I'm pretty sure I'm a Ti user, and I frequently think in motion or on paper. But in these cases I already know the essence of the big picture or the theory, and I'm trying to work out the details. I'm a very visual person. But I see everything in my head.
My mind is like in infinitely large filing cabinet. Details get filed away in folders and subfolders in drawers, and most of the time I won't remember all of them because they're hidden away- I just know about them based on where they are. It's a lot like how information is organized on a computer, and how I access the information is similar. I can compare the size of folders, look for patterns in the information, run statistical analysis, and find everything that contains a search word. Things always remind me of other things, like googling a word.
As far as the puzzle goes, I think I go about it from all angles. I can look at a bunch of random pieces and put some of them together, like an eye or a window, or I can put the whole thing together. But other times I look at the pieces and put it together in my mind, then have to figure out how the pieces fit together. Or I'll have parts of it put together and suddenly see the whole picture like an epiphany (this happens often). Or I'll see something that needs to be illustrated and create the puzzle pieces myself, like an art project from found objects. To me the whole world is a puzzle and I see its structure in my mind. I just have to find all the pieces and make the physical (illustration of the) structure of the puzzle more intricate.