The AI took a day on one of the problems so it must have generated and discarded a lot of proofs that didn't work. How could it choose which one to submit as the answer, except the objective fact of the proof passing in Lean.

I think it proves lemmas it can submit all the lemmas it proved lol

When tackling IMO problems, the hard part is coming up with a good approach to the proof. Verifying your proof (and rejecting your false attempts) is much easier. You'll know which one to submit.

(Source: I am a two-time IMO silver medalist.)

I don't see why we should take your word for it, as opposed to just asking AlphaProof to comment instead.

Well, he does have twice the amount of silver medals... And can speak the English language... Although, an AI attempting to speak with the human race entirely through an esoteric math-proofing language would be an interesting take on the whole, "humans make ET contact, interact through universal language.... pause for half of the movie, military hothead wanting to blow it out of the sky, until pretty lady mathematician runs into the oval office waving a sheet of paper... OF MATH!" trope.... But now, it's a race between you and I, to see who can write the screenplay first!

You are a human, not an AI. You know whether your idea seems related to the solution. The AI has thousands of ideas and doesn't know which are better. Graders shouldn't accept a thousand guesses grubbing for 1 point.

If verifying a good idea is easy, then the evidence shows that the AI didn't have good ideas for the other 2 problems.

we are talking about lean proofs. Given a formal statement and a proof - the lean can verify whether it's correct or not. It's like generating computer programs to solve a problem - the problem lied in generating useful solutions/sub-solutions so that the search is effective. They achieve this via using gemini as a lean proof generator aka. using a world model LLM fine tuned to generate lean proofs in a more effective manner.

Humans are even better at this as you mention - but effectively the approach is similar. Come up with lot of ideas and see what proves it.

To some extent, what they do is stronger that the other contestants, who I understand don’t formally prove their answers.

In maths, you are only solving a problem by building a formal solution that is the proof.

Eg. if it asked you to do something computationally hard (when done in a brute force way: calculators not allowed), and you put out the answer without the process using formal math knowledge, you'd get zero points. Even if it was easy to prove the answer was correct.

That's why IMO and many math exams won't take a single answer even if correct.

Partial credit is quite challenging to earn, per: https://www.imo-official.org/year_statistics.aspx?year=2024

Link isn't working for me can you summarize. What i heard ten years ago was that 1pt is quite common for a significant progress

Yeah, the progress has to be quite significant, no points are awarded for trivial observations. Thus scores are usually bimodal around 0 and 7. In the linked stats you can see that 1 point for P3/P5 was less common than full score on other problems.

Problem 2 and 5 have a lot of 1s. Sometimes thre is an interesting advance that is easirr than the full solution. Also in problem 6 the only hope for most was to get that 1 point.

I don't believe anything was graded by the IMO, Google is just giving itself 7 for anything proved in Lean (which is reasonable IMO), so they can't really try for partial credit so much as choose not to report a higher self-graded number.

From the article: "Our solutions were scored according to the IMO’s point-awarding rules by prominent mathematicians Prof Sir Timothy Gowers, an IMO gold medalist and Fields Medal winner, and Dr Joseph Myers, a two-time IMO gold medalist and Chair of the IMO 2024 Problem Selection Committee."

Fair enough. Although I think the question is whether P3/P5 were given zero points vs not evaluated vs evaluated but not published. I don't think it is surprising that Lean-verified proofs get a 7.

It is an interesting question: how this new system quantifies progress and whether it can detect that a chain of reasoning is close to the solution. Although it was just 2 points shy of a golden medal (1 point partial credit per problem not solved?), I'd suspect that the team would want a clean victory, given the magnitude of the claim. I bet next year's results would be 42/42. Deep Mind is unstoppable!