I've called this type of thinking "subconscious thought" for years (thought "ambient thought" has a nice ring to it). This is exactly why I always have at least

I've called this type of thinking "subconscious thought" for years (thought "ambient thought" has a nice ring to it). This is exactly why I always have at least two current projects to work on. When I get stuck on some problem in one project and cannot solve it in a reasonable amount of time (depends on the project, could be 5 minutes, could be a day), I will force myself to stop working on it and do a full context switch to another project. My subconscious mind grinds on the problem in the background without me having to exert any real effort thinking about it, until it finally finds a solution and raises an interrupt in my conscious mind. This process has become so effective, that if I can spot a problem coming, I say to myself, "I should figure out how to solve X", and don't think about it. A few days later when I come back to it, as if by magic, I have a solution already starting to form. This is the same phenomenon that causes you to wake up in the middle of the night with an answer to a question you were thinking about earlier in the day... usually, "who sings this blasted song?" The brain is an amazing, complicated, wonderful thing.