At school there was one (and only one) maths problem I couldn't solve; I thought it was a distance problem when it was actually a time problem - I was looking at it the wrong way.
It must have laid buried in my subconscious for 7 years - I don't think I ever thought about it, or why I couldn't solve it. Then the answer popped into my head, catching me completely by surprise. It took a minute or so to recall what on earth the original question was!
Jenny Ireland (now Crebbin) and I were chasing a boat which had broken from its moorings; as well as wondering how far down the beach it would come ashore I wondered when. That insight, that a distance problem could equally be a time problem, proved the trigger.
To this day I rely more than most on the subconscious' ability to re-organise disparate snippets into some new and insightful order. I'll often go to sleep with the snippets whirling around, knowing there's a good chance the answer (or at least the key next question) will pop out in the morning.
Difficult decision? Tricky personal issue? Make a 'pretend' decision or draft the response late at night, then go to bed. In the morning the subconscious will tell me if it's right or wrong, and if wrong, where to look next.
I'm neither the first nor the last to do this. Einstein said "The intellect has little to do on the road to discovery. There comes a leap in consciousness, call it intuition or what you will, and the solution comes to you and you don't know how or why."