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Google DeepMind Solves Unsolvable Math Problem With AI

Updated: Dec 22, 2023

Recent advances in large language models (LLMs) have made artificial intelligence more adaptable than ever before, but that comes with a drawback: lies. Generative AI tends to make things up, but Google DeepMind has devised a new LLM that sticks to mathematical truths. The company’s FunSearch can solve highly complex math problems. Miraculously, the solutions it generates aren’t just accurate; they’re entirely new solutions that no human has ever found.


FunSearch is thus named because it searches for mathematical functions, not because it’s fun. Although, some people might consider the cap set problem a real hoot—mathematicians can’t even agree on how best to solve it, making it a genuine numerical mystery. DeepMind previously made advances in AI with its Alpha models like AlphaFold (protein folding), AlphaStar (StarCraft), and AlphaGo (playing Go). These systems were not based on LLMs, but they did reveal new mathematical concepts.


With FunSearch, DeepMind started with a large language mode, a version of Google’s PaLM 2 called Codey. There’s a second LLM layer at work, which scans Codey’s output and prunes incorrect information. The team behind this work didn’t know if this approach would work, and they’re still not sure why it does, according to DeepMind researcher Alhussein Fawzi.

To start, DeepMind engineers created a Python representation of the cap set problem, but they left out the lines that described the solution. Codey’s job was to add lines that accurately solved the problem. The error-checking layer then scores the Codey solutions to see if they are accurate. In high-level math, equations may have more than one solution, but not all of them are considered equally good. Over time, the algorithm identifies the best solutions from Codey and feeds them back into the model.



DeepMind let FunSearch churn for several days long enough to generate millions of possible solutions. This allowed FunSearch to refine the code and produce better results. According to the newly published research, the AI came up with a previously unknown but correct solution to the cap set problem. DeepMind also turned FunSearch loose on another difficult bit of math called the bin packing problem, an algorithm that describes the most efficient way to pack bins. FunSearch found a faster solution than those calculated by humans.


Mathematicians are still struggling with integrating LLM technology into their work, and the work at DeepMind shows a possible path forward. The team believes this approach has potential because it generates computer code rather than the solution. This is often easier to understand and verify than raw mathematical outputs.


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