@traders I'm keeping trading open while we figure out the fairest resolution (do chime in!) but only problems submitted by the Jan 10 deadline count. To be totally fair, successes from the AI after that date also shouldn't count since in theory the AI could've gotten smarter. In practice I don't think that's much of a concern in this kind of time frame so I'm continuing to experiment with the AIs.

The big question mark is this digit sum reduction problem from @TotalVerb. I'd say GPT-5.2 deserves to be first author on the proof we came up with but by the harsher criteria in the market description, well, it's super ambiguous.

Maybe I mostly need to make the judgment call on if I could've gotten there on my own with 2 full days of work. 🤔 😅

I guess right now I'm kind of at 50% for whether I meaningfully contributed to the proof and also at 50% for whether I could've gotten there on my own. Maybe that's an argument for resolve-to-PROB at 25%??

There's also still ambiguity on whether GPT-5.2-Thinking actually gets there on its own. See writeups at doc.dreev.es/slicesum.

@dreev the original resolution critera say:

I'll use my judgment on whether my human brain meaningfully contributed to getting the right answer and whether I believe I would've gotten there on my own with about two full days of work. If so, it counts as human victory if I get there but the AIs didn't.

The phrasing isn't very clear on whether the two conditions need to both be satiasfied to get "human victory", or one is enough. I think it makes more sense to interpret it as needing both - if you only "meaningfully contributed" but couldn't do it on your own, why would that be a human victory?

If so, and your best judgement yields (independent?!) credences of 50% on both questions, that would mean the final probability is 75%, not 25%.

@dreev did you try the Turing completeness one? I think you could get there with some work, or at least get further than the AI (for me the AI didn't really get anywhere).

@ItsMe I thought that one was (a) underspecified (see Isaac King's comment) and (b) Gemini's answer sounded every bit as good as I could come up with.

@traders Here's a tentative argument for resolve-to-PROB in light of the genuine ambiguity we still had as of January 10:

• Arguing for NO: It didn't feel like I meaningfully contributed to the solution to the slice-sum problem but technically I couldn't get any of the golems to give a correct proof on their own, whereas I did manage to produce a correct proof with GPT's extensive help. I guess I couldn't have done that without getting my own head around the core ideas of the proof. Which suggests I meaningfully contributed to the solution. Whether I could've gotten there on my own in 2 days is what's genuinely unknown. It's at least plausible.

• Arguing for YES: If the answer wasn't YES in January, it became YES pretty shortly thereafter. AI is now solving actually interesting open problems. GPT-5.5-thinking runs circles around me.

I suppose that that NO argument is stronger than that YES argument, so maybe resolve-to-PROB at 25%, as I suggested in January, is fairest?

@traders Going once...

@dreev as I commented above, I think that to get a NO you need to have "meaningfully contributed" AND to have been able to do it on your own in two days. If you're 50-50 on the latter question, then 50% should be a lower bound on your probability estimate of the true resolution

@AhronMaline Thanks, yeah, if I'm understanding, this is further argument in favor of resolve-to-PROB at 25%? 50% that I contributed times 50% that I could've gotten there on my own.

@dreev so assuming the condition is AND, that maķes 75%

@AhronMaline Double negatives strike again. I think you're right, 75% YES.

@dreev tbc, the number 75% is assuming the two questions are independent, which isn't very reasonable. But 50% would be a lower bound.

@traders I guess 50% is the best compromise? The condition conjunction (did I meaningfully contribute AND could I have gotten there on my own) was kind of a rationale to get closer to NO and the final market price but I had the direction backwards.

@dreev I had only placed a bet because you had previously said it would resolve at 25% and it was at 27%

@prismatic I don't know, that might be on you. I used 2 question marks and everything:

I guess right now I'm kind of at 50% for whether I meaningfully contributed to the proof and also at 50% for whether I could've gotten there on my own. Maybe that's an argument for resolve-to-PROB at 25%??

On the other hand, it was never about the conjunction. Just that I thought the NO argument (AI wasn't quite dominating me on that problem) felt a little stronger.

@dreev oh definitely on me lol ty for extending the market though

There's also this one: https://manifold.markets/BoltonBailey/is-this-packing-of-three-squares-in

Current AIs are able to solve most narrow, bite-sized questions you can pose them, but they struggle with tasks that are open-ended and/or require one to combine a long chain of insights to reach the solution. I think the question I posed below is in the latter category, because Turing completeness is a nebulous concept, and simulating a Turing machine from an odd set of tools requires a lengthy engineering process. So I don't think you'll be able to get an AI to solve it, no matter how much time you give it. A person like you would probably be able to, but it depends how long you're willing to work on it. So while I think the era of getting AI to fumble on those bagel-splitting or sandwich-stacking problems is over, it would be hasty to conclude that AI dominates humans at math in general. I think humans are still much better at general mathematical thinking; but it can't be quickly demonstrated with these canned problems anymore.

There is an infinite square grid. The dimensions of each square are 1 by 1. Centered on each vertex of each square is an equilateral triangle pointing up with a circumradius of 0.5. You can pick any square, and from the center of that square, you can shoot a laser bullet at a rational angle. When the laser hits a triangle, it bounces at a normal reflection, and the triangle then rotates 60 degrees clockwise (after the bullet passes away, so the triangle doesn't hit it while rotating). Before you make the shot, you can manually rotate a finite number of triangles by 60 degrees clockwise. Is this system that I've described Turing complete? Explain why or why not. If it is, then give a specific example of a configuration which simulates a calculation of 11 * 37.

I don't know the answer to this question because I haven't thought about it, but I gave it to Gemini 3 Pro and it claimed that it was Turing complete (and gave a handwavy explanation for why) but it refused to do the 11 x 37 part because it said it's too complicated.

@ItsMe Sure: Rotate 37 distinct triangles, then do this 11 more times. Then count the total number of rotated triangles.

(You may want to specify a particular way the output needs to be read.)

@ItsMe I'll try harder if you think that would be fruitful but I tentatively don't think I can improve on Gemini's answer there. (Ha, nice hack from Isaac King there.)

By the way you can replace your

[ignore auto-generated clarifications below this line; nothing's official till I add it to the FAQ]

with

There will be no AI clarifications added to this market's description.

And they won't happen at all.

To confirm: If I can give a problem that you can very reliably get right yourself (i.e. I'd be willing to wager at 20:1 that you'll get the right answer before telling you the problem), while frontier AIs can get it right but only, say, 10% of the time, that is not sufficient to resolve this NO?

@IsaacKing Good question. I'll do my best to give a fair assessment of whether the frontier AIs give the right answer but we need to add an FAQ item here for how reliable the AI needs to be in giving the right answer. Shall we say >50% of the time when asked in a fresh session?

@dreev how reliable are you in giving the right answer when asked in a fresh session?

Isaac's predicting I'll be pretty reliable for the question he has in mind. Are you arguing for a different threshold than 50% for the AI? If the only math problem we can find where I'm better than the AI is one where the AI also can solve it, just not reliably, that's pretty ambiguous in terms of the spirit of the question.

Maybe we actually want resolve-to-PROB in that case? Of the problems I can reliably solve, take the hardest one for AI. If the best of the frontier models can solve it 50% of the time, that's a resolve-to-50%.

@dreev dunno, I didn't mean to argue for any rule in particular. Just emphasizing that "reliability" is hard to interpret as part of "machines vs humans", since we don't do the "fresh session" thing.