Stockfish did not teach itself to play chess. You are probably thinking of Leela Chess Zero - an open re-implementation of AlphaZero - both were given nothing but the rules of chess and a board and played millions of games against themselves until they were the strongest engine available at the time.
Stockfish's neural net evaluation model was trained on millions of its positions with its own original algorithmic evaluation function (entirely developed by humans) and search tree. The result was a much smaller model than Leela's that requires little computation (not even a GPU), paired with its already extremely efficient search/pruning algorithms that made it stronger than Leela in competitive play. Leela's evaluation function is much stronger (at one ply it has an ELO of around 2300, Stockfish is probably closer to 1800), but it requires vastly more resources and those are always bounded in a match.
Humans haven't learned as much new information about chess from Stockfish as we have from Leela.
"The proof came from a general-purpose reasoning model, not a system built specifically to solve math problems or this problem in particular, and represents an important milestone for the math and AI communities."
I would have thought a triangular grid works better than a grid of squares. You get ~3n links vs ~2n for the square grid. Curious what the AI came up with.
While the result is impressive, this blog post is extremely disappointing.
- It does not show an example of the new best solution, nor explain why they couldn't show an example (e.g. if the proof was not constructive)
- It does not even explain the previous best solution. The diagram of the rescaled unit grid doesn't indicate what the "points" are beyond the normal non-scaled unit grid. I have no idea what to take away from it.
- It's description of the new proof just cites some terms of art with no effort made to actually explain the result.
If this post were not on the OpenAI blog, I would assume it was slop. I understand advanced pure mathematics is complicated, but it is entirely possible to explain complicated topics to non-experts.
I dunno, I'm skeptical without proof. I've had the MAX+ plan for a while and I'm sorry, the quality between GPT vs Claude is night and day difference. Claude understands. GPT stumbles over every request I give it.
Is there a reason why we only hear of Erdos problems being solved? I would imagine there are a myriad of other unsolved problems in math, but every single ChatGPT "breakthrough in math" I come across on r/singularity and r/accelerate are Erdos problems.
As others have written, Erdős was a lifelong curator of mathematical problems, from high-school level problems to the types that will land you a Fields medal. Like the Collatz conjecture.
Most new math problems appear in other papers, doctoral dissertations, etc. Usually you'll find them in the "future work" / "future research" section.
So obviously in order to present and formalize these problems, you either need the author(s) to do it, or some reader. At this level of math, there are many extremely niche fields, where the papers might only be read by a small amount of people.
In short, it is a visibility problem.
But, I figure, there's some potential use in AI models to extract and present these problems, which would make them available to a larger audience.
That is exactly what Erdős did. His life revolved around math, and seeking mathematical questions.
Erdos problems are well-posed for AI — elementary statements, exact counterexample targets, extensively catalogued. selection bias: these are exactly the problems AI can actually search
The summarized chain of thought for this task (linked in the blogpost) is 125 pages. That's an insane scale of reasoning, quite akin to what Anthropic has been teasing with Mythos.
To the “LLMs just interpolate their training data” crowd:
Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules.
I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries.
So either “recombining existing material” isn’t disqualifying, or a lot of Fields Medals need to be returned.
It is the old discovery vs invention in mathematics.
Or of you prefer philosophy: Parmenides (nothing changes) vs Heraclitus (you cannot bath twice in the same river aka everything changes all the time).
Postmodernism also claimed that everything has been done already. IMO these 2 are points of view that one can adopt, not truths based on fact. So the distinction is a matter of taste or perspective, not of truth, IMO.
For anyone using LLMs heavily for coding, this shouldn't be too surprising. It was just a matter of time.
Mathematicians make new discoveries by building and applying mathematical tools in new ways. It is tons of iterative work, following hunches and exploring connections. While true that LLMs can't truly "make discoveries" since they have no sense of what that would mean, they can Monte Carlo every mathematical tool at a narrow objective and see what sticks, then build on that or combine improvements.
Reading the article, that seems exactly how the discovery was made, an LLM used a "surprising connection" to go beyond the expected result. But the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept.
Every time I interact even with OpenAI's pro model, I am forced to come to the conclusion that anything outside the domain of specific technical problems is almost completely hopeless outside of a simple enhanced search and summary engine.
For example, these machines, if scaling intellect so fiercely that they are solving bespoke mathematics problems, should be able to generate mundane insights or unique conjectures far below the level of intellect required for highly advanced mathematics - and they simply do not.
Ask a model to give you the rundown and theory on a specific pharmacological substance, for example. It will cite the textbook and meta-analyses it pulls, but be completely incapable of any bespoke thinking on the topic. A random person pursuing a bachelor's in chemistry can do this.
Anything at all outside of the absolute facts, even the faintest conjecture, feels completely outside of their reach.
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[ 1.7 ms ] story [ 141 ms ] threadFor those in academics, is OpenAI the vendor of choice?
A difficult part was constructing a chess board on which to play math (Lean). Now it's just pattern recognition and computation.
LLMs are just the beginning, we'll see more specialized math AI resembling StockFish soon.
Stockfish's neural net evaluation model was trained on millions of its positions with its own original algorithmic evaluation function (entirely developed by humans) and search tree. The result was a much smaller model than Leela's that requires little computation (not even a GPU), paired with its already extremely efficient search/pruning algorithms that made it stronger than Leela in competitive play. Leela's evaluation function is much stronger (at one ply it has an ELO of around 2300, Stockfish is probably closer to 1800), but it requires vastly more resources and those are always bounded in a match.
Humans haven't learned as much new information about chess from Stockfish as we have from Leela.
Nitpick: it’s Elo, not ELO. The name comes from the inventor’s surname and is not an acronym.
- It does not show an example of the new best solution, nor explain why they couldn't show an example (e.g. if the proof was not constructive)
- It does not even explain the previous best solution. The diagram of the rescaled unit grid doesn't indicate what the "points" are beyond the normal non-scaled unit grid. I have no idea what to take away from it.
- It's description of the new proof just cites some terms of art with no effort made to actually explain the result.
If this post were not on the OpenAI blog, I would assume it was slop. I understand advanced pure mathematics is complicated, but it is entirely possible to explain complicated topics to non-experts.
There is no universally agreed-upon "central" conjecture (like "P vs. NP" in CS), but here are some pillars:
1) https://en.wikipedia.org/wiki/Happy_ending_problem
2) https://en.wikipedia.org/wiki/Hadwiger_conjecture_(combinato...
3) https://en.wikipedia.org/wiki/Hirsch_conjecture
Most new math problems appear in other papers, doctoral dissertations, etc. Usually you'll find them in the "future work" / "future research" section.
So obviously in order to present and formalize these problems, you either need the author(s) to do it, or some reader. At this level of math, there are many extremely niche fields, where the papers might only be read by a small amount of people.
In short, it is a visibility problem.
But, I figure, there's some potential use in AI models to extract and present these problems, which would make them available to a larger audience.
That is exactly what Erdős did. His life revolved around math, and seeking mathematical questions.
Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules.
I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries.
So either “recombining existing material” isn’t disqualifying, or a lot of Fields Medals need to be returned.
Or of you prefer philosophy: Parmenides (nothing changes) vs Heraclitus (you cannot bath twice in the same river aka everything changes all the time).
Postmodernism also claimed that everything has been done already. IMO these 2 are points of view that one can adopt, not truths based on fact. So the distinction is a matter of taste or perspective, not of truth, IMO.
Mathematicians make new discoveries by building and applying mathematical tools in new ways. It is tons of iterative work, following hunches and exploring connections. While true that LLMs can't truly "make discoveries" since they have no sense of what that would mean, they can Monte Carlo every mathematical tool at a narrow objective and see what sticks, then build on that or combine improvements.
Reading the article, that seems exactly how the discovery was made, an LLM used a "surprising connection" to go beyond the expected result. But the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept.
1. Erdos 1196, GPT-5.4 Pro - https://www.scientificamerican.com/article/amateur-armed-wit...
There are a couple of other Erdos wins, but this was the most impressive, prior to the thread in question. And it's completely unsupervised.
Solution - https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba...
2. Single-minus gluon tree amplitudes are nonzero , GPT-5.2 https://openai.com/index/new-result-theoretical-physics/
3. Frontier Math Open Problem, GPT-5.4 Pro and others - https://epoch.ai/frontiermath/open-problems/ramsey-hypergrap...
4. GPT-5.5 Pro - https://gowers.wordpress.com/2026/05/08/a-recent-experience-...
5. Claude's Cycles, Claude Opus 4.6 - https://www-cs-faculty.stanford.edu/~knuth/papers/claude-cyc...
For example, these machines, if scaling intellect so fiercely that they are solving bespoke mathematics problems, should be able to generate mundane insights or unique conjectures far below the level of intellect required for highly advanced mathematics - and they simply do not.
Ask a model to give you the rundown and theory on a specific pharmacological substance, for example. It will cite the textbook and meta-analyses it pulls, but be completely incapable of any bespoke thinking on the topic. A random person pursuing a bachelor's in chemistry can do this.
Anything at all outside of the absolute facts, even the faintest conjecture, feels completely outside of their reach.