Here is what I came up with for what I ultimately learned from Polya’s book on plausible reasoning, combined with my own professional experiences over the past decade:
Real problem-solvers GUESS and then try SOMETHING. Non-problem-solvers repeatedly try to call in “experts” to solve the problem for them.
And now for some more good passages from Mathematics and Plausible Reasoning, Volume I, by George Polya:
“Many a guess has turned out to be wrong but nevertheless useful in leading to a better one.” “No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.”
-p.204 (from From How to Solve It, p.207)
Here we see the the “smallest testable case”, the key to debugging software (especially) and for fixing virtually all technology problems.
An extreme special case: Two men are seated at a table of usual rectangular shape. One places a penny on the table, then the other does the same, and so on, alternately. It is understood that each penny lies flat on the table and not on any penny previously placed. The player who puts the last coin on the table takes the money. Which player should win, provided that each plays the best possible game?
This is a time-honored but excellent puzzle. I once had the opportunity to watch a really distinguished mathematician when the puzzle was pro- posed to him. He started by saying, “Suppose that the table is so small that it is covered by one penny. Then, obviously, the first player must win.” That is, he started by picking out an extreme special case in which the solution is obvious.
Great place to start!
This problem is important (why?) but not too easy. If you cannot solve it in full generality, solve it in significant special cases; put pertinent questions that could bring you nearer to its general solution; try to restate it; try to approach it in one way or the other.
What do you do if you are stuck? Try solving a different problem. Make it more specific than the one at had. Give it more context. You may discover something about the GENERAL problem by solving some more SPECIFIC ones. If you are lost contemplating the forest, pick a tree or two to look at for a while.
Many problems may be easier than just one. We started out to solve a problem, that about the dissection of space by 5 planes. We have not yet solved it, but we set up many new problems. Each unfilled case of our table corresponds to an open question.
This procedure of heaping up new problems may seem foolish to the uninitiated. But some experience in solving problems may teach us that many problems together may be easier to solve than just one of them — if the many problems are well coordinated, and the one problem by itself is isolated. Our original problem appears now as one in an array of unsolved problems. But the point is that all these unsolved problems form an array : they are well disposed, grouped together, in close analogy with each other and with a few problems solved already. If we compare the present position of our question, well inserted into an array of analogous questions, with its original position, as it was still completely isolated, we are naturally inclined to believe that some progress has been made.
This is one of the keys to Polya’s approach. If you get stuck, you either broaden your context, or shrink it. Your problem could be too general – fill in a bunch of the blanks (not all the blanks) and give it much more context. Make it more specific and then try to solve that. Above, he says that sometimes you have to do the opposite. Making the problem more vague could expand your imagination, but another great technique is to line up a bunch of specific problems and then start figuring out what they all have in common. This will often reveal the root issue. Compare a bunch of specific case studies and patterns emerge that were not apparent when just focusing on one thing.
On dead ends (from one of the example problem walkthroughs):
The information “C is not a circle” is “purely negative.” Could you characterize C more “positively” in some manner that would give you a foothold for tackling ex. 9?
Coming to a conclusion that is “purely negative” is usually of no for moving forward. It keeps you stuck. If you were looking for dead-ends, then maybe that is acceptable. Most of the time though, things are more complex. I think this is KEY to getting along with other people. If the person you are working with is a moron or a mortal enemy, then you are stuck. Can you try describing them in at least a mixed positive way? Stretch your imagination! They are human just like you. You can find something. That will give you a foothold to relate to them and keep some of the relationship intact. A conflict mediator can do this. It may in fact be their chief ability. I believe that as Christians we are called to do this as it is essential for learning to love.
We cannot live and we cannot solve problems without a modicum of optimism.
A “modicum” is from the Latin for measure. It means a little bit. To have no optimism is suicide, both literal or on a smaller scale. Faith, Hope, and Love. They are as essential to our being as air, water, and blood.
Chance plays a role in discovery. Inductive discovery obviously depends on the observational material.
A clever person with poor data is unlikely to make any meaningful discoveries. It isn’t his fault. A novice can find something great if he hits upon the right combination of things. Serendipity will always be worth mentioning when talking about problem solving.
It is instructive to compare two lines of inquiry which look so much alike at the outset, but one of which is wonderfully fruitful and the other almost completely barren.
If one wants to understand problem solving, then need to also look at some counter examples and put their finger on what DIDN’T work.
Having solved a problem with real insight and interest, you acquire a precious possession: a pattern, a model, that you can imitate in solving similar problems. You develop this pattern if you try to follow it, if you score a success in following it, if you reflect upon the reasons of your success, upon the analogy of the problems solved, upon the relevant circumstances that make a problem accessible to this kind of solution, etc. Developing such a pattern, you may finally attain a real discovery. At any rate, you have a chance to acquire some well ordered and readily available knowledge.
The delight of learning – acquiring the “precious possession” described above!
First guess, then prove.
And finally, a completely unrelated anecdote, just as an excuse to include a picture with this post, which is already so full of long excerpts that everyone has stopped reading already:
I presented this derivation [from Archimedes] several times in my classes and once I received a compliment I am proud of. After my usual “Are there any questions?” at the end of the derivation, a boy asked: “Who paid Archimedes for this research?” I must confess that I was not prompt enough to answer: “In those days such research was sponsored only by Urania, the Muse of Science.”
Urania, more accurately, the muse of astronomy, usually depicted with a globe in her left hand. Oddly enough, Milton appeals to her in Paradise Lost.