First, I often encounter problems in my research that I understand completely. No magic or dark corners remain. Nonetheless, finding a solution remains challenging. When I do find the solution, I don't feel as if the problem has been further illuminated.

I'd really like to see an example. For me, understanding the solution is part of the problem. I read "understand completely" as "can characterize the solution to the problem as a mapping from inputs to outputs", and such a characterization should itself be a solution.