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More idle time when in a primed state.

This, I think, is very key. You take the problem, mull over it for quite some time, think of different approaches to solving it, realize that none of them might work. And then you walk away. Take a nap, cook, work out, whatever.

The brain then does this background processing and connects the threads.

Time spent on trying different solutions (and especially, experience) is critical.

Quoting Mark Twain -

Good decisions come from experience. Experience comes from making bad decisions.

I wanted to chime in and say that this is exactly how I approach difficult problems. It has worked for me over and over again, and I have trained myself to go against my natural instincts and just drop something after a while, knowing that when I come back to it, I will likely have a fresh perspective. Also, love the Mark Twain quote.

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