In grad school, I had what I'd call the classic version. I stayed up all night mentally working on a topology question about turning a 2-torus inside out. I already knew you can't flip a torus inside out in ordinary R^3 without self-intersection. So I kept moving and stretching the torus and the surrounding space in my head, trying to understand where the obstruction actually lived.

Sometime around sunrise, it clicked that if you allow the move to go through infinity(so effectively S^3), the inside/outside distinction I was relying on just collapses, and the obstruction I was visualizing dissolves. Birds were chirping, I hadn't slept, and nothing useful came out of it, but my internal model of space felt permanently upgraded. That's clearly "thinking hard" in the sense.

But there's another mode I've experienced that feels related but different. With a tough Code Golf problem, I might carry it around for a week. I'm not actively grinding on it the whole time, but the problem stays loaded in the background. Then suddenly, in the shower or on a walk, a compression trick or a different representation just clicks.

That doesn't feel "hard" moment to moment. It's more like keeping a problem resident in memory long enough for the right structure to surface.

One is concentrated and exhausting, the other is diffuse and slow-burning. They're different phenomenologically, but both feel like forms of deep engagement that are easy to crowd out.