If you have played the card game Set, you already know this problem. Each card has four attributes with three values each, and a valid set is three cards that are either all the same or all different in every attribute. The question combinatorialists ask is how many cards you can lay out with no set among them at all. The answer is 20, out of 81.
Generalise it to n attributes instead of four and you have the cap set problem: the largest subset of an n-dimensional grid of side three with no three points in a line. The exact answers are known up to dimension 6, where it is 112, and dimension 7, where it is 236. Dimension 8 is not settled, and the best anyone had constructed was 496 points.
What FunSearch did
In December 2023 Google DeepMind published a system called FunSearch in Nature. It found 512.
The design is the interesting part. FunSearch does not ask a language model for an answer. It asks for a program that constructs an answer, runs that program, scores the result, and feeds the best-scoring programs back as examples for the next round. The model supplies ideas and the evaluator supplies truth. A model that invents a plausible-sounding construction which does not work gets filtered out at the scoring step, which neatly removes the usual objection to using language models for mathematics.
The output is also a program rather than a weight matrix, so a human can read it and see what the idea was. That is a real difference from the usual machine learning result, where you get a number and no explanation.
Is sixteen points a big deal?
Taken alone, no. It is a 3.2 per cent improvement in one dimension of a problem whose asymptotic behaviour is what people actually care about.
The reason the dimension matters at all is that cap sets multiply. Combine a cap set in dimension n with itself and you get one in dimension 2n with the size squared, so a construction propagates upwards forever and the figure that counts is the size to the power one over the dimension. The cap set growth rate calculator does that conversion: 496 in dimension 8 gives a rate of 2.1724, and 512 gives 2.1810.
Here is the honest part, which the coverage at the time mostly skipped. Dimension 6 on its own gives a rate of 2.1955, which is better than either. The dimension-8 record is a record for dimension 8, not a new asymptotic bound. The best known asymptotic rate is 2.2180, from Tyrrell in 2023 using admissible sets, and the ceiling is 2.756 from Ellenberg and Gijswijt in 2016. FunSearch did contribute on the asymptotic side too, through a different construction, but the 512 headline and the asymptotic improvement are two separate things that were often reported as one.
Why it was still a landmark
Because it was the first time a large language model produced a verified new discovery in an open mathematical problem. Not a restatement of something in its training data, not a plausible-looking proof that needed checking by hand, but a construction that is either valid or not and turned out to be valid.
Everything since has followed the same shape: generate, score, keep what survives. The same system was pointed at a practical problem on the same day, covered in the piece on bin packing, and the search-and-score pattern turns up again in the minimum overlap problem.