Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2
Yuri Chervonyi, Trieu H. Trinh, Miroslav Olšák, Xiaomeng Yang, Hoang Nguyen, Marcelo Menegali, Junehyuk Jung, Vikas Verma, Quoc V. Le, Thang Luong
2025-02-07
Summary
This paper talks about AlphaGeometry2, an improved AI system developed by DeepMind that can solve complex geometry problems at a level surpassing the average gold medalist in the International Mathematical Olympiad (IMO). It's an upgrade from the original AlphaGeometry and has significantly better performance in solving challenging geometry problems.
What's the problem?
Solving high-level geometry problems, like those in the IMO, is extremely difficult for AI systems. These problems require deep mathematical reasoning and understanding, which most AI struggle with. The original AlphaGeometry could only solve about half of the IMO geometry problems from the past 25 years, which wasn't quite at the level of top human performers.
What's the solution?
The researchers improved AlphaGeometry in several ways. They expanded its language to handle more complex problems involving object movements and equations. They used a more advanced AI called Gemini for better language understanding. They also created a new way for the AI to share knowledge across different problem-solving attempts. Additionally, they improved how the AI generates practice problems for itself and how it processes information symbolically. These changes allowed AlphaGeometry2 to solve 84% of IMO geometry problems from the last 25 years, up from 54% previously.
Why it matters?
This breakthrough is significant because it shows that AI can now perform at an elite human level in a field that requires deep reasoning and creativity. It's not just about solving geometry problems; it demonstrates that AI is getting better at the kind of logical thinking needed for advanced mathematics and science. This could lead to AI systems that can help with complex scientific research or even make new mathematical discoveries. It's a step towards more general artificial intelligence that can think and reason like humans in challenging intellectual tasks.
Abstract
We present AlphaGeometry2, a significantly improved version of AlphaGeometry introduced in Trinh et al. (2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we first extend the original AlphaGeometry language to tackle harder problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with other additions, has markedly improved the coverage rate of the AlphaGeometry language on International Math Olympiads (IMO) 2000-2024 geometry problems from 66% to 88%. The search process of AlphaGeometry2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that combines multiple search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AlphaGeometry2 to 84% for all geometry problems over the last 25 years, compared to 54% previously. AlphaGeometry2 was also part of the system that achieved silver-medal standard at IMO 2024 https://dpmd.ai/imo-silver. Last but not least, we report progress towards using AlphaGeometry2 as a part of a fully automated system that reliably solves geometry problems directly from natural language input.