It is rare for artists, magicians, and other professionals to let people see how they work. Mathematicians, too, rarely expose the thought processes that led them to a discovery – not necessarily because they want to keep this a closely guarded secret, but often just because they just don’t know what the process was (or find it very difficult to describe). Poincaré famously wrote about the mysterious mechanisms of mathematical discovery, emphasizing how letting the subconscious take over helps magical insights to reveal themselves. This is also manifest in how people tend to present proofs: in almost all cases, the argument is presented in the most elegant and slick way, which usually implies that the motivations, partial ideas, false turns and mistakes are well hidden from view.
Tim Gowers is one of very, very few world-class mathematicians who are both capable and willing to share the way they think about problems (the other one I know is Terry Tao). In a recent series of posts (starting here), Gowers explains his attempts to attack the problem of finding lower bounds for circuit complexity. They are fascinating both as a great way to understand the various negative results that were proved in this direction (Razborov-Rudich-style, on why certain natural approaches are doomed to fail), and as an opportunity to see how Gowers approaches the problem.
of premises to a set 
of conclusions. Sometimes the road can be long – meaning the problem can be quite hard – even though there is another point 
in “idea space” such that the following peculiar circumstances are true: getting from 
