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

arXiv: https://arxiv.org/abs/2505.19914 · HTML: https://mlsanity.com/paper/2505.19914/

## Key facts

- Cited by 56 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 31 of its 45 references that are in the map.
- First posted on arXiv 26 May 2025.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 30 of 31)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Open-Reasoner-Zero: An Open Source Approach to Scaling Up Reinforcement Learning on the Base Model](https://mlsanity.com/paper/2503.24290/) (2025)
- [Reinforcement Learning for Reasoning in Large Language Models with One Training Example](https://mlsanity.com/paper/2504.20571/) (2025)
- [VAPO: Efficient and Reliable Reinforcement Learning for Advanced Reasoning Tasks](https://mlsanity.com/paper/2504.05118/) (2025)
- [DeepMath-103K: A Large-Scale, Challenging, Decontaminated, and Verifiable Mathematical Dataset for Advancing Reasoning](https://mlsanity.com/paper/2504.11456/) (2025)
- [A Survey of Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2312.14925/) (2023)
- [Logic-RL: Unleashing LLM Reasoning with Rule-Based Reinforcement Learning](https://mlsanity.com/paper/2502.14768/) (2025)
- [FOLIO: Natural Language Reasoning with First-Order Logic](https://mlsanity.com/paper/2209.00840/) (2022)
- [A Survey on Test-Time Scaling in Large Language Models: What, How, Where, and How Well?](https://mlsanity.com/paper/2503.24235/) (2025)
- [ZebraLogic: On the Scaling Limits of LLMs for Logical Reasoning](https://mlsanity.com/paper/2502.01100/) (2025)
- [Light-R1: Curriculum SFT, DPO and RL for Long COT from Scratch and Beyond](https://mlsanity.com/paper/2503.10460/) (2025)
- [Seed1.5-Thinking: Advancing Superb Reasoning Models with Reinforcement Learning](https://mlsanity.com/paper/2504.13914/) (2025)
- [On the Measure of Intelligence](https://mlsanity.com/paper/1911.01547/) (2019)
- [KodCode: A Diverse, Challenging, and Verifiable Synthetic Dataset for Coding](https://mlsanity.com/paper/2503.02951/) (2025)
- [Big-Math: A Large-Scale, High-Quality Math Dataset for Reinforcement Learning in Language Models](https://mlsanity.com/paper/2502.17387/) (2025)
- [What's Behind PPO's Collapse in Long-CoT? Value Optimization Holds the Secret](https://mlsanity.com/paper/2503.01491/) (2025)
- [KOR-Bench: Benchmarking Language Models on Knowledge-Orthogonal Reasoning Tasks](https://mlsanity.com/paper/2410.06526/) (2024)
- [MetaSpatial: Reinforcing 3D Spatial Reasoning in VLMs for the Metaverse](https://mlsanity.com/paper/2503.18470/) (2025)
- [Exploring Data Scaling Trends and Effects in Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2503.22230/) (2025)
- [FINEREASON: Evaluating and Improving LLMs' Deliberate Reasoning through Reflective Puzzle Solving](https://mlsanity.com/paper/2502.20238/) (2025)
- [Assessing and Enhancing the Robustness of Large Language Models with Task Structure Variations for Logical Reasoning](https://mlsanity.com/paper/2310.09430/) (2023)
- [ReasoningWeekly: A General Knowledge and Verbal Reasoning Challenge for Large Language Models](https://mlsanity.com/paper/2502.01584/) (2025)
- [A Puzzle-Based Dataset for Natural Language Inference](https://mlsanity.com/paper/2112.05742/) (2021)

## Built on by (most cited first, 30 of 56)

- [A Survey of Reinforcement Learning for Large Reasoning Models](https://mlsanity.com/paper/2509.08827/) (2025)
- [Pass@k Training for Adaptively Balancing Exploration and Exploitation of Large Reasoning Models](https://mlsanity.com/paper/2508.10751/) (2025)
- [RLVE: Scaling Up Reinforcement Learning for Language Models with Adaptive Verifiable Environments](https://mlsanity.com/paper/2511.07317/) (2025)
- [Jigsaw-R1: A Study of Rule-based Visual Reinforcement Learning with Jigsaw Puzzles](https://mlsanity.com/paper/2505.23590/) (2025)
- [Language Models that Think, Chat Better](https://mlsanity.com/paper/2509.20357/) (2025)
- [GUI-GENESIS: Automated Synthesis of Efficient Environments with Verifiable Rewards for GUI Agent Post-Training](https://mlsanity.com/paper/2602.14093/) (2026)
- [Agentic Jigsaw Interaction Learning for Enhancing Visual Perception and Reasoning in Vision-Language Models](https://mlsanity.com/paper/2510.01304/) (2025)
- [NP-Engine: Empowering Optimization Reasoning in Large Language Models with Verifiable Synthetic NP Problems](https://mlsanity.com/paper/2510.16476/) (2025)
- [Good SFT Optimizes for SFT, Better SFT Prepares for Reinforcement Learning](https://mlsanity.com/paper/2602.01058/) (2026)
- [ProtoReasoning: Prototypes as the Foundation for Generalizable Reasoning in LLMs](https://mlsanity.com/paper/2506.15211/) (2025)
- [The Era of Agentic Organization: Learning to Organize with Language Models](https://mlsanity.com/paper/2510.26658/) (2025)
- [Mirage or Method? How Model-Task Alignment Induces Divergent RL Conclusions](https://mlsanity.com/paper/2508.21188/) (2025)
- [Towards an AI Musician: Synthesizing Sheet Music Problems for Musical Reasoning](https://mlsanity.com/paper/2509.04059/) (2025)
- [Think Visually, Reason Textually: Vision-Language Synergy in ARC](https://mlsanity.com/paper/2511.15703/) (2025)
- [Pushing the Boundaries of Natural Reasoning: Interleaved Bonus from Formal-Logic Verification](https://mlsanity.com/paper/2601.22642/) (2026)
- [When Domains Interact: Asymmetric and Order-Sensitive Cross-Domain Effects in Reinforcement Learning for Reasoning](https://mlsanity.com/paper/2602.01365/) (2026)
- [Pencil Puzzle Bench: A Benchmark for Multi-Step Verifiable Reasoning](https://mlsanity.com/paper/2603.02119/) (2026)
- [Enhancing Vision-Language Model Training with Reinforcement Learning in Synthetic Worlds for Real-World Success](https://mlsanity.com/paper/2508.04280/) (2025)
- [MM-HELIX: Boosting Multimodal Long-Chain Reflective Reasoning with Holistic Platform and Adaptive Hybrid Policy Optimization](https://mlsanity.com/paper/2510.08540/) (2025)
- [Skin-R1: Clinical Knowledge-Guided Dermatological Diagnosis Using Vision-Language Models](https://mlsanity.com/paper/2511.14900/) (2025)
- [Low-rank Optimization Trajectories Modeling for LLM RLVR Acceleration](https://mlsanity.com/paper/2604.11446/) (2026)
- [DiG: Differential Grounding for Enhancing Fine-Grained Perception in Multimodal Large Language Model](https://mlsanity.com/paper/2512.12633/) (2025)
- [VisRes Bench: On Evaluating the Visual Reasoning Capabilities of VLMs](https://mlsanity.com/paper/2512.21194/) (2025)
- [TopoBench: Benchmarking LLMs on Hard Topological Reasoning](https://mlsanity.com/paper/2603.12133/) (2026)
- [CAST: Game Solvers as Turn-Level Teachers for LLM Agents](https://mlsanity.com/paper/2607.25308/) (2026)
- [Think Right: Learning to Mitigate Under-Over Thinking via Adaptive, Attentive Compression](https://mlsanity.com/paper/2510.01581/) (2025)
- [GTAlign: Game-Theoretic Alignment of LLM Assistants for Social Welfare](https://mlsanity.com/paper/2510.08872/) (2025)
- [SPHINX: A Synthetic Environment for Visual Perception and Reasoning](https://mlsanity.com/paper/2511.20814/) (2025)
- [The Self-Correction Illusion: Role Relabeling Gates Explicit Error Flagging in Large Language Models](https://mlsanity.com/paper/2606.05976/) (2026)
- [Group Pattern Selection Optimization: Let LRMs Pick the Right Pattern for Reasoning](https://mlsanity.com/paper/2601.07238/) (2026)

## Abstract (arXiv)

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