DeepMind's FunSearch coaxed a code LLM into new mathematics — cap set included

In December 2023, Google DeepMind reported in Nature that its tool FunSearch had used a large language model to crack a famous unsolved problem — the first time an LLM had discovered a solution to a long-standing scientific puzzle, producing verifiable new information that did not previously exist. "It's not in the training data — it wasn't even known," said coauthor Pushmeet Kohli. The cap set problem asks how large a set can be with no three points forming a line; mathematicians do not even agree on how to approach it, and Terence Tao once called it "perhaps my favorite open question."

The method: researchers sketched the problem in Python but left out the lines specifying how to solve it. FunSearch gets Codey — a version of PaLM 2 fine-tuned on computer code — to fill in the blanks, while a second algorithm checks and scores what it produces. The best suggestions, even incorrect ones, are saved and given back so the model can try again: "Many will be nonsensical, some will be sensible, and a few will be truly inspired," said Kohli. A couple of million suggestions and a few dozen iterations produced code generating a correct, previously unknown solution.

The design grew out of a limitation: DeepMind's earlier AlphaTensor and AlphaDev, built on AlphaZero, had beaten records in matrix multiplication and everyday algorithms, but, as researcher Bernardino Romera-Paredes put it, they were stuck in their lanes. FunSearch outputs code — a recipe rather than an answer — so it generalizes across problems and is easier to understand.

Tested on the bin packing problem, FunSearch found an approach faster than human-devised ones. The team is candid that the mechanism isn't fully understood: "we have hypotheses, but we don't know exactly why this works," said research scientist Alhussein Fawzi.

Don't ask the LLM for answers — ask it for programs, then let an automatic verifier throw away the garbage and keep the inspired few

An automatic verifier does the epistemology: wrong or nonsensical programs are scored out, so the LLM's hallucinations cost nothing but compute.

Recycling the best-so-far programs into the prompt steers later guesses — an evolutionary loop over code rather than single-shot answers.

Progress is a runnable program anyone can check, not a proof anyone must trust — which made an LLM's output scientifically acceptable.

The recipe form generalizes: the same loop moved from the cap set problem to bin packing without redesign.

Verification beats eloquence: a model famous for making things up becomes a discovery engine when a machine-checkable score does the believing for you.

The cap set result was published in Nature in December 2023 and FunSearch also produced a faster-than-human approach to bin packing, relevant from data centers to e-commerce. Tao called it "a promising paradigm" and "one possible way forward" for incorporating LLMs into mathematical research, extending DeepMind's streak of AI-found results after AlphaTensor and AlphaDev.