Google DeepMind’s AlphaGeometry2 model has outperformed human gold medalists in the International Mathematical Olympiad (IMO), solving 84% of geometry problems presented over the last 25 years.
AlphaGeometry2 builds upon its predecessor AlphaGeometry by enhancing its representation language. Google expanded its capabilities to handle a broader range of geometry problems, including those involving object movements and linear equations related to angles, ratios, and distances.
Last July, the system already reached the silver medal level in solving International Mathematical Olympiad problems.
The new achievement positions the AI system as a milestone in computational reasoning, showcasing DeepMind’s ability to push artificial intelligence into domains previously dominated by human expertise.
Its predecessor, AlphaGeometry, achieved a success rate of just 54%, marking the new system as a substantial leap forward.
This breakthrough builds upon DeepMind’s legacy of achievements, including AlphaFold 3, which transformed protein structure prediction, and AlphaGo, which mastered the ancient board game of Go.
Excited to share details of AlphaGeometry2 (AG2), part of the system that achieved silver-medal standard at IMO 2024 last July! AG2 now has surpassed the average gold-medalist in solving Olympiad geometry problems, achieving a solving rate of 84% for all IMO geometry problems… https://t.co/jAVTpNdBMu pic.twitter.com/eXHstdeVTP
— Thang Luong (@lmthang) February 7, 2025
The application of AI in competitive mathematics adds to this growing body of work, demonstrating the adaptability of DeepMind’s models in addressing a diverse array of challenges.
The system combines neural network architecture with symbolic reasoning, employing a hybrid approach that enables it to tackle problems that require both creativity and logical precision.
AlphaGeometry2 not only outperforms many human experts but also introduces techniques that could influence broader AI research and applications, including fields such as engineering and physics.
Its success is grounded in innovations like the Shared Knowledge Ensemble of Search Trees (SKEST) and optimized symbolic engines, which allow the AI to solve problems at unprecedented speeds.
In AG1, we use a simple beam search to discover proofs. In AG2, we design a novel search algorithm, called Shared Knowledge Ensemble of Search Trees (SKEST), to allow multiple beam searches to run in parallel and help each other. This was one of the reasons why we can solve IMO… pic.twitter.com/Z1078G083W
— Thang Luong (@lmthang) February 8, 2025
The Hybrid Design Behind AlphaGeometry2
At the heart of AlphaGeometry2 lies its hybrid architecture, which combines a finetuned version of DeepMind’s Gemini language model with a symbolic reasoning engine known as DDAR (Deductive Database Arithmetic Reasoning).
This collaboration enables the AI to interpret and formalize complex geometry problems, generate potential solutions, and validate these solutions through rigorous logical proofs.
According to a recently published DeepMind research paper about AlphaGeometry2, “These enhancements culminate in a substantial improvement in performance: AG2 achieves an impressive 84% solve rate on all 2000–2024 IMO geometry problems, demonstrating a significant leap forward in AI’s ability to tackle challenging mathematical reasoning tasks.”
AlphaGeometry2’s workflow is powered by SKEST, an algorithm that allows multiple problem-solving strategies to work in parallel. SKEST coordinates these approaches by creating a shared knowledge base where intermediate discoveries are pooled for mutual benefit.
The unique mechanism enhances both the efficiency and creativity of the AI, enabling it to explore multiple avenues of reasoning simultaneously.
DeepMind has also implemented substantial technical upgrades to the system’s symbolic reasoning engine. Rewritten in C++, the engine is now up to 300 times faster than its Python-based predecessor, allowing for more comprehensive problem-solving within constrained computational budgets.
These optimizations expand the range of problems AlphaGeometry2 can handle, including complex locus-type problems where objects move while maintaining specific relationships with other geometric elements.
Exceeding Human Performance in Geometry
AlphaGeometry2’s performance places it above the average IMO gold medalist, who typically solves 40 out of 50 problems in the IMO-AG-50 benchmark set.
The system solved 42 problems, marking a slight but meaningful edge over human experts. This achievement is particularly striking given the difficulty of IMO problems, which demand rigorous proofs for statements about geometric relationships on a plane.
One of the most notable aspects of AlphaGeometry2 is its ability to solve advanced geometry problems, such as those involving loci. Locus-type problems require understanding how points or objects move while preserving certain conditions, a task that combines abstract reasoning with mathematical rigor.
By successfully addressing these challenges, AlphaGeometry2 has expanded its problem coverage from 66% to 88% of IMO geometry problems.
As Kevin Buzzard, a mathematician at Imperial College London, observed, *“I imagine it won’t be long before computers are getting full marks on the IMO.”* (Nature). Such advancements suggest that AI systems like AlphaGeometry2 are not just matching human performance but potentially redefining what is achievable in mathematical problem-solving.
Innovations That Drive AlphaGeometry2
A critical factor in AlphaGeometry2’s success is its reliance on synthetic training data. DeepMind generated over 300 million synthetic theorems and proofs, covering a wide range of complexity, to train the Gemini-based language model.
This approach allowed the AI to develop a deep understanding of geometric principles and solve problems that extend beyond human-curated datasets. These synthetic datasets not only enhance problem-solving capabilities but also demonstrate the scalability of DeepMind’s AI research.
AlphaGeometry2’s symbolic reasoning engine, DDAR, plays a key role in transforming these theoretical insights into practical solutions. By verifying the logical consistency of the language model’s suggestions, DDAR ensures that each step in the problem-solving process adheres to strict mathematical rules.
This combination of neural adaptability and logical precision sets AlphaGeometry2 apart from more traditional large language models.
Despite its remarkable performance, the system has limitations. It struggles with problems involving inequalities, non-linear equations, and variable point counts—areas that require even more advanced reasoning capabilities. According to DeepMind’s research, “Until model speed is improved and hallucinations are completely resolved, tools like symbolic engines will remain essential for math applications.”
Implications Beyond Competitive Mathematics
AlphaGeometry2’s success shows the potential of hybrid AI systems in solving highly specialized problems.
Beyond competitive mathematics, its applications could extend to fields such as engineering, where geometric proofs are critical for structural design, or physics, where complex models often rely on intricate geometric calculations.
By combining symbolic reasoning with neural networks, AlphaGeometry2 paves the way for AI systems capable of addressing challenges that require both precision and creativity.
DeepMind’s broader AI advancements provide valuable context for understanding the significance of AlphaGeometry2. Earlier projects like AlphaFold, which revolutionized the field of protein structure prediction, illustrate how targeted AI solutions can drive progress across disciplines.
Similarly, AlphaGo demonstrated the potential of AI to master strategic reasoning, while large language models like Gemini have introduced innovative ways to tackle abstract problems.
Future Prospects and Challenges
The development of AlphaGeometry2 has reignited debates within the AI research community about the role of hybrid systems in solving complex problems. While large language models like Gemini or OpenAI’s GPT models excel at generating human-like text, they often falter when faced with tasks requiring formal reasoning or logical consistency.
AlphaGeometry2 bridges this gap by integrating symbolic reasoning, offering a potential blueprint for the next generation of AI systems.
However, challenges remain. The reliance on symbolic engines introduces computational overhead, and the system’s inability to handle certain problem types highlights the need for further innovation. As researchers refine the model, integrating advanced reasoning methods and faster algorithms will be key to overcoming these limitations.
For readers interested in the latest developments in AI, DeepMind’s ongoing efforts, including the recent open-sourcing of AlphaFold 3, demonstrate the company’s dedication to expanding the boundaries of what AI can achieve.
