This one is crazy. I made up a silly topology problem which I guessed wouldn't be in a textbook (given X create a shape with Euler characteristic X) and set it to work. Its first effort was a program that randomly generated shapes, calculated X and hoped it was right. I went and figured out a solution and gave it a clue.

Watching it "think" through the answer is surreal and something I haven't felt since watching GPT-2 write punchlines for jokes.

Here's the lightbulb moment it "realizes" what the clue meant:

> For example, if you have a covering space of degree n, then the Euler characteristic of the covering space is n times the Euler characteristic of the base space.

> But I'm not sure how that relates here.

> Alternatively, perhaps it's about considering the Euler characteristic of a quotient space.

> But again, that might be too advanced for this problem.

> Maybe I'm overcomplicating things.

> Let me go back to the beginning.

> The problem says: "Consider the Euler characteristic of the tetrahemihexahedron and of a genus g torus. Then consider the additive property of the Euler characteristic. Then try again."

> So, perhaps it's simply about adding their Euler characteristics together.

I'm under no illusions about how LLMs are built and operate, but this really feels somehow more "real" and person-y than previous generations, even when you coax them into an answer.

I'm going to go and try having GPT-4o roleplay a student solving the problem and see if it's genuinely different. I've been getting impressive answers from o1, but just coldly getting the result is much more robot than human.