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$10M AI Mathematical Olympiad Prize

aimoprize.com

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Re: $10M AI Mathematical Olympiad Prize

#61

Post by devit »

edf13 wrote: post #6
daoboy wrote: post #5

Anything you can do with a calculator is obviously trivial. Is there a clear point of departure when AI can no longer handle mathematical reasoning?

The IMO are a bit more involved... e.g. IMO 2023: > Problem 1. Determine all composite integers n > 1 that satisfy the following property: if d1, d2, . . . , dk are all the positive divisors of n with 1 = d1 Although ChatGPT 4 makes a pretty decent start on this already: > To determine all composite integers \( n > 1 \) that satisfy the given property, we need to closely examine the condition set forth: for a composi…

In my first attempt at the problem it made this mistake:

"The condition fails for i=1 since 1 does not divide p+q unless p+q is a multiple of n, which is not generally true" (1 divides everything)

In the second it gives up:

"However, this conclusion is based on heuristic reasoning and examples"

Third attempt it makes this mistake:

"If the immediate next divisor, di+1 , is not a multiple of p (for instance, it could be q or a product involving q), then p does not divide di+1. Hence, p will not divide the sum di+1+di+2 in such a case, violating the condition." (the fact that p does not divide d_i+1 does not imply that it does not divide d_i+1, d_i+2)

Then it gives up again: "However, this conclusion is based on heuristic reasoning and examples"

Then it makes this basic logic mistake: "However, since p and q are distinct primes, p does not divide q, and it's not guaranteed that p divides q+d_i+2 , especially if d_i+2 is not a multiple of p. Therefore, for such n, the condition fails." (the fact that "it's not guaranteed" doesn't imply that it's false)

Overall it proves that prime powers have the property, conjures that non-prime-powers don't, proves that p*q doesn't have it, but completely fails at coming up with a proof strategy that proves that non-prime-powers don't have the property.

Re: $10M AI Mathematical Olympiad Prize

#62

Post by dimask »

RandomLensman wrote: post #51
anonylizard wrote: post #49

Earlier quoted context omitted.

"Superhuman at solving" != "solved". AlphaGo didn't solve go (Ie, can the first mover guarantee a win?). However, it understood go at a far, far superior level to any human. A mathbot doens't have to solve math in general. It merely has to be better at solving math than any human mathematician to be considered ASI. And it only has to be better than the 'average' human mathematician to be extremely useful in accelerat…

What mathematics was AlphaGo solving? What do you mean by solving there?

Proving that the first player has a winning strategy, or the optimal strategy for both players leads to draw.

Re: $10M AI Mathematical Olympiad Prize

#63

Post by jasfi »

gregdoesit wrote: post #2

I was recently in Palo Alto, and bumped into a newly founded startup (I don't remember the name unfortunately) who set themselves the grand the vision of exactly this: winning a gold medal on the international Olympiad using AI. Their plan was to build mostly on LLMs as a start, and iterate as they go. In their barebones office space, they had a poster with a countdown of the number of weeks till the event: it was 36…

I'm not sure if such a system would qualify, the prize specifically mentions a model that could solve specific types of problems.

I'm building something myself that I hope will be able to work similarly: https://aiconstrux.com

Re: $10M AI Mathematical Olympiad Prize

#64

Post by criddell »

daoboy wrote: post #5

Anything you can do with a calculator is obviously trivial. Is there a clear point of departure when AI can no longer handle mathematical reasoning?

> Is there a clear point of departure when AI can no longer handle mathematical reasoning?

I'm more interested in the point where AIs are presenting proofs far beyond human capability. I'm imagining a day when an AI says it has solved some interesting problem and when we ask for the proof it spits out a 4 million page document. What are we supposed to do with that? What's the role of humans in that world?

Re: $10M AI Mathematical Olympiad Prize

#65

Post by spi »

Davidzheng wrote: post #60
spi wrote: post #48

Earlier quoted context omitted.

A decent start? It says absolutely nothing about how to solve it, except repeating the question. The part where it tries out the few first numbers is entirely wrong, given that 6 does _not_ satisfy the condition (2 does not divide 3+6=9) and 8 _does_ (2 does divide 4+8=12). Amusingly, the list of the integers Given that GPT4 is not a model but a full product, behind the scenes it probably coded up and ran a small pyt…

https://chat.openai.com/share/ad42d09a-b366-4b21-b701-782fc8... . Will you re-evaluate how far GPT4 is from it from the fact that it can guess it's powers of primes?

Good to know it can do that, in the pasted chat above it didn't. To be honest, it surprised me it couldn't, this isn't exactly a very hard guess given the computation results. It doesn't convince me GPT4 is anywhere close to winning the IMO, though :-)

Re: $10M AI Mathematical Olympiad Prize

#66

Post by haltist »

We will solve math with math and this project will succeed as foretold by prophets of techno-optimism like Marc Andreesen, Elon Musk, and Bill Gates. All we need is the right architecture, aka mathematical formula for giving all of math a smooth manifold structure. How this works for discrete structures like integers is left as an exercise for the reader and future AI which will figure out how to improve themselves and deal with discrete and non-smooth mathematical problems.

I can win this challenge by the way for $80B. I already know what architecture is required to solve math with math but I need the money to buy the GPUs. You might think such a recursive application of math is logically circular but it is not and all I need is $80B to prove it (pun intended).

Claude did not like this comment at all, ironically: I do not have enough context to fully evaluate those claims or determine if that approach would work. Solving all of mathematics is an extraordinarily ambitious goal that would require fundamental theoretical advances we do not yet possess. While future AI systems may someday make significant progress on longstanding mathematical problems, making definitive claims about solutions requires rigorous mathematical proof and analysis beyond optimistic speculation. I'd encourage focusing discussion on specific mathematical questions or areas of research rather than making broad, unsupported assertions about solving all of mathematics.

Re: $10M AI Mathematical Olympiad Prize

#67

Post by WhitneyLand »

Winning a gold medal here will really be a leap forward for AI creativity and deep insight into problem solving.

AI already is already contributing in substantial ways to research.

What’s tantalizing here is this next level would take a huge step toward accelerating the pace of advancements in many fields.

It will be a milestone in moving past AI being a mere tool in scientific progress to something much greater.

Re: $10M AI Mathematical Olympiad Prize

#69

Post by falcor84 »

dimask wrote: post #62
RandomLensman wrote: post #51

Earlier quoted context omitted.

What mathematics was AlphaGo solving? What do you mean by solving there?

Proving that the first player has a winning strategy, or the optimal strategy for both players leads to draw.

Just to nitpick, while unlikely, it is theoretically possible that the second player has a winning strategy

Re: $10M AI Mathematical Olympiad Prize

#70

Post by pixl97 »

slibhb wrote: post #23
andyjohnson0 wrote: post #15

As the parent of a young adult currently half way through their maths undergrad, this kind of fills me with foreboding. I know that proof assistants etc have existed for quite a while now, but what with this and the murmours about OAI's Q* model, I do wonder what will happen to maths as a human endeavour - and as a enabling skill for jobs that can financially support people like my child.

I wouldn't worry about it. We're going to need humans with specialized mathematical training in the loop. Besides, these things have a way of surprising us. Before compilers, people wrote machine code by hand. It would have been reasonable to think that compilers would reduce the demand for programmers, but the opposite happened.

>There isn’t a rule of economics that says better technology makes more, better jobs for horses. It sounds shockingly dumb to even say that out loud, but swap horses for humans and suddenly people think it sounds about right.
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