasoning Gym. The paper might consider analysis to set it apart.\n\n\nA1: Thank you for the thoughtful comment. Enigmata and Reasoning Gym were developed independently as contemporary works, so our original submission did not include a detailed comparison. We appreciate the suggestion and will add one in the revision.\n\nThe two benchmarks differ in focus: Reasoning Gym emphasizes broad cognitive coverage and is primarily designed for supervised evaluation, whereas Enigmata targets knowledge-free, puzzle-style reasoning and supports RL-based training via automatic generation, verifiable rewards, and controllable difficulty. We view the two as complementary in goals and design. Enigmata's unique value lies in enabling scalable training and fine-grained analysis of how LLMs acquire and generalize core reasoning skills."}, {"review_id": "Q0LcWZVXb4", "reviewer": "Reviewer_Ubqb", "summary": "The authors present a new dataset of puzzle tasks for the evaluation of Large Reasoning Models (LRM), along with an accompanying benchmark and training procedure for fine-tuning a model on the task. The dataset consists of 36 puzzle tasks across 7 high-level categories, and most tasks also feature a generator with a scalable difficulty parameter. The authors demonstrate that current state-of-the-art LRMs do not solve the ENIGMATA benchmark and also show that their fine-tuned open-weight model performs comparably to the best closed-weight models on the benchmark. The paper concludes with a deeper analysis of the fine-tuned model, including various ablation studies.", "questions": "- How would you propose addressing the low performance of models in the Graph and Sequential categories? Is this something that can be solved with more data?\n\n- What is the main benefit of using the ENIGMATA benchmark over other similar reasoning benchmarks?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "This is a strong paper. The authors introduce a broad dataset of puzzle tasks and effectively demonstrate that it represents a reasonable challenge for current models. I’m also impressed with the thoroughness of the training procedure for ENIGMATA-Model and the accompanying experiments, which help immensely in understanding the effects of the various decisions / hyperparameters. My sense is that the benchmark could easily prove useful to the larger research community. However, I must also admit that I’m not the most knowledgeable about the various reasoning benchmarks that already exist -- it may be worth the authors providing additional explanation of why the ENIGMATA benchmark is a useful test suite in the context of other (potentially more challenging) evaluations like ARC-AGI 2.\n\nI would also strongly encourage the authors to include some indication of the statistical significance of their experiments. In the checklist, the authors indicate that this information is present in Section 5.3 -- however, this section only concerns the ablation studies and does not mention anything about variance or statistical significance. The tables and figures also appear to be missing error bars / indications of variance. I don’t feel that this critique sinks the paper, given the scale of the other contributions, but I again would urge the authors to address it.\n\nI also encourage the authors to cite this relevant paper on the automatic generation of programming puzzles:\n\n[1] Pourcel, Julien, et al. \"ACES: generating diverse programming puzzles with autotelic language models and semantic descriptors.\" Neurips (2024).", "was_revised": true, "final_justification": "The authors clarified my outstanding questions and I feel the paper remains a strong candidate for publication. As such, I am not changing my score.", "rebuttal": "Thank you for the positive and encouraging feedback! We are especially grateful for your recognition of our benchmark design, thorough experimentation, and the potential value of Enigmata to the broader research community.\n\n## Here are the responses to your concerns\n\n> W1: Benchmark Comparison – Why Enigmata over existing benchmarks like ARC-AGI 2? I'm not deeply familiar with all reasoning benchmarks—could the authors clarify why Enigmata is useful in the context of other (potentially more challenging) test suites?\n\nA1: Thank you for the thoughtful question. We agree that ARC-AGI and similar benchmarks are valuable for evaluating reasoning capabilities. However, they are primarily designed for evaluation, with small and static datasets, minimal training examples, and limited task diversity, making them less suitable for developing or analyzing reasoning capabilities through learning.\n\nIn contrast, Enigmata supports both training and evaluation. It provides a broad set of automatically generated, knowledge-free puzzle tasks across diverse reasoning types, with verifiable rewards, difficulty scaling, and RL-compatible interfaces. These features enable scalable training, curriculum learning, and structured generalization analysis.\n\nImportantly, our experiments show that models trained on Enigmata improve on ARC-AGI and other challenging tasks, demonstrating its utility as a foundation for building generalizable reasoning abilities. We will clarify this distinction more explicitly in the revised version.\n\n> W2: Statistical Significance and Error Bars. The authors mention significance in Section 5.3, but it only covers ablations. Tables and figures seem to lack error bars or variance estimates.\n\nA2: We appreciate this careful observation. In fact, our evaluation protocol already incorporates repeated sampling to ensure statistical reliability. Specifically, we evaluate each AIME problem 32 times and all other benchmarks 4 times, reporting average performance across these runs. This setup is designed to reduce the effect of stochastic decoding (we use temperature = 0.7) and improve the stability of reported metrics.\nIn response to your helpful suggestion, we have now conducted additional statistical significance tests and included error bars or confidence intervals in key tables and figures in the revised version, making variance across runs more explicit.\nThe following table summarizes the results of one-sample t-tests on several datasets, based on our final model checkpoints. Each test compares the model's mean accuracy against a null hypothesis of 0.5 (random chance):\n\n| Dataset             | Mean Accuracy | Std Error | 95% CI Lower | 95% CI Upper | t-statistic (vs 0.5) | p-value | Significance (α=0.05) |\n| :------------------ | :------------ | :-------- | :----------- | :----------- | :------------------- | :------ | :-------------------- |\n| AIME        | 0.5375        | 0.0161    | 0.5059       | 0.5691       | 2.3291               | 0.0201  | Significant           |\n| ARC-AGI     | 0.3646        | 0.0157    | 0.3338       | 0.3953       | -8.6397              | 0.0000  | Significant           |\n| Enigmata-Eval | 0.6562        | 0.0157    | 0.6255       | 0.6870       | 9.9708               | 0.0000  | Significant           |\n| KORBench   | 0.6260        | 0.0290    | 0.5689       | 0.6831       | 4.3478               | 0.0000  | Significant           |\n\n> W3: Citation of ACES (Pourcel et al., NeurIPS 2024). I encourage the authors to cite this related work on the automatic generation of programming puzzles.\n\nA3: Thank you for the pointer! We will cite the ACES paper (Pourcel et al., NeurIPS 2024) and briefly compare it with Enigmata. \n\n## Here are the responses to your question\n\n> Q1: Low Performance in Graph and Sequential Domains. Can this be solved with more data? \n\nA1: Great question. We observe that both Graph and Sequential domains require multi-step planning, structured memory, and sometimes intermediate tool use—challenges that current models (even SOTA) struggle with. Our current hypothesis is two-fold:\n- For Sequential puzzles, we plan to explore multi-turn RL approaches, enabling models to iteratively refine and revise partial plans, rather than relying on single-pass generation. This aligns better with the inherent temporal and decision-state dependencies of these tasks.\n- For Graph tasks, we are investigating the use of graph-centric supervised fine-tuning, which may better capture relational structures and multi-hop dependencies.\nWhile scaling up training data may provide incremental gains, we believe that improvements in architectural bias and temporal credit assignment will be essential to make substantial progress in these domains.\n\n> Q2: What is the main benefit of Enigmata over other reasoning benchmarks?\n\nA2: Thank you for the question. Enigmata stands out by combining scalability, diversity, and trainability in a unified benchmark. Compared to existing puzzle reasoning datasets, it covers the largest number of task categories (7) and provides a substantially larger number of tasks (36), provides automatic instance generation and verification, and uniquely supports both reinforcement learning and curriculum-based training through the RLVR paradigm.\n\nAs summarized in Table 2, Enigmata is the only benchmark that simultaneously offers diversity, scalability, verifiable reward signals, and structured training support—making it especially suitable for studying how reasoning capabilities can be acquired and generalized through learning."}, {"review_id": "xFBPV4R0O9", "reviewer": "Reviewer_X8gF", "summary": "This paper introduces ENIGMATA, designed to enhance Large Language Models' logical reasoning capabilities, particularly for solving puzzles. The ENIGMATA suite consists of three main components: ENIGMATA-Data, a dataset of 36 diverse puzzle tasks across seven categories, each equipped with an automated generator for scalable, difficulty-controlled examples and a rule-based verifier for automatic evaluation; ENIGMATA-Eval, a benchmark derived from this data; and ENIGMATA-Model, a training recipe combining rejection fine-tuning and multi-task Reinforcement Learning with Verifiable Rewards (RLVR). The authors demonstrate that their trained model, Qwen2.5-32B-ENIGMATA, achieves good performance on ENIGMATA-Eval, ARC-AGI, and out-of-domain puzzle benchmarks, while maintaining strong mathematical reasoning abilities. The work aims to provide a unified, controllable framework for advancing logical reasoning in LLMs.", "questions": "Please refer to the weaknesses.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "[Strengths]\n\nENIGMATA-Data is featuring 36 distinct puzzle tasks across 7 broad categories. \n\nThe proposed ENIGMATA-Model recipe, which combines rejection fine-tuning (RFT) to establish reasoning patterns and multi-task Reinforcement Learning (RL) for general reasoning skills, is good. \n\nThe trained Qwen2.5-32B-ENIGMATA model demonstrates good performance.\n\nThe model seems to exhibit good generalization to out-of-domain puzzle benchmarks (KOR-Bench) and maintain good performance in mathematical reasoning (AIME 24).\n\n\n\n[Weaknesses]\nThe paper admits that 6 of the 36 tasks draw from \"fixed pools\" and some instances \"rely on manually collected and annotated data rather than auto-generation\".  Would this inconsistency undermine the claim of universal scalability and fully automated data generation/verification across the entire ENIGMATA suite?\n\nDespite implementing multi-stage RL to mitigate task conflicts, the paper notes \"more volatile response length under Multi-Stage RL suggests instability in generation\". This indicates that the proposed training strategies might not fully resolve the inherent challenges of simultaneously optimizing for diverse reasoning skills.\n\nThe analysis explicitly states that \"code utilization actually hindered performance on puzzle tasks\" and that \"current models fail to effectively use code without executing it for complex reasoning tasks\" This probably reveals a technical limitation.", "was_revised": true, "final_justification": "Thanks for the authors rebuttal, which addresses my most concerns. I keep my positive score.", "rebuttal": "Thank you for the encouraging and constructive feedback! We appreciate your recognition of the Enigmata benchmark, our proposed Enigmata-Model training paradigm, and the model's demonstrated generalization ability.\n\n## Here are the responses to your concerns\n\n> W1: Inconsistent Data Generation — 6 of 36 tasks rely on fixed pools or manual annotation. Would this undermine the claim of universal scalability and automation?\n\nA1: Thank you for this question. We would like to clarify that the vast majority of Enigmata tasks (30 out of 36) are fully auto-generated with configurable parameters, controllable difficulty, and programmatic verification. These auto-generated environments cover all seven domains and form the core of our training and evaluation pipeline.\n\nFor the remaining six tasks that rely on fixed pools or partial manual annotation, we chose to include them because they exhibit unique structural challenges that are currently difficult to procedurally generate with high quality. These tasks still provide valuable diversity and difficulty for evaluating general reasoning, even if not fully automated.\nRather than undermining scalability, we believe this design reflects a practical trade-off between automation and task complexity. We are actively working toward expanding auto-generation coverage for these remaining categories.\n\n> W2: Multi-Stage RL Training Instability. The paper notes that response length is more volatile under multi-stage RL, suggesting possible instability when optimizing diverse reasoning tasks.\n \nA2: Thank you for raising this point. We would like to clarify that the observed variability in response length is not a sign of training failure, but rather a reflection of the different inductive behaviors of the two training paradigms analyzed in Section 4.3. We compare two strategies under the same total training budget:\n- Multi-Stage RL: better at acquiring task-specific skills, especially for complex tasks not seen in the SFT phase, such as ARC-AGI and Enigmata.\n- Mix-Training RL: shows superior generalization to OOD tasks, thanks to broader exposure and more diverse reasoning supervision.\n\n> W3: Code Utilization Fails to Improve Reasoning. The paper states that code utilization actually hindered performance, and current models fail to use code effectively without execution.\n\nA3: Yes, and thank you for pointing this out. In our experiments, we investigated whether prompting models to write code (**without execution feedback**) would help in complex symbolic puzzles. We found that this did not improve and sometimes worsened performance, likely because without an actual interpreter or reasoning agent loop, models lacked feedback on correctness.\nThis aligns with our broader hypothesis: current LLMs, when used purely as reasoning models, cannot fully leverage coding prompts without execution. We are actively working on extensions toward agent-based settings, where generated code can be executed, verified, and refined, leading to more meaningful gains. We believe Enigmata can serve as a strong resource for developing such RL-augmented or tool-augmented agents."}]

Author rebuttal:

Meta-review: The paper introduces a collection of puzzles, called ENIGMATA, for training and evaluating the reasoning capability of LLMs. ENIGMATA contains 36 puzzle tasks across 7 categories in total. Besides the benchmark dataset, the paper also studies different training strategies (combinations of RFT, mix-training & multi-stage training RL) and demonstrates the significant improvement of reasoning capability after the proposed training solutions on ENIGMATA, ARC-AGI1/2. 

The dataset introduced in this paper is regarded as highly valuable by all reviewers. The study of different training strategies is also insightful.
At the same time, reviewers also point out some room to improve in experiments. Especially, only QWen2.5-32B model is studied, more models from other vendors (e.g., DeepSeek V3, Llama models). More in-depth investigation into the standalone impact of puzzle data on reasoning capabilities on other general tasks will be a good plus.
Decision: Accept (spotlight)