On problem solving, with Polya

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.

Polya’s plausible reasoning

Heuristic: (adj.) Enabling a person to discover or learn something for themselves.

The next handful of blog posts will all be notes and observations from Mathematics and Plausible Reasoning, Volume I by the great math pedagogue George Polya. The book was lent to me by a good friend and mentor at work. Ironically, I spent two years working for a teaching lab named after Polya that (despite it’s other merits) didn’t actually follow Polya’s philosophy in nearly any regard.

This is also the first book I’ve ever read that is nearly 50% equations. Many of these I had to skim through as they involved higher math than I am not readily familiar with (differential calculus, analytical geometry, etc.). Some of the examples involving solid geometry and probability were more accessible. I didn’t care about the math though – I was interested in what Polya had to say about the nature of problem-solving in general. Fortunately, there was a lot to be gleaned from his examples. The bits of history sprinkled here and there were also fascinating. I was reminded of the Newton biography I read last year. My respect for Newton continues to soar.

I love a book that starts out by clarifying word definitions. This is what captivated me in N.T. Wright’s big books. It seems that in many of my favorite works, the introduction or preface is often the most helpful section of all. Polya begins with some great comments on the nature of explanations and conjecture:

Strictly speaking, all our knowledge outside mathematics and demonstrative logic (which is, in fact, a branch of mathematics) consists of conjectures. There are, of course, conjectures and conjectures. There are highly respectable and reliable conjectures as those expressed in certain general laws of physical science. There are other conjectures, neither reliable nor respectable, some of which may make you angry when you read them in a newspaper. And in between there are all sorts of conjectures, hunches, and guesses.

We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning.

The difference between the two kinds of reasoning is great and manifold. Demonstrative reasoning is safe, beyond controversy, and final. Plausible reasoning is hazardous, controversial, and provisional. Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs. Demonstrative reasoning has rigid standards, codified and clarified by logic (formal or demonstrative logic), which is the theory of demonstrative reasoning. The standards of plausible reasoning are fluid, and there is no theory of such reasoning that could be compared to demonstrative logic in clarity or would command comparable consensus.

-George Polya, Mathematics and Plausible Reasoning, Volume I, p. v

Biologists, climatologists, archeologists, economists, psychologists and especially sociologists have got to stop pretending that what they do is rigid “demonstrative” reasoning. They imagine themselves to be hard scientists like physicists (to some degree) or mathematicians. It’s not the same. That’s OK, but be honest about it. The standards are fluid (ever changing!) and even the best work is riddled with guesses, some supported substantially and others not so much. Biologists often laugh at theologians for not being real scientists, but in reality they are both on exactly the same playing field.

I do not believe that there is a foolproof method to learn guessing. At any rate, if there is such a method, I do not know it, and quite certainly I do not pretend to offer it on the following pages. The efficient use of plausible reasoning is a practical skill and it is learned, as any other practical skill, by imitation and practice. I shall try to do my best for the reader who is anxious to learn plausible reasoning, but what I can offer are only examples for imitation and opportunity for practice.

In what follows, I shall often discuss mathematical discoveries, great and small. I cannot tell the true story how the discovery did happen, because nobody really knows that. Yet I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it, in short, everything that deserves imitation. Of course, I shall try to impress the reader; this is my duty as teacher and author. Yet I shall be perfectly honest with the reader in the point that really matters: I shall try to impress him only with things which seem genuine and helpful to me.

p. vi

I have to come to really enjoy the way that authors humble themselves (or not!) in the first few pages of their books. The authors that do not are immediately suspect to me. Someone that can’t laugh at themselves or immediately recognize how large insignificant their work is are not, in my opinion, spiritually or psychologically healthy enough to be thinking clearly (or gracefully) about the topic at hand.

“I shall try to make up a likely story.” If only so many other books came with this much-needed disclaimer! I love it!

Most parts of this book have been presented in my lectures, some parts several times. In some parts and in some respects, I preserved the tone of oral presentation. I do not think that such a tone is advisable in printed presentation of mathematics in general, but in the present case it may be appropriate, or at least excusable.


I’m going to disagree a bit here. I think the oral tone IS acceptable in print. I have found it to frequently be a clear mode of writing. This is maybe because I am a slow reader that “reads aloud” to myself in my head. It may be inefficient, but I actually prefer it in many cases. Transcriptions of speeches can be messy and superfluous, but they can also be crystal clear and de-gunked in a way that unspoken prose cannot rise to.

The advanced reader who skips parts that appear to him too elementary may miss more than the less advanced reader who skips parts that appear to him too complex.


A good warning! (And encouraging to this reader, who definitely skipped some of the complex parts.)