AI Math Whiz: Solving Unsolvable Problems

Artificial intelligence has officially moved beyond writing emails and generating images. It is now solving mathematical riddles that have stumped researchers for decades. In a major scientific breakthrough, Google DeepMind’s AI system has discovered new solutions to the “cap set problem,” a notorious challenge in combinatorics. This marks the first time a Large Language Model (LLM) has surpassed human mathematicians in discovering verifiable, novel information for an open scientific problem.

The Cap Set Problem Explained

To understand this breakthrough, you first need to understand the problem. The cap set problem comes from a branch of mathematics called combinatorics (specifically Ramsey theory). It involves plotting points on a high-dimensional grid. The goal is to identify the largest set of points where no three points form a straight line.

If this sounds abstract, picture the card game “Set.” In that game, players look for three cards that satisfy specific patterns. The cap set problem is essentially a high-dimensional version of finding the maximum number of cards you can lay down without creating a valid “set.”

For years, mathematicians have struggled to calculate the upper limits of these sets as dimensions increase. The problem is so complex that renowned mathematician Terence Tao once described it as his favorite open question.

The Specific Breakthrough

DeepMind’s AI focused on dimension 8. Before this study, the best solution human mathematicians had found for dimension 8 was a set size of 496. The AI system, known as FunSearch (short for Function Search), discovered a construction for a set of size 512.

This was not an approximation. It was a concrete, verifiable mathematical fact that broke a record standing for nearly 20 years.

Meet FunSearch: The Engine Behind the Discovery

Unlike standard chatbots like ChatGPT or Gemini, FunSearch does not simply guess the answer. Standard LLMs are prone to “hallucinations,” where they confidently state incorrect facts. In high-level mathematics, a hallucination renders the result useless.

FunSearch solves this by pairing two distinct components:

1. The Creative Writer: It uses a pre-trained LLM called Codey (a version of Google’s PaLM 2 model). Codey generates computer code that attempts to solve the problem.
2. The Strict Judge: An automated evaluator runs the code to see if it actually works. If the code fails or produces an invalid answer, it is immediately discarded.

This creates an evolutionary loop. The system starts with a rough sketch of a program. Codey suggests changes to the code. The evaluator tests them. Only the best, working programs are fed back into Codey to be improved further. After millions of iterations, FunSearch evolved a piece of code that human experts had never conceived.

Why "White Box" AI Matters

One of the most significant aspects of this discovery is how the answer was delivered.

In the past, AI breakthroughs like AlphaGo (which mastered the game of Go) acted as “black boxes.” They could tell you what move to make, but they couldn’t explain why. They provided the output, not the reasoning.

FunSearch is different. It produces computer code that human mathematicians can read and inspect. When FunSearch solved the cap set problem, it didn’t just spit out a number. It provided a short, concise Python program that generated the solution. This allows scientists to look at the code and understand the logic behind the discovery, effectively teaching humans new strategies.

Practical Application: The Bin Packing Problem

DeepMind did not stop at abstract dots on a graph. To prove FunSearch wasn’t a one-trick pony, researchers applied it to the “bin packing problem.”

This is a classic logistical challenge: how do you pack items of various sizes into the fewest number of bins? This has massive real-world implications, from loading shipping containers to scheduling jobs in a data center to maximize server efficiency.

FunSearch managed to generate algorithms that outperformed the standard heuristics (rules of thumb) currently used in the industry. Because the output was code, the result could be immediately integrated into existing logistical software systems to improve efficiency.

The Future of AI in Science

The success of FunSearch, published in the journal Nature in late 2023, signals a shift in how science is conducted. AI is no longer just a tool for processing data; it is becoming a collaborator capable of genuine creativity.

By searching through the “function space” (the universe of possible computer programs) rather than just looking for data patterns, systems like FunSearch act as a force multiplier for human intelligence. They handle the brute-force trial and error, handing over clean, verifiable code that pushes the boundaries of what we know is possible.

Frequently Asked Questions

Did the AI actually prove a mathematical theorem? Not exactly. FunSearch discovered a “counterexample” or a new lower bound. It found a specific set of 512 points, proving that a set of that size is possible. It did not provide a formal proof that 512 is the maximum limit, but it disproved the previous assumption that the limit was lower.

Can I use FunSearch on my computer? FunSearch is a research methodology developed by Google DeepMind rather than a consumer app you can download today. However, the underlying method uses LLMs similar to those available to developers, and the code generated for the cap set problem has been made public for researchers.

Does this mean AI is better than mathematicians? No. AI is faster at checking millions of possibilities. The “prompts” and the setup of the problem were designed by human scientists. The result is best viewed as a collaboration where the AI acts as a tireless research assistant that never sleeps and makes zero calculation errors when verified by the evaluator.

What is the risk of the AI hallucinating the answer? FunSearch eliminates hallucinations through its evaluator. Because the output is computer code, it is either right or wrong. The evaluator runs the code; if the code is broken or yields an incorrect mathematical result, it is rejected before a human ever sees it.