The Unsolvable Problem
In the world of pure mathematics, some problems are so famously difficult they achieve legendary status. One such puzzle is the 'cap set problem'. In simple terms, it asks for the largest possible group of points you can have in a high-dimensional space
where no three points form a straight line. Think of it like a complex, multi-dimensional game of tic-tac-toe where you try to place as many markers as possible without getting three in a row. For decades, this problem in a field called extremal combinatorics has vexed mathematicians because the number of potential combinations is astronomically large, making it impossible to solve through sheer computing power.
A New Type of Collaborator
Enter FunSearch, an AI system developed by Google's DeepMind. Unlike many AI models that are trained on existing human knowledge, FunSearch is designed to discover entirely new knowledge. It pairs a creative Large Language Model (LLM) with an automated evaluator. The LLM, named Codey, generates potential solutions in the form of computer code, while the evaluator meticulously checks the AI's work, filtering out incorrect ideas or 'hallucinations'. This iterative back-and-forth allows the system to start with a basic understanding of a problem and 'evolve' its solutions, building upon its best ideas to uncover novel insights. The name FunSearch comes from its method of searching for mathematical 'functions'.
Human Guidance Meets Machine Creativity
This isn't a story of a machine working in isolation. The breakthrough on the cap set problem was a true collaboration. Researchers guided the AI, providing it with the initial framework and knowledge about the problem. The AI then took over the heavy lifting, exploring countless avenues and unconventional pathways that a human mathematician might dismiss as too time-consuming or unlikely to yield results. In this partnership, the humans provide the strategic direction and conceptual understanding, while the AI provides a powerful, tireless engine for exploration and pattern recognition. It's a model that leverages the best of both worlds: human intuition and machine scale. This process has been described as giving mathematicians a new kind of telescope to see patterns that were previously invisible.
The 'Stunning' Result
The collaboration paid off spectacularly. FunSearch discovered new and larger cap sets than had ever been found before, providing solutions that went beyond the best-known results developed by human mathematicians. The AI didn't just find an answer; it generated computer programs that clearly showed how it constructed the solutions, offering a transparent pathway for human experts to verify and understand the new discovery. This marked the first time an LLM has been used to find novel solutions to a long-standing open problem in mathematics, a significant milestone that demonstrates AI's potential to not just process information, but to contribute to genuine scientific discovery.
A New Era for Scientific Discovery
The implications of this breakthrough extend far beyond a single math puzzle. The successful application of FunSearch to the cap set problem serves as a powerful proof of concept for a new way of conducting research. The same system has also been used to find more efficient solutions for the 'bin-packing problem', a logistical challenge with widespread real-world applications, such as improving efficiency in data centres. This demonstrates the system's versatility and points toward a future where AI collaborators could help accelerate discoveries in fields as diverse as medicine, materials science, and physics, helping to solve some of humanity’s most complex and pressing challenges.
