How to Solve It – and by It – I Mean *Anything*

I’m so smitten with learning that I fear I may be addicted to it. Imagine my delight when I learned about How To Solve It – a book about the very nature of learning how to solve problems. Written in 1945 by noted Hungarian mathematician George Pólya, How To Solve It is a book written for math students on how to solve math problems, but it was equally written for math teachers on how to teach them to do so.

How To Solve It is not just a book about math but is actually a book about how to solve problems – any problems – and it became the foundation for the field of heuristics. Based on Polya’s prodigious knowledge and historic research, the book has a curious structure, with several introductions to the book, an outline of his four-step framework for solving problems, and then a dictionary of heuristics: which is actually an explanation of 67 different concepts to aid in solving problems.

Diagram from How to Solve It

For a math book, its is a joy to read. Polya’s voice is funny, knowing, and sincere. You can tell he spent years in the classroom coaching students of all kinds, and while the book is absolutely about math, it is equally universal, as he said In the late 1960s: “I believe the most important part of thinking that is developed in mathematics is the right attitude in tackling and solving problems. We have problems in everyday life. We have problems in science. We have problems in politics. We have problems everywhere. The right attitude for problem solving may be slightly different from one domain to another, but we have only one head, and therefore it is natural that there should be one general set of tactics to tackling all kinds of problems. My personal opinion is that the main point in mathematics teaching is to develop these tactics of problem solving.”

For the majority of the book Polya teaches his process by solving geometry examples or finding patterns in number sequences but often switches it up using non-math examples applying the same heuristic for solving a crossword puzzle or understanding ancient Egyptian engineering. It’s here that How to Solve It really comes into its own, as Polya’s method results in a series of questions that can be applied to almost any problem.

After reading (and re-reading) How to Solve It, I think Polya’s heuristics can absolutely help our thinking in unraveling problems in data visualization. This (rather long) article is split into 4 parts. I present my analysis of the book before an explanation of the concepts because I didn’t want to get lost in too many details – and there are a lot of them. Part II focuses on the book and process itself, Part III on it’s legacy, and Part IV is a very long annotated list of the 64 heuristics terms, concepts, and key protagonists.

Before I get into the details, here’s the whole book – read it for yourself! https://notendur.hi.is/hei2/teaching/Polya_HowToSolveIt.pdf

Part I: Analysis of Polya’s heuristic methodology

Polya’s process versus design thinking

I’ve been reading How to Solve It as a supplement to guiding the design process. While math and data are connected, mathematics and design have hardly been cozy. It’s interesting to consider Polya’s process of solving mathematical problems in comparison to design thinking. 

How to Solve It was written and published in Zurich in 1940 before the first English language version was published in 1945. Design thinking evolved in the 1960s as an approach to exploring production issues in industrial design, but both arguably drew from the same well of the scientific method with similar outcomes.  

Polya’s method takes us inside the creative process by establishing a series of questions to understand the problem – which, for our needs, I think of as the communications objective. Polya puts more emphasis on the earliest phase of the discovery process, focusing on learning, research, and understanding the goal (and data) before the mechanics of doing the work and testing for accuracy.

The discovery phase in design thinking has evolved into a loosely collected series of design exercises by whole industries of practitioners to discover the right solution to the right problem. These exercises largely crowd-source the research and problem solving which inevitably results in a degree of groupthink as an outcome. The actual creative process may be supported through these kinds of design thinking exercises, but it’s just not the same thing. Hopefully the iterative process catches any group-think indulgences through prototyping and testing the result with users – or at least it should.

It’s certainly arguable that Polya’s heuristics may not be as comparable to the wide range of “wicked problems” that design thinking is usually used for, but certainly, the same kind of mindset is at play.  Where design thinking is largely used to understand complex problems, Polya is codifying the types of internal questions that manifest through the creative process itself. 

For example, the Double Diamond approach is a standard way of thinking about the 4 phases of design thinking. The diamond represents a series of diverging and converging activities to complete a project or task. After playing around with it for a while, I’ve come to think of Polya’s process as more of a staircase up through the Problem phases of the Double Diamond approach while the second and third steps – devising a plan & carrying out a plan – are 2 sides of the same coin, as the planning of the act and the follow-through result in the 4th step – Checking/Looking back. Stating it more colloquially, once you understood the problem, and figured out to solve it, it’s all downhill from there.

Diagram by the author showing the superimposition of the Double Diamond and Polya’s process.

What does it all mean? Well, I think Polya’s method could be more effective for small groups or individuals as it takes less input from others and establishes a framework to translate the data and design to known patterns and semiotics of visual culture. 

How might Polya’s heuristic process fit into a dataviz methodology? 

Data visualization is light on process and I loved reading this book in search of clues to how we might use these heuristics to make our work better, more intuitive, or more creative. Polya again makes an apt comment: “Mathematics is interesting in so far as it occupies our reasoning and inventive powers… To make such steps comprehensible by suitable remarks or by carefully chosen questions and suggestions takes a lot of time and effort; but it may be worthwhile.” 

Part II: Polya’s method

Ok, let’s dig into the actual book. Here’s an outline directly from the beginning of the book. 

Here’s a summarized text version (which I tweaked a smidge):

 1. Understand the Problem 

  1. What do you know?
  2. What don’t you know?
  3. Draw a diagram of the problem and notate the various parts.

2. Devise a Plan

  1. What associations does the problem remind you of? What kinds of similar problems can you think of?
  2. Can you decompose the problem or think of it differently?
  3. Can you add auxiliary information to make sense of the problem?
  4. If you can’t solve the problem, can you solve a related problem? (My favorite!)

3. Carry Out the Plan

(Literally, do the math.)

4. Look Back

  • Can you check the result? 
  • Does your answer make sense?
  • Can you use your solution for something else?

After this, Polya spends the rest of the book walking through the 67 heuristics. I’ll post an annotated list at the end (in part IV), but there are many wonderful concepts nested between the math-solving approaches.

In addition, Polya also explains the nature of heuristics:

The aim of heuristic[s] is to study the methods and rules of discovery and invention… 

as well as heuristic reasoning:

…not regarded as final and strict but as provisional, …whose purpose is to discover the solution of the present problem. 

Throughout the book, Polya’s method is used over and over again as a question-and-answer format to reveal the structured nature of the process. While this may feel obvious for some, the rigor of the approach is refreshing as it gives voice to the natural process many of us lean on to help us untangle the messy world around us. Understanding the power of analogy, research, composition and decomposition, validation, and even chance events is somehow reassuring as defined.

PART III: The legacy of How to Solve It

Teaching as a way of living

In the late 1960s, Polya was speaking on his process to a class of elementary school math teachers. His comments were delivered with the same precise and clever voice found in the book. He speaks a lot about how kids can be mentally engaged with math through practice and then passes along a list of ideas for teachers to do the same: 

Diagram from How to Solve It

Teaching is not a science; it is an art. If teaching were a science there would be a best way of teaching and everyone would have to teach like that. Since teaching is not a science, there is great latitude and much possibility for personal differences. In an old British manual there was the following sentence, “Whatever the subject, what the teacher really teaches is himself.” 

This really resonates with me, so I’ve adjusted it substituting the word ‘teaching’ for ‘dataviz:’

[Dataviz] is not a science; it is an art. If [dataviz] were a science there would be a best way of teaching and everyone would have to [practice] like that. Since [dataviz] is not a science, there is great latitude and much possibility for personal differences. In an old British manual there was the following sentence, “Whatever the subject, what the [practitioner] really [designs] is [themselves].” 

Polya also lists out some general advice, and again, I think this is directly relevant for us as dataviz practitioners: 

  • Be interested in your subject.
  • Know your subject.
  • Know about the ways of learning: The best way to learn anything is to discover it by yourself.
  • Try to read the faces of your students, try to see their expectations and difficulties, put yourself in their place.
  • Give them not only information, but ‘know-how,’ attitudes of mind, the habit of methodical work.
  • Look out for such features of the problem at hand as may be useful in solving the problems to come — try to disclose the general pattern that lies behind the present concrete situation.
  • Do not give away your whole secret at once – let the students guess before you tell it – let them find out by themselves as much as is feasible.
  • Suggest it, do not force it down their throats.

What we design, what we share with others, is rarely the subject matter of our design and insights, but rather a reflection of who we are as people. Polya’s humanism radiates with the care of a teacher’s fondness for the future. As Polya says, “Teaching to solve problems is education of the will,” and I wonder if that isn’t a life lesson for all of us.

How to Solve It: Modern Heuristics

The influence of Polya’s book has rippled across fields of study.  Heuristics is now a practice that can be seen across many fields – chiefly in cognitive psychology, philosophy, economics, and computer science/AI research. 

Zbigniew Michalewicz and David B. Fogel’s well-known book on modern heuristics for algorithm design even nabs the title from Polya. Their book, How to Solve It: Modern Heuristics begins, “Gyorgy Polya’s How to Solve It stands as one of the most important contributions to the problem-solving literature in the twentieth century. Even now, as we move into the new millennium, the book continues to be a favorite among teachers and students for its instructive heuristics…  Essentially, the book is an encyclopedia of problem-solving methods to be carried out by hand, but more than that, it is a treatise on how to think about framing and attacking problems.”

As I said, I’ve been reading How to Solve It as a supplement to guiding the design process for dataviz. It’s our process that helps us to accomplish how we help others understand the story of the data. Polya’s technique may be a further rubric to guide us on our creative journey.

Does Polya’s logic fit into a dataviz mindset? Will these ideas resonate with our community? Could an augmented version of How to Solve It be re-conceived for individuals following a design thinking approach? 

What do you think? I’ve outlined a lot of ideas in this article, but let’s talk about it! Add your comments on social media or email me! I’d love to discuss it more!

Part IV: Selected quotes from “A SHORT DICTIONARY OF HEURISTICS”

In case you want to learn more, I’ll include selected quotes from Polya across his 67 definitions of heuristics here. Certainly, I don’t expect anyone to read this cliff notes version as a stand-in for the original, but it’s probably much easier to scan through them to become acquainted with the concepts. Who knows – it might help you in your creative process or inspire your process!

The contents page of the book shows its structure but also provides a handy list of the heuristics
  • Analogy

Similar objects agree with each other in some respect, analogous objects agree in certain relations of their respective parts…

Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements. Analogy is used on very different levels. People often use vague, ambiguous, incomplete, or incompletely clarified analogies, but analogy may reach the level of mathematical precision. All sorts of analogy may play a role in the discovery of the solution and so we should not neglect any sort.

Inference by analogy appears to be the most common kind of conclusion, and it is possibly the most essential kind. It yields more or less plausible conjectures which may or may not be confirmed by experience and stricter reasoning. The chemist, experimenting on animals in order to foresee the influence of his drugs on humans, draws conclusions by analogy. But so did a small boy I knew. His pet dog had to be taken to the veterinary, and he inquired:

“Who is the veterinary?”

“The animal doctor.”

“Which animal is the animal doctor?”

  • Auxiliary elements

There are various reasons for introducing auxiliary elements. We are glad when we have succeeded in recollecting a problem related to ours and solved before. It is probable that we can use such a problem but we do not know yet how to use it. For instance, the problem which we are trying to solve is a geometric problem, and the related problem which we have solved before and have now succeeded in recollecting is a problem about triangles. Yet there is no triangle in our figure; in order to make any use of the problem recollected we must have a triangle; therefore, we have to introduce one, by adding suitable auxiliary lines to our figure. 

  • Auxiliary problem

… a problem which we consider, not for its own sake, but because we hope that its consideration may help us to solve another problem, our original problem. The original problem is the end we wish to attain, the auxiliary problem a means by which we try to attain our end.

Yet we should not fail to look around for more good problems when we have succeeded in solving one. Good problems and mushrooms of certain kinds have something in common; they grow in clusters. Having found one, you should look around; there is a good chance that there are some more quite near.

  • Bolzano, Bernard (1781-1848)

He writes about this part of his work: “I do not think at all that I am able to present here any procedure of investigation that was not perceived long ago by all men of talent; and I do not promise at all that you can find here anything quite new of this kind. But I shall take pains to state in clear words the rules and ways of investigation which are followed by all able men, who in most cases are not even conscious of following them. Although I am free from the illusion that I shall fully succeed even in doing this, I still hope that the little that is presented here may please some people and have some application afterwards.”

  • Bright idea

…or “good idea,” or “seeing the light,” is a colloquial expression describing a sudden advance toward the solution. The coming of a bright idea is an experience familiar to everybody but difficult to describe…

  • Can you check the result?

It is clear that our nonmathematical knowledge cannot be based entirely on formal proofs. The more solid part of our everyday knowledge is continually tested and strengthened by our everyday experience. Tests by observation are more systematically conducted in the natural sciences. Such tests take the form of careful experiments and measurements, and are combined with mathematical reasoning in the physical sciences. 

  • Can you derive the result differently?

When the solution that we have finally obtained is long and involved, we naturally suspect that there is some clearer and less roundabout solution: Can you derive the result differently? Can you see it at a glance?… Having found a proof, we wish to find another proof as we wish to touch an object after having seen it.

  • Can you use the result?

Exploit your success! Can you use the result, or the method, for some other problem?

  • Carrying out

To conceive a plan and to carry it through are two different things.

We may use provisional and merely plausible arguments when devising the final and rigorous argument as we use scaffolding to support a bridge during construction. When, however, the work is sufficiently advanced we take off the scaffolding, and the bridge should be able to stand by itself. In the same way, when the solution is sufficiently advanced, we brush aside all kinds of provisional and merely plausible arguments, and the result should be supported by rigorous argument alone.

In short, the new unknown should be a sort of stepping stone. A stone in the middle of the creek is nearer to me than the other bank which I wish to arrive at and, when the stone is reached, it helps me on toward the other bank.

Trying to prove formally what is seen intuitively and to see intuitively what is proved formally is an invigorating mental exercise. 

  • Condition

A condition is called redundant if it contains superfluous parts. It is called contradictory if its parts are mutually opposed and inconsistent so that there is no object satisfying the condition.

  • Contradictory

Thus, if a condition is expressed by more linear equations than there are unknowns, it is either redundant or contradictory; if the condition is expressed by fewer equations than there are unknowns, it is insufficient to determine the unknowns; if the condition is expressed by just as many equations as there are unknowns it is usually just sufficient to determine the unknowns but may be, in exceptional cases, contradictory or insufficient.

  • Corollary

a theorem which we find easily in examining another theorem just found. The word is of Latin origin; a more literal translation would be “gratuity” or “tip.”

  • Could you derive something useful from the data?

We have to find the connection between the data and the unknown. We may represent our unsolved problem as open space between the data and the unknown, as a gap across which we have to construct a bridge. We can start constructing our bridge from either side, from the unknown or from the data.

Diagram from How to Solve It
  • Could you restate the problem?

These questions aim at suitable VARIATION OF THE PROBLEM.

  • Decomposing and recombining

    You have an impression of the object as a whole but this impression, possibly, is not definite enough. A detail strikes you, and you focus your attention upon it. Then, you concentrate upon another detail; then, again, upon another. Various combinations of details may present themselves and after a while, you again consider the object as a whole but you see it now differently. You decompose the whole into its parts, and you recombine the parts into a more or less different whole.

    If you go into detail you may lose yourself in details. Too many or too minute particulars are a burden on the mind. They may prevent you from giving sufficient attention to the main point, or even from seeing the main point at all. Think of the man who cannot see the forest for the trees.

    Of course, we do not wish to waste our time with unnecessary detail and we should reserve our effort for the essential. The difficulty is that we cannot say beforehand which details will turn out ultimately as necessary and which will not.

    Therefore, let us, first of all, understand the problem as a whole. Having understood the problem, we shall be in a better position to judge which particular points may be the most essential. Having examined one or two essential points we shall be in a better position to judge which further details might deserve closer examination. 

    After having decomposed the problem, we try to recombine its elements in some new manner. Especially, we may try to recombine the elements of the problem into some new, more accessible problem which we could possibly use as an auxiliary problem.

    There are, of course, unlimited possibilities of recombination. Difficult problems demand hidden, exceptional, original combinations, and the ingenuity of the problem-solver shows itself in the originality of the combination. There are, however, certain usual and relatively simple sorts of combinations, sufficient for simpler problems, which we should know thoroughly and try first, even if we may be obliged eventually to resort to less obvious means.

    There is a formal classification in which the most usual and useful combinations are neatly placed. In constructing a new problem from the proposed problem, we may

    • Definition

    …a term is a statement of its meaning in other terms which are supposed to be well known

    • Descartes, René (1596-1650)

    …great mathematician and philosopher, planned to give a universal method to solve problems but he left unfinished his Rules for the Direction of the Mind. The fragments of this treatise, found in his manuscripts and printed after his death, contain more—and more interesting—materials concerning the solution of problems than his better known work Discours de la Méthode although the “Discours” was very likely written after the “Rules.” The following lines of Descartes seem to describe the origin of the “Rules”: “As a young man, when I heard about ingenious inventions, I tried to invent them by myself, even without reading the author. In doing so, I perceived, by degrees, that I was making use of certain rules.”

    • Determination, hope, success

    It would be a mistake to think that solving problems is a purely “intellectual affair”; determination and emotions play an important role. Lukewarm determination and sleepy consent to do a little something may be enough for a routine problem in the classroom. But, to solve a serious scientific problem, will power is needed that can outlast years of toil and bitter disappointments.

    • Diagnosis

    We are here particularly concerned with the student’s efficiency in solving problems. How can we characterize it? We may derive some profit from the distinction of the four phases of the solution. In fact, the behavior of the students in the various phases is quite characteristic.

    Incomplete understanding of the problem, owing to lack of concentration, is perhaps the most widespread deficiency in solving problems. With respect to devising a plan and obtaining a general idea of the solution two opposite faults are frequent. Some students rush into calculations and constructions without any plan or general idea; others wait clumsily for some idea to come and cannot do anything that would accelerate its coming. In carrying out the plan, the most frequent fault is carelessness, lack of patience in checking each step. Failure to check the result at all is very frequent; the student is glad to get an answer, throws down his pencil, and is not shocked by the most unlikely results.

    • Did you use all the data?

    Have we got what we need? Is our conception adequate? Did you use all the data? Did you use the whole condition? 

    • Do you know a related problem?

    In fact, when solving a problem, we always profit from previously solved problems, using their result, or their method, or the experience we acquired solving them. And, of course, the problems from which we profit must be in some way related to our present problem.

    There is usually no difficulty at all in recalling formerly solved problems which are more or less related to our present one. On the contrary, we may find too many such problems and there may be difficulty in choosing a useful one. 

    • Draw a figure

    The beginner should construct as many figures as they can in order to acquire a good experimental basis… Yet, for the purpose of reasoning, carefully drawn free-hand figures are usually good enough, and they are much more quickly done. Of course, the figure should not look absurd; lines supposed to be straight should not be wavy, and so-called circles should not look like potatoes.

    If there are many details, we cannot imagine all of them simultaneously, but they are all together on the paper. A detail pictured in our imagination may be forgotten; but the detail traced on paper remains, and, when we come back to it, it reminds us of our previous remarks, it saves us some of the trouble we have in recollecting our previous consideration.

    • Examine your guess

    Your guess may be right, but it is foolish to accept a vivid guess as a proven truth—as primitive people often do. Your guess may be wrong. But it is also foolish to disregard a vivid guess altogether—as pedantic people sometimes do. Guesses of a certain kind deserve to be examined and taken seriously: those which occur to us after we have attentively considered and really understood a problem in which we are genuinely interested. Such guesses usually contain at least a fragment of the truth although, of course, they very seldom show the whole truth. Yet there is a chance to extract the whole truth if we examine such a guess appropriately.

    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.

    Diagram from How to Solve It
    • Figures

    Figures are not only the object of geometric problems but also an important help for all sorts of problems in which there is nothing geometric at the outset. Thus, we have two good reasons to consider the role of figures in solving problems.

    • Generalization

    Passing from the consideration of one object to the consideration of a set containing that object; or passing from the consideration of a restricted set to that of a more comprehensive set containing the restricted one.

    The more general problem may be easier to solve. This sounds paradoxical but, after the foregoing example, it should not be paradoxical to us. The main achievement in solving the special problem was to invent the general problem. After the main achievement, only a minor part of the work remains. Thus, in our case, the solution of the general problem is only a minor part of the solution of the special problem.

    • Have you seen it before?

    It is possible that we have solved before the same problem that we have to do now, or that we have heard of it, or that we had a very similar problem. These are possibilities which we should not fail to explore. We try to remember what happened. Have you seen it before? Or have you seen the same problem in a slightly different form? Even if the answer is negative such questions may start the mobilization of useful knowledge.

    ….We cannot know, of course, in advance which parts of our knowledge may be relevant; but there are certain possibilities which we should not fail to explore. Thus, any feature of the present problem that played a role in the solution of some other problem may play again a role. Therefore, if any feature of the present problem strikes us as possibly important, we try to recognize it. What is it? Is it familiar to you?

    • Here is a problem related to yours and solved before

    This is good news; a problem for which the solution is known and which is connected with our present problem, is certainly welcome. It is still more welcome if the connection is close and the solution simple. There is a good chance that such a problem will be useful in solving our present one.

    The intention of using a certain formerly solved problem influences our conception of the present problem. Trying to link up the two problems, the new and the old, we introduce into the new problem elements corresponding to certain important elements of the old problem. 

    • Heuristic

    …or “ars inveniendi” was the name of a certain branch of study, not very clearly circumscribed, belonging to logic, or to philosophy, or to psychology, often outlined, seldom presented in detail, and as good as forgotten today. The aim of heuristic is to study the methods and rules of discovery and invention. A few traces of such study may be found in the commentators of Euclid; a passage of PAPPUS is particularly interesting in this respect. The most famous attempts to build up a system of heuristic are due to DESCARTES and to LEIBNITZ, both great mathematicians and philosophers. Bernard BOLZANO presented a notable detailed account of heuristic. The present booklet is an attempt to revive heuristic in a modern and modest form. 

    • Heuristic reasoning

    Reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building.

    Heuristic reasoning is good in itself. What is bad is to mix up heuristic reasoning with rigorous proof. What is worse is to sell heuristic reasoning for rigorous proof.

    .…A heuristic argument presented with taste and frankness may be useful; it may prepare for the rigorous argument of which it usually contains certain germs. But a heuristic argument is likely to be harmful if it is presented ambiguously with visible hesitation between shame and pretension.

    • If you cannot solve the proposed problem

    Do not let this failure afflict you too much but try to find consolation with some easier success, try to solve first some related problem; then you may find courage to attack your original problem again. Do not forget that human superiority consists in going around an obstacle that cannot be overcome directly, in devising some suitable auxiliary problem when the original one appears insoluble.

    • Induction and mathematical induction

    Induction is the process of discovering general laws by the observation and combination of particular instances. It is used in all sciences, even in mathematics. Mathematical induction is used in mathematics alone to prove theorems of a certain kind. It is rather unfortunate that the names are connected because there is very little logical connection between the two processes. There is, however, some practical connection; we often use both methods together. 

    • Inventor’s paradox

    The more ambitious plan may have more chances of success.

    This sounds paradoxical. Yet, when passing from one problem to another, we may often observe that the new, more ambitious problem is easier to handle than the original problem. More questions may be easier to answer than just one question. The more comprehensive theorem may be easier to prove, the more general problem may be easier to solve.

    The paradox disappears if we look closer at a few examples. The more ambitious plan may have more chances of success provided it is not based on mere pretension but on some vision of the things beyond those immediately present.

    • Is it possible to satisfy the condition?

    It is good to foresee any feature of the result for which we work. When we have some idea of what we can expect, we know better in which direction we should go. Now, an important feature of a problem is the number of solutions of which it admits. Most interesting among problems are those which admit of just one solution; we are inclined to consider problems with a uniquely determined solution as the only “reasonable” problems. Is our problem, in this sense, “reasonable”? If we can answer this question, even by a plausible guess, our interest in the problem increases and we can work better.

    • Leibnitz, Gottfried Wilhelm (1646-1716)

    Great mathematician and philosopher, planned to write an “Art of Invention” but he never carried through his plan. Numerous fragments dispersed in his works show, however, that he entertained interesting ideas about the subject whose importance he often emphasized. Thus, he wrote: “Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves.”

    • Lemma

    Lemma means “auxiliary theorem.” The word is of Greek origin; a more literal translation would be “what is assumed.”

    • Look at the unknown

    This is old advice; the corresponding Latin saying is: “respice finem.” That is, look at the end. Remember your aim. Do not forget your goal. Think of what you are desiring to obtain. Do not lose sight of what is required. Keep in mind what you are working for. Look at the unknown. Look at the conclusion. The last two versions of “respice finem” are specifically adapted to mathematical problems, to “problems to find” and to “problems to prove” respectively.

    Focusing our attention on our aim and concentrating our will on our purpose, we think of ways and means to attain it. What are the means to this end? How can you attain your aim? How can you obtain a result of this kind? What causes could produce such a result? Where have you seen such a result produced? What do people usually do to obtain such a result? And try to think of a familiar problem having the same or a similar unknown. And try to think of a familiar theorem having the same or a similar conclusion. Again, the last two versions are specifically adapted to “problems to find” and to “problems to prove” respectively.

    • Modern heuristic

    …endeavors to understand the process of solving problems, especially the mental operations typically useful in this process… Experience in solving problems and experience in watching other people solving problems must be the basis on which heuristic is built. In this study, we should not neglect any sort of problem, and should find out common features in the way of handling all sorts of problems; we should aim at general features, independent of the subject matter of the problem. The study of heuristic has “practical” aims; a better understanding of the mental operations typically useful in solving problems could exert some good influence on teaching, especially on the teaching of mathematics.

    A spread from How to Solve It
    • Notation

    Speaking and thinking are closely connected, the use of words assists the mind. Certain philosophers and philologists went a little further and asserted that the use of words is indispensable to the use of reason.

    Speaking and thinking are closely connected, the use of words assists the mind. Certain philosophers and philologists went a little further and asserted that the use of words is indispensable to the use of reason.

    Signs must be, first of all, unambiguous. It is inadmissible that the same symbol denote two different objects in the same inquiry. If, solving a problem, you call a certain magnitude a you should avoid calling anything else a which is connected with the same problem. Of course, you may use the letter a in a different meaning in a different problem.

    • Pappus

    …an important Greek mathematician, lived probably around A.D. 300. In the seventh book of his Collectiones, Pappus reports about a branch of study which he calls analyomenos. We can render this name in English as “Treasury of Analysis,” or as “Art of Solving Problems,” or even as “Heuristic”; the last term seems to be preferable here. A good English translation of Pappus’s report is easily accessible7; what follows is a free rendering of the original text:

    “The so-called Heuristic is, to put it shortly, a special body of doctrine for the use of those who, after having studied the ordinary Elements, are desirous of acquiring the ability to solve mathematical problems, and it is useful for this alone. It is the work of three men, Euclid, the author of the Elements, Apollonius of Perga, and Aristaeus the elder. It teaches the procedures of analysis and synthesis.

    “In analysis, we start from what is required, we take it for granted, and we draw consequences from it, and consequences from the consequences, till we reach a point that we can use as starting point in synthesis. For in analysis we assume what is required to be done as already done (what is sought as already found, what we have to prove as true). We inquire from what antecedent the desired result could be derived; then we inquire again what could be the antecedent of that antecedent, and so on, until passing from antecedent to antecedent, we come eventually upon something already known or admittedly true. This procedure we call analysis, or solution backwards, or regressive reasoning.

    “But in synthesis, reversing the process, we start from the point which we reached last of all in the analysis, from the thing already known or admittedly true. We derive from it what preceded it in the analysis, and go on making derivations until, retracing our steps, we finally succeed in arriving at what is required. This procedure we call synthesis, or constructive solution, or progressive reasoning.

    “Now analysis is of two kinds; the one is the analysis of the ‘problems to prove’ and aims at establishing true theorems; the other is the analysis of the ‘problems to find’ and aims at finding the unknown.

    “If we have a ‘problem to prove’ we are required to prove or disprove a clearly stated theorem A. We do not know yet whether A is true or false; but we derive from A another theorem B, from B another C, and so on, until we come upon a last theorem L about which we have definite knowledge. If L is true, A will be also true, provided that all our derivations are convertible. From L we prove the theorem K which preceded L in the analysis and, proceding in the same way, we retrace our steps; from C we prove B, from B we prove A, and so we attain our aim. If, however, L is false, we have proved A false.

    “If we have a ‘problem to find’ we are required to find a certain unknown x satisfying a clearly stated condition. We do not know yet whether a thing satisfying such a condition is possible or not; but assuming that there is an x satisfying the condition imposed we derive from it another unknown y which has to satisfy a related condition; then we link y to still another unknown, and so on, until we come upon a last unknown z which we can find by some known method. If there is actually a z satisfying the condition imposed upon it, there will be also an x satisfying the original condition, provided that all our derivations are convertible. We first find z; then, knowing z, we find the unknown that preceded z in the analysis; proceeding in the same way, we retrace our steps, and finally, knowing y, we obtain x, and so we attain our aim. If, however, there is nothing that would satisfy the condition imposed upon z, the problem concerning x has no solution.”

    • Pedantry and mastery

    To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. Some pedants are poor fools; they never did understand the rule which they apply so conscientiously and so indiscriminately. Some pedants are quite successful; they understood their rule, at least in the beginning (before they became pedants), and chose a good one that fits in many cases and fails only occasionally.

    To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery.

    The questions and suggestions of our list may be helpful both to problem-solvers and to teachers. But, first, they must be understood, their proper use must be learned, and learned by trial and error, by failure and success, by experience in applying them. Second, their use should never become pedantic. You should ask no question, make no suggestion, indiscriminately, following some rigid habit. Be prepared for various questions and suggestions and use your judgment. You are doing a hard and exciting problem; the step you are going to try next should be prompted by an attentive and open-minded consideration of the problem before you. You wish to help a student; what you say to your student should proceed from a sympathetic understanding of his difficulties.

    And if you are inclined to be a pedant and must rely upon some rule learn this one: Always use your own brains first.

    Teaching to solve problems is education of the will. Solving problems which are not too easy for him, the student learns to persevere through unsuccess, to appreciate small advances, to wait for the essential idea, to concentrate with all his might when it appears. If the student had no opportunity in school to familiarize himself with the varying emotions of the struggle for the solution his mathematical education failed in the most vital point.

    • Practical problems
    Diagram from How to Solve It

    Practical problems are different in various respects from purely mathematical problems, yet the principal motives and procedures of the solution are essentially the same. 

    There is a widespread opinion that practical problems need more experience than mathematical problems. This may be so. Yet, very likely, the difference lies in the nature of the knowledge needed and not in our attitude toward the problem. In solving a problem of one or the other kind, we have to rely on our experience with similar problems and we often ask the questions: Have you seen the same problem in a slightly different form? Do you know a related problem?

    In solving a mathematical problem, we start from very clear concepts which are fairly well ordered in our mind. In solving a practical problem, we are often obliged to start from rather hazy ideas; then, the clarification of the concepts may become an important part of the problem. Thus, medical science is in a better position to check infectious diseases today than it was in the times before Pasteur when the notion of infection itself was rather hazy. Have you taken into account all essential notions involved in the problem? This is a good question for all sorts of problems but its use varies widely with the nature of the intervening notions.

    In a perfectly stated mathematical problem all data and all clauses of the condition are essential and must be taken into account. In practical problems we have a multitude of data and conditions; we take into account as many as we can but we are obliged to neglect some. Take the case of the designer of a large dam. He considers the public interest and important economic interests but he is bound to disregard certain petty claims and grievances. The data of his problem are, strictly speaking, inexhaustible. For instance, he would like to know a little more about the geologic nature of the ground on which the foundations must be laid, but eventually he must stop collecting geologic data although a certain margin of uncertainty unavoidably remains.

    • Problems to find, problems to prove

    The aim of a “problem to find” is to find a certain object, the unknown of the problem.

    The unknown is also called “quaesitum,” or the thing sought, or the thing required. “Problems to find” may be theoretical or practical, abstract or concrete, serious problems or mere puzzles. We may seek all sorts of unknowns; we may try to find, to obtain, to acquire, to produce, or to construct all imaginable kinds of objects. In the problem of the mystery story the unknown is a murderer. In a chess problem the unknown is a move of the chessmen. In certain riddles the unknown is a word. In certain elementary problems of algebra the unknown is a number. In a problem of geometric construction the unknown is a figure.

    The principal parts of a “problem to find” are the unknown, the data, and the condition.

    The aim of a “problem to prove” is to show conclusively that a certain clearly stated assertion is true, or else to show that it is false. We have to answer the question: Is this assertion true or false? And we have to answer conclusively, either by proving the assertion true, or by proving it false.

    A witness affirms that the defendant stayed at home a certain night. The judge has to find out whether this assertion is true or not and, moreover, he has to give as good grounds as possible for his finding. Thus, the judge has a “problem to prove.” Another “problem to prove” is to “prove the theorem of Pythagoras.” We do not say: “Prove or disprove the theorem of Pythagoras.” It would be better in some respects to include in the statement of the problem the possibility of disproving, but we may neglect it, because we know that the chances for disproving the theorem of Pythagoras are rather slight.

    If a “problem to prove” is a mathematical problem of the usual kind, its principal parts are the hypothesis and the conclusion of the theorem which has to be proved or disproved.

    • Progress and achievement

    Have you made any progress? What was the essential achievement? We may address questions of this kind to ourselves when we are solving a problem or to a student whose work we supervise. Thus, we are used to judge, more or less confidently, progress and achievement in concrete cases. The step from such concrete cases to a general description is not easy at all. Yet we have to undertake this step if we wish to make our study of heuristic somewhat complete and we must try to clarify what constitutes, in general, progress and achievement in solving problems.

    …Another aspect of the progress of our work is that our mode of conception changes. Enriched with all the materials which we have recalled, adapted to it, and worked into it, our conception of the problem is much fuller at the end than it was at the outset. Desiring to proceed from our initial conception of the problem to a more adequate, better adapted conception, we try various standpoints and view the problem from different sides. 

    • Puzzles

    What the questions and suggestions of the list can do is to “keep the ball rolling.” When, discouraged by lack of success, we are inclined to drop the problem, they may suggest to us a new trial, a new aspect, a new variation of the problem, a new stimulus; they may keep us thinking.

    • Reductio ad absurdum and indirect proof

    Reductio ad absurdum shows the falsity of an assumption by deriving from it a manifest absurdity. “Reduction to an absurdity” is a mathematical procedure but it has some resemblance to irony which is the favorite procedure of the satirist. Irony adopts, to all appearance, a certain opinion and stresses it and overstresses it till it leads to a manifest absurdity.

    Indirect proof establishes the truth of an assertion by showing the falsity of the opposite assumption. Thus, indirect proof has some resemblance to a politician’s trick of establishing a candidate by demolishing the reputation of his opponent.

    Both “reductio ad absurdum” and indirect proof are effective tools of discovery which present themselves naturally to an intent mind. Nevertheless, they are disliked by a few philosophers and many beginners, which is understandable; satirical people and tricky politicians do not appeal to everybody. We shall first illustrate the effectiveness of both procedures by examples and discuss objections against them afterwards.

    It must be confessed that “reductio ad absurdum” as a means of exposition is not an unmixed blessing. Such a “reductio,” especially if it is long, may become very painful indeed for the reader or listener. All the derivations which we examine in succession are correct but all the situations which we have to face are impossible. Even the verbal expression may become tedious if it insists, as it should, on emphasizing that everything is based on an initial assumption; the words “hypothetically,” “supposedly,” “allegedly” must recur incessantly, or some other device must be applied continually. We wish to reject and forget the situation as impossible but we have to retain and examine it as the basis for the next step, and this inner discord may become unbearable in the long run.

    Yet it would be foolish to repudiate “reductio ad absurdum” as a tool of discovery. It may present itself naturally and bring a decision when all other means seem to be exhausted as the foregoing examples may show.

    • Redundant

    See CONDITION. {I can’t help but see this as a math joke – JF}

    • Routine problem

    Routine problems, even many routine problems, may be necessary in teaching mathematics but to make the students do no other kind is inexcusable. Teaching the mechanical performance of routine mathematical operations and nothing else is well under the level of the cookbook because kitchen recipes do leave something to the imagination and judgment of the cook but mathematical recipes do not.

    • Rules of discovery

    The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.

    It may be good to be reminded somewhat rudely that certain aspirations are hopeless. Infallible rules of discovery leading to the solution of all possible mathematical problems would be more desirable than the philosophers’ stone, vainly sought by the alchemists. Such rules would work magic; but there is no such thing as magic. To find unfailing rules applicable to all sorts of problems is an old philosophical dream; but this dream will never be more than a dream.

    A reasonable sort of heuristic cannot aim at unfailing rules; but it may endeavor to study procedures (mental operations, moves, steps) which are typically useful in solving problems. Such procedures are practiced by every sane person sufficiently interested in his problem. They are hinted by certain stereotyped questions and suggestions which intelligent people put to themselves and intelligent teachers to their students. A collection of such questions and suggestions, stated with sufficient generality and neatly ordered, may be less desirable than the philosophers’ stone but can be provided. The list we study provides such a collection.

    A spread from How to Solve It
    • Rules of style

    The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.

    • Rules of teaching

    The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach.

    First things come first. The author of this book does not think that all rules of conduct for teachers are completely useless; otherwise, he would not have dared to write a whole book about the conduct of teachers and students. Yet it should not be forgotten that a teacher of mathematics should know some mathematics, and that a teacher wishing to impart the right attitude of mind toward problems to his students should have acquired that attitude himself.

    • Separate the various parts of the condition

    Our first duty is to understand the problem. Having understood the problem as a whole, we go into detail. We consider its principal parts, the unknown, the data, the condition, each by itself. When we have these parts well in mind but no particularly helpful idea has yet occurred to us, we go into further detail. We consider the various data, each datum by itself.

    Having understood the condition as a whole, we separate its various parts, and we consider each part by itself.

    We see now the role of the suggestion that we have to discuss here. It tends to provoke a step that we have to take when we are trying to see the problem distinctly and have to go into finer and finer detail. It is a step in DECOMPOSING AND RECOMBINING.

    Separate the various parts of the condition. Can you write them down? We often have opportunity to ask this question when we are SETTING UP EQUATIONS.

    • Setting up equations

    To set up equations means to express in mathematical symbols a condition that is stated in words; it is translation from ordinary language into the language of mathematical formulas. The difficulties which we may have in setting up equations are difficulties of translation.

    In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.

    It is very much the same in setting up equations. In easy cases, the verbal statement splits almost automatically into successive parts, each of which can be immediately written down in mathematical symbols. In more difficult cases, the condition has parts which cannot be immediately translated into mathematical symbols. If this is so, we must pay less attention to the verbal statement, and concentrate more upon the meaning. Before we start writing formulas, we may have to rearrange the condition, and we should keep an eye on the resources of mathematical notation while doing so.

    • Signs

    As Columbus and his companions sailed westward across an unknown ocean they were cheered whenever they saw birds. They regarded a bird as a favorable sign, indicating the nearness of land. But in this they were repeatedly disappointed. They watched for other signs too. They thought that floating seaweed or low banks of cloud might indicate land, but they were again disappointed. One day, however, the signs multiplied. On Thursday, the 11th of October, 1492, “they saw sandpipers, and a green reed near the ship. Those of the caravel Pinta saw a cane and a pole, and they took up another small pole which appeared to have been worked by iron; also another bit of cane, a land-plant, and a small board. The crew of the caravel Niña also saw signs of land, and a small branch covered with berries. Everyone breathed afresh and rejoiced at these signs.” And in fact the next day they sighted land, the first island of a New World.

    Our undertaking may be important or unimportant, our problem of any kind—when we are working intensely, we watch eagerly for signs of progress as Columbus and his companions watched for signs of approaching land. We shall discuss a few examples in order to understand what can be reasonably regarded as a sign of approaching the solution.

    In a well-constructed chess problem there is no superfluous piece. Therefore, we have to take into account all chessmen on the board; we have to use all the data. The correct solution does certainly use all the pieces, even that apparently superfluous white knight. In this last respect, the new move that I contemplate agrees with the correct move that I am supposed to find. The new move looks like the correct move; it might be the correct move.

    Thanks to Mary Aviles for the edits and conversations on Design Methodology

    Jason Forrest is a data visualization designer and writer living in New York City. He is the director of the Data Visualization Lab for McKinsey and Company. In addition to being on the board of directors of the Data Visualization Society, he is also the editor-in-chief of Nightingale: The Journal of the Data Visualization Society. He writes about the intersection of culture and information design and is currently working on a book about pictorial statistics.