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ScaleDiff: Scaling Difficult Problems for Advanced Mathematical Reasoning
Qizhi Pei, Zhuoshi Pan, Honglin Lin, Xin Gao, Yu Li, Zinan Tang, Conghui He, Rui Yan, Lijun Wu
TL;DR
Automated synthesis of difficult mathematical problems is costly to scale and can produce problems of limited difficulty. ScaleDiff identifies difficult problems, trains a generator to create more, and distills and filters solutions; fine-tuning on its dataset improves results against strong baselines across five benchmarks.
Problem
Automated mathematical-problem synthesis faces high computational costs, complex prompting, and limits in the difficulty of generated problems.
Source: §1 · p. 2Method
ScaleDiff identifies difficult problems with an adaptive model, trains DiffGen-8B to generate more, then distills and filters solutions to build its dataset.
Source: §1 · p. 2Results
ScaleDiff outperforms OpenThinker3 and AceReason-Nemotron on average across five benchmarks and achieves an 11.3% relative gain over AM-Qwen3-Distilled-7B.
Source: §1 · pp. 2–3Takeaways & Limitations
On AIME’24 and AIME’25, accuracy improves as the number of difficult problems in the augmenting dataset increases.
Source: §1 · p. 3Assessing the mathematical correctness and solvability of generated problems remains non-trivial and is left for future work.
Source: §2.3 · p. 5
Abstract
from arXiv · showhide
Large Reasoning Models (LRMs) have shown impressive capabilities in complex problem-solving, often benefiting from training on difficult mathematical problems that stimulate intricate reasoning. Recent efforts have explored automated synthesis of mathematical problems by prompting proprietary models or large-scale open-source models from seed data or inherent mathematical concepts. However, scaling up these methods remains challenging due to their high computational/API cost, complexity of prompting, and limited difficulty level of the generated problems. To overcome these limitations, we propose ScaleDiff, a simple yet effective pipeline designed to scale the creation of difficult problems. We efficiently identify difficult problems from existing datasets with only a single forward pass using an adaptive thinking model, which can perceive problem difficulty and automatically switch between "Thinking" and "NoThinking" modes. We then train a specialized difficult problem generator (DiffGen-8B) on this filtered difficult data, which can produce new difficult problems in large scale, eliminating the need for complex, per-instance prompting and its associated high API costs. Fine-tuning Qwen2.5-Math-7B-Instruct on the ScaleDiff-Math dataset yields a substantial performance increase of 11.3% compared to the original dataset and achieves a 65.9% average accuracy on AIME'24, AIME'25, HMMT-Feb'25, BRUMO'25, and MATH500, outperforming recent strong LRMs like OpenThinker3. Notably, this performance is achieved using the cost-efficient Qwen3-8B model as a teacher, demonstrating that our pipeline can effectively transfer advanced reasoning capabilities without relying on larger, more expensive teacher models. Furthermore, we observe a clear scaling phenomenon in model performance on difficult benchmarks as the quantity of difficult problems increases. Code: https://github.com/QizhiPei/ScaleDiff.
1 Introduction
ScaleDiff targets the cost, prompting complexity, and limited difficulty of automated hard-problem generation by identifying difficult examples and training a dedicated generator. Fine-tuning on its dataset improves benchmark performance, with accuracy rising as the amount of difficult training data increases.
1 Introduction
- Automated generation of difficult math problems remains hard to scale because of computational cost, complex prompting, and limited generated-problem difficulty.Difficult problems are sought because they typically require intricate reasoning, while many competition- or olympiad-level problems are hand-crafted by experts.Source: §1 · p. 2
- ScaleDiff uses an adaptive model to identify difficult problems in one forward pass, trains DiffGen-8B on them, and filters distilled solutions for its dataset.Qwen3-8B produces the long chain-of-thought solutions for generated problems.Source: §1 · p. 2
- 11.3% relative performance gain over AM-Qwen3-Distilled-7B accompanies average outperformance of OpenThinker3 and AceReason-Nemotron across five benchmarks.The benchmarks are AIME’24, AIME’25, HMMT-Feb’25, BRUMO’25, and MATH500.Source: §1 · pp. 2–3
- On AIME’24 and AIME’25, accuracy improves as the number of difficult problems in the augmenting dataset increases.The paper describes this as a scaling phenomenon and notes the possibility of further gains as larger, more challenging datasets become available.Source: §1 · p. 3
2 Method
ScaleDiff identifies difficult problems from existing datasets, trains a generator to create more, and distills and filters solutions for a training set.
2.1 Difficult Problem Identification
- AdaptThink’s adaptive mode choice becomes a binary difficulty label: NoThinking is simple, otherwise difficult, identified from one generated token in one forward pass.Its objective favors NoThinking responses only while maintaining expected accuracy at or above a fixed reference model’s level.Source: §2.1 · pp. 3–4
2.1.1 Effectiveness of Difficulty
- 59.2 overall for full data (558K); difficult data (192K) averages 56.6 versus 45.1 for equal-sized simple data and 53.3 for random data.Difficult data also averages 56.6 versus 48.9 for a much larger simple subset (366K); the full dataset is nearly three times larger than the difficult subset, but its advantage is only 2.6 points.Source: Table 1 · p. 4, §2.1.1 · pp. 4–5
2.2 Difficult Problem Generator
- DiffGen-8B learns from identified difficult problems using problem tokens rather than solutions, then generates new problems at scale by adjusting decoding parameters.The generator targets problems of comparable complexity, not problem solving.Source: §2.2 · p. 5
2.3 Solution Distillation and Filtration
- About 88% of generated problems are classified as difficult; after teacher-solution distillation and two-stage rule/model filtering, approximately 43% of initial samples are removed.The authors leave assessment of generated-problem mathematical correctness and solvability to future work, focusing instead on distilled-solution quality.Source: §2.3 · p. 5
3 Experiments
Experiments evaluate ScaleDiff against mathematical reasoning baselines and examine the effects of its data construction choices. ScaleDiff approaches its Qwen3-8B teacher, while ablations show lower average performance when response filtration or difficult-problem identification is removed.
3.1 Experimental Setup
- The 558K-instance initial dataset yields 192K difficult problems for DiffGen-8B, whose generated and filtered data expands ScaleDiff-Math to 1.7M problem-solution pairs.Evaluation uses Pass@1 across five benchmarks, averaging results over 10 seeds for four benchmarks and 3 for MATH500.Source: §3.1 · p. 6
3.2 Main Results
- 65.9% average accuracy for ScaleDiff-7B approaches Qwen3-8B’s 68.9%, and ScaleDiff surpasses several strong SFT/RL LRMs across five math benchmarks.Table 2 reports Pass@1 accuracy on AIME’24, AIME’25, HMMT-Feb’25, BRUMO’25, and MATH500.Source: Table 2 · p. 7, §3.2 · pp. 7–8
- 11.3% relative performance gain over AM-Qwen3-Distilled-7B accompanies ScaleDiff’s increase in both training-data difficulty and volume.The authors describe this difficulty-and-volume increase as “hiking” the dataset.Source: §3.2 · p. 7
3.3 Ablation Study
- With fine-tuning data fixed at 192K samples, removing response filtration lowers average performance from 58.4 to 55.3, and also removing difficult-problem identification lowers it to 50.4.The authors attribute the filtration drop to noisy, repetitive, or low-value samples and the identification drop to a greater share of trivial generated problems.Source: §3.3 · p. 8, Table 3 · p. 8
4 Analysis
Increasing difficult-data augmentation improves performance on AIME’24 and AIME’25, while MATH500 shows no gain. The analysis also finds that teacher choice has little effect on difficult-response lengths and that generated problems tend to elicit longer responses.
4.1 Impact of Data Scaling
- As augmentation grows, AIME’24 and AIME’25 performance improves without saturating, while MATH500 shows no improvement.The tested augmentation sizes are 1/2, 1, and 2 times the original dataset.Source: §4.1 · pp. 8–9, Figure 3 · p. 9
4.2 Effect of Teacher Model
- Qwen3-235B-A22B yields slightly better performance than Qwen3-8B, but the difference is not substantial.The authors choose Qwen3-8B as the more cost-efficient teacher.Source: §4.2 · p. 9, Table 4 · p. 9
4.3 Difficulty of Generated Problems
- AdaptThink’s difficulty levels strongly correlate with response length: difficult problems shift solution lengths toward longer token counts.Simple problems have a sharp density peak at very short token lengths.Source: §4.3 · p. 9, Figure 4 · p. 10
- For difficult problems, choosing between teacher models from the Qwen3 family has little impact on response-length distributions, with both models producing similar patterns.Source: §4.3 · p. 10
- 58.4 average performance comes from an equal amount of generated difficult data with rule and model filtration, versus 56.6 for the original difficult dataset.The authors note that generated problems tend to induce longer responses than original difficult problems.Source: §4.3 · p. 10
- Model filtration removes relatively easier problems from the rule-filtered set, retaining problems with greater difficulty and longer reasoning traces.The comparison is between distributions for generated problems with rule filtration and with both rule and model filtration.Source: §4.3 · p. 10
5 Related Work
Prior mathematical-data work selects or augments problems and solutions, but difficulty control and the synthesis of sufficiently challenging new problems remain concerns.
5.1 Mathematical Data for LRMs
- Mathematical SFT data comes from textbooks, exams, and websites, or from answer, question, and persona-based augmentation.Question augmentation can broaden topical coverage and problem structures, but requires validation of synthesized questions and solutions [21].Source: §5.1 · p. 10
5.2 Difficulty-aware Data Selection and Synthesis
- Difficulty-aware selection methods retain challenging existing problems, but do not generate new examples and often assess difficulty through model-specific fail rates.The passage names S1 [69], Light-R1 [70], AceReason [13], and DeepMath-103K [10].Source: §5.2 · p. 11
- Synthesis approaches include solution augmentation, skill-based generation, difficulty-matched prompting, preference optimization, and concept–explanation generation.The cited methods include DART-Math, MATH2 [71], DAST [72], ScaleQuest [73], and MathSmith [27]; the passage says generated-question difficulty remains limited for most methods.Source: §5.2 · p. 11
6 Conclusion
ScaleDiff constructs difficult mathematical problems at scale and improves LRM reasoning benchmarks. Increasing quantities of difficult training problems correspond to better performance on challenging benchmarks.
6 Conclusion
- ScaleDiff combines AdaptThink identification, a dedicated DiffGen-8B generator, and rule and model filtering to construct ScaleDiff-Math.The conclusion describes the pipeline as a strategy for scaling difficult mathematical problem construction.Source: §6 · p. 11
- Fine-tuning on ScaleDiff-Math improves results over strong SFT- and RL-based baselines across five mathematical reasoning benchmarks.The benchmarks are AIME’24, AIME’25, HMMT-Feb’25, BRUMO’25, and MATH500.Source: §6 · p. 11
- Increasing the quantity of difficult problems consistently improves performance on challenging benchmarks.The conclusion presents this as evidence for the value of difficulty-aware augmentation.Source: §6 · p. 11