I have a feeling this should belong in this section.
Quite a few times, it so happens that I can't figure something out. No matter how much time I spend reflecting on it, nothing comes. Then, I close my eyes and relax, and my brain does its own little thing: it throws all the information in a problem "out there", gathers a bunch of it together, and then reaches a conclusion without me even knowing the process. I remember this situation a few years ago in a math class in which I had to find something - it might have been a side length or something of that sort - and I did a slight miscalculation somewhere (which I couldn't spot). I went over to the teacher and began to explain what I did; I was focused on two things: (1), talking to the teacher, and (2), the details of the triangle. It was checking every detail of the triangle to find a solution, and then, BAM!, it figured out the answer, without me even having had to reason it out.
Or another situation that happened two weeks ago (these happen much more often, but these two examples are the ones I remember): I was in my German class, and I was asked to figure out if a noun in a certain sentence is accusative or dative. I didn't pause to think for a second and reflect on it: I singled out (not consciously) every single element of the sentence and then everything was put together to figure out what the case was. I didn't think about it in the fashion of "Well, so-and-so in this sentence is so-and-so, so thus it must be X", but I figured out the answer without even having to reason about it.
I'm curious as to what function - or combination of functions - does this. Does anybody have a clue?