There's research on intuition and the unconscious mind to back this up. Apparently, for a lot of problems the best way to solve it is by thinking very hard about it and then letting it go. Your unconscious mind will go on, especially when you're doing activities such as showering (which has little distractions), and all of a sudden you will have the solution.

I use this technique all the time, if I can't think of a solution for a hard problem I'll just leave it and go for a run, take a shower or work an another problem.

Unfortunately, I only have read a book from the Dutch psychologist Ap Dijksterhuis, which is unfortunately in Dutch, but I'm sure there are people here who will have some references in English.