The Dawn of a New Era in AI-Driven Mathematical Discovery
In a landmark achievement for artificial intelligence in mathematics, researchers from Peking University and the Beijing Institute for General Artificial Intelligence (BIGAI) have unveiled TongGeometry, a groundbreaking neuro-symbolic system capable of both proposing and solving Olympiad-level geometry problems. This innovation not only surpasses previous state-of-the-art models from US developers but also marks the first time an AI has autonomously generated problems deemed worthy for high-stakes competitions, including one selected for China's 2024 National High School Mathematics League and two shortlisted for a top US civil Olympiad in the same year.
TongGeometry represents a paradigm shift from passive problem-solving to active mathematical creation, functioning more like a seasoned coach than a mere student. By modeling geometry as a structured tree search process, the system navigates vast combinatorial spaces with unprecedented efficiency, using consumer-grade hardware to achieve superhuman performance on benchmarks like the IMO-AG-30 dataset. This breakthrough, detailed in a peer-reviewed paper published in Nature Machine Intelligence, underscores China's rising dominance in AI research within higher education institutions.
Understanding Olympiad-Level Geometry: Why It's a Tough Nut for AI
The International Mathematical Olympiad (IMO), often called the 'World Cup of Math,' challenges the brightest high school students worldwide with problems demanding deep creativity, rigorous proof, and spatial intuition. Geometry problems, in particular, blend numerical precision with visual reasoning, requiring auxiliary constructions—additional lines or circles—to unlock proofs. Historically, AI struggled here due to the 'path explosion' in search spaces, where equivalent configurations (via rotation or reflection) multiply exponentially.
Prior systems like Google DeepMind's AlphaGeometry, released in early 2024, made strides by solving 25 of 30 IMO geometry problems but relied on massive compute clusters (246 CPUs + 4 V100 GPUs) and struggled with symmetry and efficiency. TongGeometry addresses these pain points head-on, solving all 30 problems in under 38 minutes on just 32 CPU cores and one RTX 4090 GPU, while generating novel challenges that human experts validated as Olympiad-caliber.
This context highlights why TongGeometry's dual capability—discovering theorems and proving them—is revolutionary for advancing general artificial intelligence (AGI) logical reasoning.
Meet TongGeometry: The Neuro-Symbolic Powerhouse from Peking University
Developed by a team led by first author Chi Zhang and including notable figures like Yixin Zhu, an assistant professor at Peking University's School of Psychological and Cognitive Sciences, TongGeometry integrates neural networks with symbolic reasoning. The system's core is a guided tree search that represents Euclidean plane geometry as a Markov process on a finite tree of actions: drawing points, lines, and circles, which trigger deductive rules for facts like congruence or collinearity.
Step-by-step, here's how it works:
- Canonical Representation: Prunes redundant paths by normalizing diagrams, compressing the 1056 possible configurations.
- Deductive Database: Applies rules for angle chasing and similarity without algebraic crutches.
- LLM Guidance: Fine-tuned policy and value networks (based on models like DeepSeek-Coder) suggest auxiliaries and estimate proof steps.
- Symmetry Enforcement: Dynamically maps equivalent transformations, prioritizing elegant problems.
"TongGeometry can precisely capture high-quality problems that meet the aesthetic standards of human mathematicians," explains Chi Zhang, emphasizing the 'duality' where proof complexity exceeds construction simplicity.
The codebase is openly available on GitHub, inviting global collaboration.
Crushing Benchmarks: TongGeometry Outperforms US Rivals
On the IMO-AG-30 benchmark—30 geometry problems from 23 IMO years—TongGeometry achieved a perfect 100% solve rate, eclipsing the average IMO gold medalist's performance and AlphaGeometry's partial success. It also tackled the broader MO-TG-225 dataset at 81.3%, with its deductive core alone solving 55.2% via pure logic.
Compared to AlphaGeometry:
| Metric | TongGeometry | AlphaGeometry |
|---|---|---|
| Solve Time (IMO-AG-30) | <38 min | Hours (est.) |
| Compute | 32 CPU + 1 GPU | 246 CPU + 4 V100 GPUs |
| Proof Simplicity | Higher (fewer steps) | Complex |
| Problem Generation | 6.7B problems | None |
This efficiency stems from a 'small data, big task' paradigm, simulating human intuition without billions of labeled examples. For full technical details, see the original paper.
From Synthetic Billions to Real-World Olympiad Glory
Training on just 196 past Olympiad problems as priors, TongGeometry generated 6.7 billion synthetic challenges over 30 days on 10,368 CPU cores, filtering for those needing auxiliaries and symmetry. Of these, 4.1 billion featured geometric symmetry, ensuring Olympiad-like elegance.
Remarkably, three AI-proposed problems passed muster with human experts:
- One featured in the 2024 Chinese Mathematical Olympiad (Beijing District) national team qualifier.
- Two shortlisted for the 2024 US Ersatz Math Olympiad, a prestigious practice event for top US talents.
This milestone validates AI's creative potential, bridging lab research to competitive reality.
Peking University: A Hub for Cutting-Edge AI Research in China
Peking University (PKU), one of China's elite institutions, hosts multiple TongGeometry contributors across its Institute for Artificial Intelligence, School of Intelligence Science and Technology, and School of Psychological and Cognitive Sciences. Collaborating with BIGAI—a think tank pioneering general AI—this project exemplifies interdisciplinary excellence in Chinese higher education.
Song-Chun Zhu, a senior author, brings vision from his leadership roles, while Yixin Zhu notes: "This path... evolves through internal logic, [key] to AGI." Such innovations position PKU as a global leader, attracting top talent. Researchers and students interested in similar pursuits can find opportunities via higher ed research jobs or faculty positions at leading universities.
Implications for Mathematics Education and Higher Ed Careers
TongGeometry's deployment as an IMO coach—curating personalized problems—promises to transform math education in Chinese universities and beyond. It adjusts difficulty via proof length, aiding students from novices to Olympiad hopefuls. In higher ed, this accelerates theorem discovery, freeing researchers for novel proofs.
For aspiring academics, mastering neuro-symbolic AI opens doors in booming fields. Check how to craft a winning academic CV or explore China university jobs. A preliminary app already supports training, hinting at scalable edtech.
- Enhances curriculum with infinite problem banks.
- Democratizes elite training via affordable hardware.
- Boosts research productivity in AI-math hybrids.
Future Outlook: TongGeometry Paves the Way for AGI in Science
Looking ahead, the 'Tong' series aims to tackle algebra and beyond, balancing exploration with exploitation for truly novel discoveries. Challenges remain, like reducing symmetry bias, but its open-source nature fosters rapid iteration.
In China, this bolsters national AGI ambitions, with PKU and BIGAI leading. Globally, it challenges US dominance, urging balanced investment in ethical AI. For professionals, it's a call to upskill—visit higher ed career advice or rate my professor for insights into top programs.
For more on China-focused opportunities, explore university jobs and postdoc roles.
Photo by Bangyu Wang on Unsplash
This TongGeometry breakthrough not only redefines AI capabilities but elevates Chinese higher education on the world stage, inspiring the next generation of mathematicians and AI pioneers.
