Just to throw in my two cents worth in to the Intuitively Obvious bucket,
 when I was a math student at Towson State University we were given a final
 exam that involved proving that two N dimesional matrices were related in 
 a given way. I started with the first matrix and used every theorem that I
 could remember trying to reach the second, but I got stuck halfway through.
 Working feverishly on a piece of scrap paper, I started on the second matrix,
 but couldn't work it back to the first. In a flash of inspiration, I set the
 two intermediate results equal to each other and copied the second set of 
 equations backwards onto the tail of the first. When I got the paper back,
 there was a C which was crossed out and replaced by an A, the midpoint of
 my equations was underlined, with a note saying - At first I doubted that 
 this step was intuitively obvious, but after thinking about it for several
 hours, I decided that it was.