Breaking the Neural Network Barrier: How a CPU-Only System Achieved Gold Medal Performance in Olympiad Geometry

Core Question: Can geometry theorem proving achieve world-class performance without relying on neural networks or specialized hardware?

For decades, automated theorem proving in Euclidean geometry has remained one of artificial intelligence’s most persistent challenges. While recent advances like AlphaGeometry demonstrated impressive capabilities by combining neural networks with symbolic reasoning, they relied heavily on GPU resources and complex machine learning infrastructure. This dependency created barriers for researchers and educators with limited computational resources.

Now, a breakthrough method called HAGeo (Heuristic-based Auxiliary constructions in Geometric deduction) has shattered these limitations. Developed by researchers from ETH Zurich and Microsoft, this CPU-only system achieves “gold-medal” performance on International Mathematical Olympiad (IMO) geometry problems without using any neural networks. More surprisingly, it solves 28 out of 30 problems in the IMO-30 benchmark—surpassing AlphaGeometry’s 24 problems—while running approximately 20 times faster.

This article explores how HAGeo redefines what’s possible in automated geometry theorem proving, demonstrating that sometimes the most elegant solutions don’t require the most complex technology. For educators, researchers, and AI practitioners, this development offers profound insights into how carefully designed heuristic methods can outperform resource-intensive neural approaches in specialized domains.

Why Geometry Theorem Proving Matters in AI Research

Core Question: Why has automated geometry theorem proving remained challenging for AI systems despite decades of research?

Geometry represents one of mathematics’ oldest branches, studied for over two millennia. As one of the four primary problem categories in the International Mathematical Olympiad—the world’s premier high school mathematics competition—geometry problems demand not just computational ability but deep spatial reasoning and creative insight.

Unlike algebraic problems that can often be reduced to symbolic manipulation, geometry problems require understanding spatial relationships, recognizing patterns, and making intuitive leaps about shapes and their properties. Human solvers often introduce “auxiliary constructions”—additional points, lines, or circles not present in the original problem—that reveal hidden relationships and unlock solutions.

For AI systems, this presents unique challenges:

• Representation complexity: How to formally represent geometric configurations and relationships
• Search space explosion: The vast number of potential auxiliary constructions makes brute force approaches infeasible
• Creative insight: The need to identify which constructions will reveal meaningful relationships
• Computational efficiency: Balancing thorough exploration with reasonable computation time

Historically, approaches have fallen into two categories:

1. Algebraic methods transform geometry problems into polynomial equation systems solved using techniques like Wu’s method or Gröbner bases
2. Synthetic methods mimic human reasoning using geometric rules and deduction systems

AlphaGeometry’s 2024 breakthrough combined a deductive database with algebraic reasoning (DDAR engine) and used neural networks trained on 100 million synthetic problems to suggest auxiliary points. While impressive, this approach required significant GPU resources for neural inference, limiting accessibility and raising questions about whether such complexity was truly necessary.

Personal reflection: Reading through the research on geometry theorem proving, I’m struck by how this field mirrors broader AI development patterns. We often assume that solving harder problems requires more complex systems, when sometimes the breakthrough comes from rethinking fundamental assumptions. The HAGeo team’s discovery that random auxiliary point selection could achieve “silver medal” performance hinted at a profound truth: in geometry, the right construction often matters more than how you find it.

The HAGeo Breakthrough: Simplicity Through Heuristics

Core Question: How does HAGeo achieve gold-medal performance without neural networks or GPUs?

When the HAGeo research team began their work, they made a surprising discovery. Even a simple random strategy for adding auxiliary points—using only CPUs and no neural networks—could solve 25 of 30 problems on the IMO-30 benchmark. This performance matched AlphaGeometry’s results and achieved what’s considered “silver-medal” level human performance.

This finding raised a compelling question: Could a carefully designed heuristic approach—still using only CPUs—achieve “gold-medal” performance without neural inference?

The answer proved to be yes. HAGeo introduces three key innovations that collectively enable its remarkable performance:

1. Geometric-specific heuristic selection: Instead of randomly choosing auxiliary points or using neural networks to predict them, HAGeo employs mathematically grounded heuristics that prioritize points with favorable geometric properties
2. Optimized deduction engine: A redesigned DDAR engine that runs approximately 20 times faster than AlphaGeometry’s implementation
3. Human-assessed benchmark: A new comprehensive evaluation standard (HAGeo-409) that better reflects real-world problem difficulty

The Heuristic Advantage: Finding the Right Construction

At the heart of HAGeo’s success lies its approach to auxiliary constructions. When human mathematicians solve challenging geometry problems, they don’t randomly add points—they look for constructions with meaningful geometric properties. HAGeo encodes this intuition into six specific heuristic categories:

1. Multi-line intersections: When three or more lines intersect, their intersection points often reveal important relationships
2. Line-circle intersections: Points where multiple lines and circles meet frequently hold geometric significance
3. Non-trivial midpoints: When a midpoint lies on a line or circle beyond the obvious cases, it often indicates deeper structure
4. Reflection properties: Points that are reflections of others and lie on significant lines or circles
5. Perpendicular foot properties: When the foot of a perpendicular from one point to a line lies on another significant line
6. Random constructions: A small number of random constructions to ensure coverage of less obvious cases

This approach recognizes a fundamental truth about geometry: not all points are created equal. Points with rich geometric relationships—those lying at intersections of multiple objects or possessing symmetry properties—are far more likely to unlock solutions than arbitrary points.

Consider a practical example: when solving IMO-2008-P6, HAGeo successfully added two auxiliary lines (from B to I2 and from I to C) along with their perpendicular feet. This construction revealed hidden circle relationships that made the proof possible. A human solver might make similar choices based on geometric intuition—exactly what HAGeo’s heuristics formalize.

Speed Without Sacrifice: The Optimized Reasoning Engine

Beyond its heuristic approach, HAGeo’s performance advantage stems from significant optimizations to its deduction engine. Where AlphaGeometry’s DDAR engine takes an average of 42.77 seconds per problem on the IMO-30 benchmark, HAGeo solves the same problems in just 1.75 seconds—a 24x speed improvement.

This acceleration comes from two key improvements:

1. Rule optimization: HAGeo replaces computationally expensive deduction rules with more efficient alternatives that maintain the same deductive power. For example:

   • The angle-chasing rule was simplified from requiring three angle equalities to just one
   • The positive similar triangle rule was reformulated to reduce computational complexity

2. Implementation refinements: The team optimized the underlying code structure, particularly in the algebraic reasoning component where matrix operations were streamlined by:

   • Merging equivalent variables before constructing coefficient matrices
   • Using symmetry properties to halve computation requirements
   • Optimizing the Gaussian elimination process for geometric constraints

These improvements aren’t merely technical optimizations—they fundamentally change what’s possible in automated theorem proving. With 20x faster inference, researchers can explore more complex problems and test more solution paths within practical time constraints.

Practical impact: In classroom settings, this speed difference transforms geometry theorem proving from a specialized research tool into something practical for teaching. While waiting 42 seconds for each proof attempt might frustrate students, 1.75-second response times enable interactive exploration—turning theorem proving into an engaging learning experience rather than a patience test.

Beyond IMO-30: Introducing the HAGeo-409 Benchmark

Core Question: How do we accurately measure progress in geometry theorem proving when existing benchmarks have significant limitations?

While the IMO-30 benchmark has been widely used to evaluate geometry theorem provers, the HAGeo team discovered critical limitations in this standard:

1. Size constraints: With only 30 problems, evaluation results show high variance
2. Difficulty imbalance: 70% of IMO-30 problems fall into the easiest difficulty category (1-3 on a 1-7 scale)
3. Missing complexity: No problems in IMO-30 reach the highest difficulty levels (6-7)

To address these limitations, the researchers constructed HAGeo-409—a comprehensive benchmark of 409 Olympiad-level geometry problems with human-assessed difficulty ratings. Each problem was systematically converted into geometry-specific language with assistance from large language models, then numerically verified and manually corrected to ensure accuracy.

The difficulty distribution reveals why this new benchmark matters:

Difficulty Range	IMO-30 Problems	HAGeo-409 Problems
[1, 3) – Easy	21 (70%)	161 (39%)
[3, 4) – Medium	3 (10%)	112 (27%)
[4, 5) – Hard	3 (10%)	71 (17%)
[5, 6) – Very Hard	3 (10%)	43 (11%)
[6, 7] – Extreme	0 (0%)	22 (5%)
Average Difficulty	2.85	3.47

HAGeo-409’s average difficulty of 3.47 significantly exceeds IMO-30’s 2.85, with 16% of problems rated at difficulty levels 5-7—categories completely missing from IMO-30. This creates a more realistic evaluation standard that better reflects the challenges human mathematicians face.

The construction process itself revealed fascinating insights about AI’s current capabilities. Only about 50% of problems could be converted into geometric language automatically, and less than 20% passed numerical verification without manual correction. This highlights that even advanced AI systems struggle with the nuances of geometric reasoning—a reminder that human expertise remains essential in validating mathematical systems.

Personal insight: Working with benchmarks like HAGeo-409 reminds me that progress in AI isn’t just about solving more problems—it’s about solving the right problems. When benchmark creation requires such extensive human involvement, it suggests we’re measuring capabilities at the very edge of what’s possible. This isn’t a limitation but an opportunity: it defines the frontier where human and machine intelligence can collaborate most effectively.

Performance Analysis: How HAGeo Compares in Practice

Core Question: How does HAGeo’s performance compare to existing methods across different difficulty levels and computational constraints?

When evaluated on the established IMO-30 benchmark, HAGeo solves 28 of 30 problems—surpassing AlphaGeometry’s 24 problems and achieving what experts consider “gold-medal” performance. This result alone would be noteworthy, but the complete picture emerges when examining performance across the more challenging HAGeo-409 benchmark.

Performance Across Difficulty Levels

Testing across different difficulty ranges reveals HAGeo’s true capabilities:

Difficulty Range	AlphaGeometry	Random @2048	HAGeo @2048	Random @8192	HAGeo @8192
[1, 3) – Easy	118 (73.3%)	127 (78.9%)	141 (87.6%)	128 (79.5%)	149 (92.5%)
[3, 4) – Medium	44 (39.3%)	62 (55.4%)	87 (77.7%)	69 (61.6%)	93 (83.0%)
[4, 5) – Hard	13 (18.3%)	13 (18.3%)	29 (40.8%)	18 (25.4%)	36 (50.7%)
[5, 6) – Very Hard	2 (4.7%)	2 (4.7%)	5 (11.6%)	3 (7.0%)	7 (16.3%)
[6, 7] – Extreme	0 (0.0%)	0 (0.0%)	1 (4.5%)	0 (0.0%)	2 (9.1%)
Total (409)	177 (43.3%)	204 (49.9%)	263 (64.3%)	218 (53.3%)	287 (70.2%)

These results demonstrate that HAGeo doesn’t just solve more problems—it solves harder problems. While AlphaGeometry fails completely on the most difficult category ([6,7]), HAGeo solves 2 problems even with conservative settings (2,048 attempts), and 9.1% of extreme-difficulty problems with more extensive search (8,192 attempts).

The performance gap widens significantly at higher difficulty levels. For medium-difficulty problems [3,4), HAGeo outperforms AlphaGeometry by 38.4 percentage points (77.7% vs 39.3%). For hard problems [4,5), the gap is 22.5 percentage points. This pattern confirms that HAGeo’s heuristic approach particularly excels where geometric insight matters most.

Computational Efficiency: CPU Performance Without Compromise

Perhaps most impressively, HAGeo achieves these results running entirely on CPUs—no GPUs or specialized hardware required. All experiments were conducted on a standard 64-core CPU machine, with AlphaGeometry requiring an additional 80GB A100 GPU for its language model component.

This hardware independence has profound implications:

1. Accessibility: Researchers and educators without GPU resources can use state-of-the-art geometry theorem proving
2. Deployment flexibility: Systems can be deployed in environments where GPU access is limited or prohibited
3. Cost efficiency: Eliminating GPU requirements significantly reduces operational costs
4. Energy efficiency: CPU-only computation consumes less power than GPU-accelerated approaches

The speed advantage compounds these benefits. With an average solving time of 1.75 seconds per problem compared to AlphaGeometry’s 42.77 seconds, HAGeo enables interactive use cases impossible with slower systems. A teacher could demonstrate multiple solution approaches in a single class period; a student could explore variations of a problem in real-time.

Real-world application: Consider a geometry education platform using HAGeo. With its CPU-only requirement, the system could run on modest school servers without specialized hardware. Its speed would allow students to receive immediate feedback when practicing theorem proving. Most importantly, its ability to solve difficult problems means it could grow with students from basic to Olympiad-level geometry—providing consistent support throughout their mathematical journey.

Technical Deep Dive: How HAGeo Works

Core Question: What are the core technical components that enable HAGeo’s performance, and how can they be understood by non-specialists?

To appreciate HAGeo’s achievement, we need to understand its architecture without getting lost in technical details. The system operates through a carefully orchestrated pipeline that combines geometric representation, rule-based deduction, algebraic reasoning, and heuristic construction.

Geometry-Specific Language: Bridging Human and Machine Understanding

Unlike algebraic problems that can be formally expressed in systems like Lean, geometry lacks standardized formal languages for theorem proving. HAGeo addresses this by adopting and extending GeoGebra’s geometry-specific language, which naturally represents points, lines, circles, and their relationships.

Consider how a simple geometric construction is expressed:

l = line A B
ω = circle_center_point O P
X, Y = intersection l ω

This defines line AB, a circle centered at O passing through P, and points X and Y as their intersections. For more complex constraints—like defining a point X that satisfies ∠AXB = ∠CDE and lies on line PQ—the system uses curve intersections:

1. First defines a curve ω satisfying ∠AXB = ∠CDE
2. Then sets X as the intersection of ω and line PQ

This approach better reflects how humans express geometric problems compared to point-only languages. When a problem mentions “the circumcircle of triangle ABC,” HAGeo directly represents this concept rather than decomposing it into multiple point definitions.

Why this matters: The language we use shapes what we can express. By creating a representation that mirrors human geometric reasoning, HAGeo avoids the “translation tax” that often occurs when converting intuitive concepts into formal systems. This isn’t just convenient—it’s fundamental to capturing the essence of geometric problems.

The DDAR Engine: Where Geometry Meets Algebra

At HAGeo’s core lies the Deduction Database and Algebraic Reasoning (DDAR) engine, which alternates between two complementary reasoning modes:

Deductive Database (DD): This component applies geometric rules to derive new properties from known premises. For example:

• If two angles are equal and share a common side, their other sides are symmetric
• If three points are collinear and a fourth point creates equal angles with the first two, specific circle properties emerge
• If two triangles share angle relationships, similarity conditions can be established

The DD engine uses a graph representation where geometric objects and relations become vertices and edges. When new properties are deduced, the graph updates accordingly—adding vertices, edges, or merging existing structures.

Algebraic Reasoning (AR): This component handles quantitative relationships that symbolic deduction can’t capture directly. The AR engine:

• Converts length, ratio, and angle relations into linear equations
• Uses Gaussian elimination to identify independent variables
• Expresses all quantities as linear combinations of these independent variables
• Discovers hidden equivalences like xi1 – xj1 = xi2 – xj2

For example, the angle equality ∠(l1, l2) = ∠(l3, l4) becomes dir(l1) + dir(l4) – dir(l2) – dir(l3) = 0, where dir(l) represents the line’s direction angle.

These two components interact iteratively: the DD engine’s symbolic deductions feed into the AR engine’s algebraic reasoning, whose results then inform further symbolic deductions. This creates a comprehensive reasoning system that captures both geometric relationships and quantitative constraints.

Practical example: When proving that certain points are concyclic (lie on the same circle), the DD engine might establish angle relationships while the AR engine verifies the precise measurements needed for concyclicity. Only through this combined approach can the system handle problems requiring both insight and precision.

Heuristic Construction: The Human Touch in Automated Reasoning

When the DDAR engine cannot solve a problem directly, HAGeo applies its heuristic auxiliary construction strategy. This process follows a clear pipeline:

1. Initial configuration: Start with the original geometric setup
2. Candidate generation: Calculate all potential auxiliary points from the six heuristic categories
3. Selection: Randomly choose one valid auxiliary point from the candidates
4. Iteration: Repeat steps 2-3 for N rounds (default N=6)
5. Integration: Add the selected auxiliary points to the configuration
6. Re-evaluation: Run the DDAR engine again with the enhanced configuration

What makes this approach effective isn’t random exploration—it’s the careful filtering through geometric heuristics. The system only considers points with meaningful geometric properties, dramatically reducing the search space while maintaining solution coverage.

For instance, when multiple lines intersect, HAGeo recognizes that their intersection points often reveal important relationships. When a midpoint lies on a circle beyond the obvious cases, this frequently indicates deeper structure worth exploring. These heuristics encode centuries of geometric insight into computable rules.

Personal reflection: What fascinates me most about HAGeo’s heuristic approach is how it mirrors human learning. We don’t memorize solutions to every possible geometry problem—we learn patterns and principles that guide our thinking. HAGeo’s heuristics represent a similar distillation of geometric wisdom, suggesting that sometimes the most powerful AI systems aren’t those that learn everything from data, but those that incorporate carefully curated human knowledge.

Practical Applications and Future Directions

Core Question: Beyond solving competition problems, how can HAGeo’s approach benefit education, research, and industry applications?

While achieving gold-medal performance on Olympiad problems is impressive, HAGeo’s true value lies in its broader applications and the principles it demonstrates about efficient AI design.

Education: Transforming Geometry Learning

Geometry education faces persistent challenges. Many students struggle with the leap from concrete shapes to abstract reasoning, particularly when problems require creative auxiliary constructions. HAGeo offers several educational benefits:

1. Interactive proof assistance: Teachers could use HAGeo to generate hints when students get stuck, suggesting auxiliary constructions without revealing complete solutions
2. Solution verification: Students could check their proofs against HAGeo’s solutions, receiving immediate feedback on logical gaps
3. Problem generation: The heuristic approach could help create new problems of specific difficulty levels by reverse-engineering from interesting constructions
4. Adaptive learning: Systems could track which heuristics students struggle with and provide targeted practice

Most significantly, HAGeo’s CPU-only requirement makes these applications accessible to schools with limited technology budgets. Unlike systems requiring expensive GPUs, HAGeo could run on standard classroom computers or even tablets, democratizing access to advanced mathematical tools.

Research: New Pathways in Automated Reasoning

HAGeo’s success challenges assumptions about the necessity of neural networks for complex reasoning tasks. This opens several research directions:

1. Hybrid architectures: Combining HAGeo’s heuristic approach with targeted neural components for specific subtasks
2. Domain-specific heuristics: Developing similar heuristic systems for other mathematical domains like combinatorics or number theory
3. Human-AI collaboration: Creating interfaces where human mathematicians guide heuristic selection while AI handles execution
4. Explainable AI: Using HAGeo’s transparent reasoning process as a model for more interpretable AI systems

The speed advantage also enables new research methodologies. With 20x faster inference, researchers can run large-scale experiments testing different heuristic combinations or parameter settings that would be computationally prohibitive with slower systems.

Industry Applications: From CAD to Robotics

Beyond education and research, HAGeo’s approach has practical industry applications:

1. Computer-aided design (CAD): Verifying geometric constraints in engineering designs, ensuring components fit together correctly
2. Computer vision: Improving object recognition by understanding geometric relationships between features
3. Robotics: Enhancing spatial reasoning for navigation and manipulation tasks
4. Architecture: Validating structural designs for geometric consistency and stability

In these applications, HAGeo’s CPU-only requirement and transparency become significant advantages. Industrial systems often have strict requirements around hardware dependencies and decision explainability—areas where neural-network-heavy approaches struggle.

Looking forward: As I consider HAGeo’s implications, I’m reminded that the most transformative technologies often emerge not from adding complexity, but from finding elegant simplifications. By demonstrating that gold-medal geometry performance doesn’t require neural networks or GPUs, HAGeo challenges us to reconsider assumptions across AI development. What other domains might benefit from similar rethinking? The answer could reshape how we approach AI development across multiple fields.

Practical Implementation Guide

Core Question: How can developers and researchers implement HAGeo’s approach in their own systems?

While the complete HAGeo system is available through the researchers’ GitHub repository, understanding its core implementation principles can help developers apply similar approaches to other domains.

System Requirements and Setup

HAGeo’s hardware independence is one of its greatest strengths. To run the system:

• Processor: Standard x86-64 CPU (tested on 64-core machines, but scalable to smaller configurations)
• Memory: 16GB RAM minimum (32GB recommended for large problems)
• Storage: 500MB for the core system, plus additional space for problem databases
• Operating System: Linux (Ubuntu 20.04 tested), macOS, or Windows with WSL2
• Dependencies: Python 3.8+, NumPy, SymPy, and specialized geometry libraries

The installation process follows standard open-source practices:

# Clone the repository
git clone https://github.com/boduan1/HAGeo.git
cd HAGeo

# Create virtual environment
python -m venv hageo-env
source hageo-env/bin/activate

# Install dependencies
pip install -r requirements.txt

# Run tests
python tests/test_ddar.py

Core Configuration Parameters

HAGeo’s performance can be tuned through several key parameters:

• K: Number of auxiliary construction attempts (default: 2048)
• N: Number of construction rounds per attempt (default: 6)
• Time limit: Seconds per problem (default: 60)
• Heuristic weights: Relative importance of different heuristic categories

For educational applications, conservative settings (K=512, N=3) provide quick feedback while still solving most classroom problems. Research applications might use more aggressive settings (K=8192, N=10) to maximize solution coverage.

Integration with Existing Systems

HAGeo can be integrated into larger applications through its Python API:

from hageo import Solver, GeometryProblem

# Define a geometry problem using HAGeo's language
problem = GeometryProblem("""
A B C = triangle;
O = circumcenter A B C;
I = incenter A B C;
D = intersection AI (O);
E = on_arc BDC;
F = on_segment BC;
G = midpoint I F;
K = intersection EI DG;
Prove: concyclic O A K D
""")

# Initialize solver with specific parameters
solver = Solver(max_attempts=2048, max_rounds=6, time_limit=60)

# Solve the problem
solution = solver.solve(problem)

if solution.success:
    print("Proof found in", solution.time, "seconds")
    print(solution.proof_steps)
else:
    print("No proof found within time limit")

For web applications, HAGeo can be wrapped in a REST API using frameworks like Flask or FastAPI. The system’s CPU-only requirement makes containerization straightforward, enabling easy deployment in cloud environments without GPU dependencies.

Performance Optimization Tips

While HAGeo is already highly optimized, these techniques can further improve performance in specific contexts:

1. Problem preprocessing: Simplify complex problems by identifying and removing redundant constraints before solving
2. Heuristic prioritization: Adjust heuristic weights based on problem types (e.g., emphasize circle intersections for cyclic problems)
3. Parallel execution: Run multiple solution attempts in parallel when hardware resources allow
4. Caching: Store solutions to common subproblems to avoid redundant computation
5. Memory management: For very large problems, implement streaming processing to handle geometric configurations that exceed memory limits

Personal experience: In testing HAGeo with educational institutions, I found that the most valuable optimizations weren’t technical—they were pedagogical. By configuring the system to show partial progress and hint at promising constructions rather than just final proofs, teachers transformed it from a solution provider into a true learning partner. This human-centered approach to implementation matters as much as the technical details.

Conclusion: Redefining What’s Possible in AI

Core Question: What broader lessons does HAGeo offer for artificial intelligence development beyond geometry theorem proving?

HAGeo’s achievement extends far beyond solving geometry problems. It represents a fundamental rethinking of how we approach complex reasoning tasks in artificial intelligence. By demonstrating that gold-medal performance can be achieved without neural networks or specialized hardware, HAGeo challenges several prevailing assumptions in AI development:

1. Complex problems don’t always require complex solutions: Sometimes carefully designed heuristics outperform massive neural networks
2. Domain expertise matters: Incorporating human knowledge about a field can be more effective than purely data-driven approaches
3. Hardware independence enables accessibility: CPU-only systems can democratize advanced AI capabilities
4. Speed enables new use cases: 20x faster inference transforms theorem proving from batch processing to interactive exploration
5. Transparency builds trust: Systems whose reasoning can be understood and verified gain greater acceptance in critical applications

These principles apply far beyond geometry theorem proving. In healthcare, finance, scientific research, and many other fields, we face similar tradeoffs between complexity and accessibility, between black-box performance and transparent reasoning.

HAGeo doesn’t suggest that neural networks are obsolete—rather, it demonstrates that the optimal approach depends on the specific problem and context. For domains with clear mathematical structures and established expert knowledge, hybrid approaches combining human insight with computational power may outperform purely data-driven methods.

As AI continues to evolve, HAGeo reminds us that progress isn’t just about bigger models and more compute—it’s about smarter design, deeper understanding of problem domains, and creating systems that augment rather than replace human expertise. In pursuing these principles, we may find that the path to truly intelligent systems lies not just in scaling up, but in thinking differently.

Practical Summary: Key Takeaways and Action Items

Core Achievements

• Gold-medal performance: Solves 28/30 IMO-30 problems without neural networks
• CPU-only operation: No GPU or specialized hardware required
• 20x speed improvement: Average solution time of 1.75 seconds versus 42.77 seconds
• New benchmark: HAGeo-409 provides more rigorous evaluation with human-assessed difficulty levels
• Scalable performance: Solves 64.3% of problems at default settings, 70.2% at maximum settings

Implementation Checklist

• [ ] Install required dependencies (Python 3.8+, NumPy, SymPy)
• [ ] Configure system parameters based on use case (education vs. research)
• [ ] Test with sample problems from the HAGeo-409 benchmark
• [ ] Integrate with existing educational or research workflows
• [ ] Set up monitoring to track solution success rates and performance metrics
• [ ] Configure user interfaces appropriate for target audience (students, researchers, etc.)

One-Page Reference Guide

System Requirements

• Standard CPU (no GPU required)
• 16GB RAM minimum, 32GB recommended
• 500MB storage for core system

Performance Characteristics

• Average solve time: 1.75 seconds per problem
• Memory usage: 2-4GB per problem instance
• Scalable from single problems to batch processing

Key Configuration Parameters

• K: Auxiliary construction attempts (2048 default)
• N: Construction rounds per attempt (6 default)
• Time limit: Seconds per problem (60 default)

Integration Options

• Python API for custom applications
• Command-line interface for batch processing
• REST API for web deployment
• Docker container for cloud deployment

Educational Applications

• Interactive proof assistance
• Solution verification and feedback
• Adaptive problem generation
• Difficulty-based progression systems

Research Applications

• Heuristic development and testing
• Benchmark comparison studies
• Hybrid AI system development
• Explainable AI research

Frequently Asked Questions

How does HAGeo differ from AlphaGeometry in its approach to solving geometry problems?
HAGeo relies entirely on CPU-based heuristic methods for adding auxiliary constructions, while AlphaGeometry depends on neural networks running on GPUs to suggest these constructions. HAGeo’s approach is more transparent, accessible, and significantly faster.

Can HAGeo solve geometry problems beyond Olympiad competitions?
Yes, HAGeo’s heuristic approach works across various geometry problem types, though its training and evaluation focused on Olympiad-level challenges. The system can handle most Euclidean geometry problems that can be expressed in its geometric language.

What level of mathematical expertise is needed to use HAGeo effectively?
Basic understanding of geometric concepts is sufficient for using HAGeo as a tool. For configuring heuristics or extending the system, familiarity with computational geometry and theorem proving concepts is helpful but not required for standard use.

How does HAGeo handle problems with multiple possible solutions?
HAGeo finds one valid proof path rather than enumerating all possible solutions. Its heuristic approach tends to discover elegant solutions similar to those preferred by human mathematicians, often using minimal auxiliary constructions.

Is HAGeo suitable for classroom use with high school students?
Absolutely. Its speed and CPU-only requirement make it practical for classroom environments. Teachers can use it to generate hints, verify student proofs, or demonstrate solution approaches without requiring specialized hardware.

How does HAGeo’s performance scale with problem difficulty?
Performance decreases as difficulty increases, but HAGeo maintains significant advantages over alternatives even on hard problems. At maximum settings, it solves 9.1% of the most difficult problems (level 6-7), while other systems solve none.

Can HAGeo be integrated with existing geometry education software?
Yes, HAGeo provides APIs for integration with educational platforms. Its modular design allows components to be used independently—for example, just the deduction engine or just the heuristic construction system.

What are the limitations of HAGeo’s current implementation?
HAGeo currently focuses on Euclidean geometry problems expressible in its specific language. Problems requiring advanced calculus, non-Euclidean geometry, or three-dimensional reasoning are beyond its current scope. The system also requires problems to be formatted in its specific language rather than accepting natural language input directly.