Letting the pieces fall into place
When was the last time you played with a puzzle - a jigsaw or one of those wire twists that ask to be untangled in specific ways? They require some analysis, of course. There are lots of logical approaches that make it possible to work at a puzzle until it is solved.
I don't think that explains the enduring appeal of puzzles. I think what we all like when we play with a puzzle is that moment of surprise and delight as something slips into place with unexpected ease. We don't get to that moment by working for it: we get to that moment while our hands are working and our minds have wandered for a moment. Suddenly we find that we are holding the piece that we need. Suddenly, we find that the knot has loosened.
When my sons were younger, they liked to buy the wire puzzles to entertain themselves on car trips. I remember playing with a new one absent-mindedly, sure that I could not solve it. Until the pieces slid apart in my hands - and I had to figure out how to restore the know before handing it back to my son!
Puzzles are solved most often by the mind at play.