Once you set that nine-dot problem into a larger frame, you could solve it in a routine way, with only a little thought. What lets you reformulate such complex scenes so easily — once you've thought of doing it? It must be that your mind is constantly preparing ways to do such things by building up connections between different kinds of descriptions. Then, when you finally change your view to find another way to look at things, you can apply a lifetime of experience as easily as turning on a switch.
This takes us back to the question of when to try to be a Reductionist or a Novelist. How do you decide when to quit after investing a great deal of effort in doing something a certain way? It would be bad to discard all that work just before you might find the answer — but there's no way to be sure when that will happen. Should people always try to break their well established, self-made mental bonds and try to think in less constricted ways? Of course not. It will usually do more harm than good.
However, when you're really stuck, you may as well try wilder ways to find some new ideas. You might even consider using one of the systematic, therapylike disciplines that go under names like brainstorming, lateral thinking, meditation, and so forth. These can help, when people get badly stuck, by encouraging the search for new formulations. However, when you switch to unfamiliar views of things you may get new ideas, but you also put yourself in the situation of a novice; you become less able to judge which new ideas are likely to be compatible with any of your older skills.
In any case, you must not be too quick to think, How stupid it was not to see that right away! Remember the principle of exceptions: it may be rash to change yourself too much just to accommodate a single strange experience. To take every exception seriously is to risk the loss of general rules that previous experience has shown to work most frequently. You must also be particularly wary of methods you can always use:
Quit what you're doing. Find an easier problem. Take a rest. You'll feel better afterward. Simply wait. Eventually the situation will change. Start over again. Things may work out better the next time.
Such methods are too general; they're things that one can always do, but they do not apply especially well to any particular problem. Sometimes they can help us get unstuck, but they must be barred from usual thought — or at least be given low priority. It isn't any accident that the things that we can always do are just the ones we should rarely do.