In Mathematical Creation (1908), Henri Poincaré famously suggested that mathematical creation largely takes place in the subconscious, when you’re thinking about something else. He famously recounts how he put in hard conscious work trying to solve a problem and had little success until he left for a geological excursion:

Just at this time I left Caen, where I was then living, to go on a geological excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty.

Then there are all the heureka moment legends about physics breakthroughs: Archimedes takes a bath and realises what it takes for an object to float, Newton sees an apple falling from a tree and thinks of his law of universal gravitation, Einstein daydreams while walking to the patent office and gets one step closer to general relativity.

This largely agrees with my own experience. I sometimes come up with solutions to exercises while out for walks, runs or bicycle rides.

In short, a useful skill for scientists is knowing when to take a walk. Therefore it’s natural to ask: how do I know I’m stuck on the right thing? This is a non-trivial question.

Oftentimes, I’ll think I’m stuck, only to discover – often to my great frustration – that the difficulty isn’t inherent to the problem. But of course, this is a good thing: at least I can partially fix my own shortcomings. If I’m tired, I can get to bed earlier; if I have shaky prerequisites, I can spend some time doing background reading; if I haven’t gotten to know the problem, I can consider toy examples or vary hypotheses¹. There’s usually something I can do.

What I’ve described is problem solving at a meta-level. While meta-level strategies like sleeping sound silly, they usually won’t occur to someone grappling with a problem. I assume the best one can do is to make such strategies habitual, so one can exclude being falsely stuck.

Other more creative solutions might be possible too. For example, you could put stickies on listing human failure modes on one’s desk. Or you could create an idea graveyard, a record of dead ends. Presumably this makes it easier noticing when you’re going in circles.

If you get stuck on the right thing – if you can formulate the crux in one crisp sentence – then you’re done. Your conscious self is done: go for a walk.