I still remember the day in seventh grade when, alone in the classroom, I found the geometry problems section in the appendix of one of our books and started to work them out, tentatively at first, but increasingly confidently as they started falling into place. Joyfully too -- bitten by the bug and still infected after all these years. The problem notebooks become buried under files with more pressing business, always to be brought back to the top after a time to receive another solution.
The preoccupation can become an ongoing part of your psyche. Ramanujan used to say that a Hindu goddess came to him in dreams with his formulas and he would write them down first thing in the morning. Poincaré has a story (as I remember it) that he had hit a stone wall on a problem after considerable application. He decided he needed a break and took a trip to the country by bus. When disembarking, as his foot rose from the last step and before it fell on the ground, the solution came to him. I was waiting for a friend at a bar once and drew a diagram of a geometry problem I'd been working on unsuccessfully on a napkin, not even applying myself, just doodling absent-mindedly, when the solution revealed itself.
