I enjoy the way a problem's appearance changes as you get deeper into it. I'm pretty deliberate and systematic in tackling problems, but I usually find that my plan changes dramatically as my understanding of a problem changes.
I start working on a problem, I poke at it a little, and make my first effort. I notice a new wrinkle, and adjust my plan to accommodate that. In the end, I find that I have a pretty good grasp of the problem, which feels great.