This was the prompt I gave o1-preview:

There is a geometric thought experiment that is often used to demonstrate the counterintuitive shape of high-dimensional phenomena. We start with a 4×4 square. There are four blue circles, with a radius of one, packed into the box. One in each corner. At the center of the box is a red circle. The red circle is as large as it can be, without overlapping the blue circles. When extending the construct to 3D, many things happen. All the circles are now spheres, the red sphere is larger while the blue spheres aren’t, and there are eight spheres while there were only four circles.

There are more than one way to extend the construct into higher dimensions, so to make it more rigorous, we will define it like so: An n-dimensional version of the construct consists of an n-cube with a side length of 4. On the midpoint between each vertex and the center of the n-cube, there is an n-ball with a radius of one. In the center of the n-cube there is the largest n-ball that does not intersect any other n-ball.

At what dimension would the red ball extend outside the box?

Response: "[...] Conclusion: The red ball extends outside the cube when n≥10n≥10."

It calculated it with a step-by-step explanation. This is the first time I'm actually pretty stunned. It analysed the problem, created an outline. Pretty crazy.