> Every Mathematician Has Only a Few Tricks
    > 
    > A long time ago an older and well-known number theorist made some disparaging remarks about Paul Erdös’s work.
    > You admire Erdös’s contributions to mathematics as much as I do,
    > and I felt annoyed when the older mathematician flatly and definitively stated
    > that all of Erdös’s work could be “reduced” to a few tricks which Erdös repeatedly relied on in his proofs.
    > What the number theorist did not realize is that other mathematicians, even the very best,
    > also rely on a few tricks which they use over and over.
    > Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory.
    > I have made a point of reading some of these papers with care.
    > It is sad to note that some of Hilbert’s beautiful results have been completely forgotten.
    > But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory,
    > it was surprising to verify that Hilbert’s proofs relied on the same few tricks.
    > Even Hilbert had only a few tricks!
    > 
    > - Gian-Carlo Rota - "Ten Lessons I Wish I Had Been Taught"

> "a broken clock is right twice a day."

The combinatorial nature of trying things randomly means that it would take millennia or longer for light-speed monkeys typing at a keyboard, or GPUs, to solve such a problem without direction.

By now, people should stop dismissing RL-trained reasoning LLMs as stupid, aimless text predictors or combiners. They wouldn’t say the same thing about high-achieving, but non-creative, college students who can only solve hard conventional problems.

Yes, current LLMs likely still lack some major aspects of intelligence. They probably wouldn’t be able to come up with general relativity on their own with only training data up to 1905.

Neither did the vast majority of physicists back then.

Yeah, they're great at interpolation - they'll just never be worth much at extrapolation.

> "a broken clock is right twice a day"

and homo sapiens, glancing at the clock when it happens to be right, may conjure an entire zodiac to explain it.

A stopped clock.

A broken clock can be broken in ways which result in it never being correct.

Wait, what do you mean "LLMs are still absolutely useless at actual maths computation"? I rely on them constantly for maths (linear algebra, multivariable calc, stat) --- literally thousands of problems run through GPT5 over the last 12 months, and to my recollection zero failures. But maybe you're thinking of something more specific?

The ultimate generalist

Also just the sheer value of brute force.

80 hours! 80 hours of just trying shit!