# Exploring Mathematical Extrapolation of Large Language Models with Synthetic Data

arXiv: https://arxiv.org/abs/2406.02100 · HTML: https://mlsanity.com/paper/2406.02100/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 25 of its 36 references that are in the map.
- First posted on arXiv 4 Jun 2024.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)
- Synthetic data and self-improvement for LLMs / [Synthetic reasoning data and bootstrapping](https://mlsanity.com/synth/synthetic-reasoning-data-and-bootstrapping/)

## Builds on (most cited first, 25 of 25)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Generative Agents: Interactive Simulacra of Human Behavior](https://mlsanity.com/paper/2304.03442/) (2023)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [DeepSeek-Coder: When the Large Language Model Meets Programming -- The Rise of Code Intelligence](https://mlsanity.com/paper/2401.14196/) (2024)
- [Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Tasks](https://mlsanity.com/paper/2211.12588/) (2022)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [DeepSeek-VL: Towards Real-World Vision-Language Understanding](https://mlsanity.com/paper/2403.05525/) (2024)
- [AGIEval: A Human-Centric Benchmark for Evaluating Foundation Models](https://mlsanity.com/paper/2304.06364/) (2023)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://mlsanity.com/paper/2309.05653/) (2023)
- [Llemma: An Open Language Model For Mathematics](https://mlsanity.com/paper/2310.10631/) (2023)
- [ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving](https://mlsanity.com/paper/2309.17452/) (2023)
- [CMATH: Can Your Language Model Pass Chinese Elementary School Math Test?](https://mlsanity.com/paper/2306.16636/) (2023)

## Built on by (most cited first, 6 of 6)

- [VersaPRM: Multi-Domain Process Reward Model via Synthetic Reasoning Data](https://mlsanity.com/paper/2502.06737/) (2025)
- [Complete Chess Games Enable LLM Become A Chess Master](https://mlsanity.com/paper/2501.17186/) (2025)
- [Generalists vs. Specialists: Evaluating LLMs on Highly-Constrained Biophysical Sequence Optimization Tasks](https://mlsanity.com/paper/2410.22296/) (2024)
- [A Lightweight Multi Aspect Controlled Text Generation Solution For Large Language Models](https://mlsanity.com/paper/2410.14144/) (2024)
- [Stroke Modeling Enables Vectorized Character Generation with Large Vectorized Glyph Model](https://mlsanity.com/paper/2511.11119/) (2025)
- [When Tokenization is Secretly Output Supervision](https://mlsanity.com/paper/2609.01386/) (2026)

## Abstract (arXiv)

Large Language Models (LLMs) have shown excellent performance in language understanding, text generation, code synthesis, and many other tasks, while they still struggle in complex multi-step reasoning problems, such as mathematical reasoning. In this paper, through a newly proposed arithmetical puzzle problem, we show that the model can perform well on multi-step reasoning tasks via fine-tuning on high-quality synthetic data. Experimental results with the open-llama-3B model on three different test datasets show that not only the model can reach a zero-shot pass@1 at 0.44 on the in-domain dataset, it also demonstrates certain generalization capabilities on the out-of-domain datasets. Specifically, this paper has designed two out-of-domain datasets in the form of extending the numerical range and the composing components of the arithmetical puzzle problem separately. The fine-tuned models have shown encouraging performance on these two far more difficult tasks with the zero-shot pass@1 at 0.33 and 0.35, respectively.