the little polya

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The Little Polya

A Small Compilation of George Polya’s HeuristicTechniques

Compiled and Rewritten by A. L. Bruce



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Contents

Preface…………………………………………………………………………...4

Introduction………….…………………………………………………………6

The Heuristic Method of Polya’s How to Solve It……………………… …7

I. Understanding the Problem……………………………………………….10

1. Condition………………………………………………………………….11

2. Can You Derive Something Useful From the Data?.......................11

3. Definitions………………………………………………………………..12

4. Did You Use All the Data?...............................................................13

5. Is it Possible to Satisfy the Condition?............................................13

6. Separating the Condition and Setting up Equations……….……..147. Symmetry…………………………………………………………………16

8. Notation…………………………………………………………………..16

II. Devising a Plan……………………………………………………………..18

1. Analogy…………………………………………………………………...19

2. Auxiliary Problems and Auxiliary Elements……………………….19

3. Decomposing and Recombining……………………………………….21

4. Figures……………………………………………………………………22

5. Look at the Unknown…………………………………………………..23

6. Working Backwards…………………………………………………….24



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III. Carrying Out the Plan…………………………………………………...26

1. Carrying Out…………………………………………………………….27

2. Generalization…………………………………………………………..27

3. Specialization……………………………………………………………28

4. Induction and Mathematical Induction………………….………….28

5. Test by Dimension……………………………………………………...29

IV. Looking Back………………………………………………………………31

1. Can You Check the Result?.............................................................32

2. Can You Derive the Result differently?..........................................32

3. Can You Use the Result?.................................................................33

V. Other Techniques…………………………………….…………………….35

1. Practical Problems……………………………………………………..36

2. Problems to Find and Problems to Prove…………………………..37

3. Progress and Achievement…………………………………………...37



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PrefacePrefacePrefacePreface

In the process of composing this companion book to Polya’s How to Solve It , I have become aware of many nuances, both in Polya’s heuristic itself, and inthe use of a book such as this. First and foremost I want to tell the reader that thisis by no means a self-contained and self-explanatory work. I have subtitled it “asmall compilation of George Polya’s heuristic techniques,” yes, but it is not acomplete compilation. In fact, I was highly selective in the material I chose toinclude. Polya’s “short dictionary of heuristic,” from which all of these techniquesare taken, is some 72 articles long; certainly to merely rewrite the entire dictionarywould be pointless –why not just read Polya’s original work? Instead I have onlyincluded the material that I think are rigorous and algorithmic techniques, or (in

most of the cases) heuristic knowledge that is absolutely essential. Much more onthis is said in the introduction, to which I now refer the reader who is curious aboutmy selection of the articles.

On that topic, as evidenced by my statement that this is not a self containedwork, is the fact that one cannot possible hope to find this of any use unless it isused in conjunction with How to Solve It . This I cannot stress enough. Most of myreaders would probably not be drawn to this work if they had no knowledge of Polya’s heuristic, but to those who do not, I recommend reading the whole of Howto Solve It as prerequisite to this. For those readers who have knowledge of Polya’sheuristic but either do not have a copy of How to Solve It or haven’t read the wholeof it, consider the purchase of a copy and reading the whole of it a good decision.In my opinion, anyone who considers themself learned in the mathematical artshould have at least a good, and desirably a masterful knowledge of Polya.

At this point I would like to discuss what the purpose of this book is. Inshort, it is nothing other than a reference for those who are practitioners of Polya’sideas. I have found that there is a true need to be able to reference a giventechnique at some point in one’s studies, or the even greater need for some easy tofind and useful advice on how one might go about solving a problem which provesto be exceedingly difficult. How to Solve It itself is in my opinion not quite aseffective for this on-the-spot type of scenario.

The sections of his book are divided into many parts, all of which areextremely valuable to the student first encountering Polya’s ideas, but whichbecome unwieldy in the face of a quick way to find and use them “in the heat of battle” so to speak. The closest any section comes to being sufficiently helpful inthis respect is the afore mentioned “short dictionary of heuristic,” but even that isburdened down with sections which are nothing more that general thoughts on



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heuristic and the psychology of problem solving. Once again, these are brilliantand truly invaluable to the student first studying heuristic, but they make the whole

much more difficult to navigate on-the-go. The relevant entries are many timesmixed with analogies, metaphors and involved examples, meant to aid in thenovice’s understanding, but which make the solid information about the techniquehard to find. Let there be no misunderstanding, I quite frequently have usedexamples in this book, many of which are taken from Polya himself (as I will citemost of the time), but the ones I have included are meant to show simply what itwould be extremely difficult to explain in the abstract. The main point of this book was to boil the dictionary down, and in doing so to establish a reference for theentire book, since the dictionary constituted the main part of his thought, as well as195 of the 253 pages. I have also included a short piece in on the general heuristicprocess developed in the first and second parts of the book to make sure all themain aspects were covered.

I think that the work itself is pretty well suited to the job of on-the-spotreference where How to Solve It may prove to too cumbersome, but it is in no waya substitute. If one is interested in any particular technique one should refer to thearticle’s counterpart in How to Solve It as every entry here does have one (thenames are usually the same in both). Lastly there may be some articles in How toSolve It which may be of use in here which are not included. I understand that nomatter how impartial I try to be in their section, discretion is subjective. I plan toput out further editions of this work which will likely contain additions to these.This is by no means a work which is incapable of changing with time.

Adam L. Bruce,

Eureka, August 2009



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IntroductionIntroductionIntroductionIntroduction

Most esteemed friends and colleagues:

If you are reading this little book, it’s probably because I have given it toyou in the hope that you may benefit by it greatly. It contains what I believe to bethe core of George Polya’s heuristic techniques as they are applied to variousmathematical and scientific problems, which are arranged in logical sections forthe reader. I hope it will serve as a good reference.

I have divided it into five main sections each representing a step in Polya’s

heuristic process. For example, you’ll find that the techniques under thesupersection of “Devising a Plan” have to do with creating a strategy to solving theproblem and the techniques under “Looking Back” have to do with learning fromthe problem. The fifth section is devoted to the techniques that aren’t specific toany one step in the process, but which are still valuable on the whole. Many of those in the fifth section may not be algorithmic techniques, but general ideasabout various heuristic topics

All of these are my reworking of selected articles from the “Short Dictionaryof Heuristic” found in How to Solve It . Their selection I have based on the

following criteria:

1. Is it a self contained technique?2. Is it applicable to an interdisciplinary set of problems?3. Is it helpful as a specific tool in heuristic thinking?

Just as the author of a book on the Calculus would include a section on logarithmicdifferentiation but may or may not include a biography of Gauss, I have chosen toinclude only those essential techniques which have a specific heuristic function andin so doing feel that I have made it a lot more useful.

In bringing all of these elements together I hope to have created a smallreference of heuristics for anyone who like myself can consider themself a“disciple” of Polya –at least in his approach to problem solving. I think this will bevery handy for that purpose and will fill the need for a readily available place thatone can find various heuristic techniques. I will not lie that I have mostly done thisfor myself, but I feel that others who are also familiar with the Polyan heuristicprocess will undoubtedly find this useful.



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The Heuristic Method of How to Solve ItThe Heuristic Method of How to Solve ItThe Heuristic Method of How to Solve ItThe Heuristic Method of How to Solve It

I have decided to include a short piece on the basic heuristic process setforth by Polya. It consists of four steps, each one designed to carry the practitionerto the next, and ultimately through the whole of the problem.

1. Understanding the Problem:

It’s almost useless to attempt to solve a problem if you don’t fist understandwhat it is asking you to accomplish, and therefore understanding the problem is thefirst step of the process. One asks themselves such questions as “what is/are the

unknown(s)?”, “what are the data?”, and “what is/are the condition(s)?”. Thesequestions help to create a thorough understanding of the problem. Other techniquessuch as drawing a figure to act as a visual representation can also aid in theunderstanding of the problem; also, the introduction of suitable notation for thenext step is crucial, and will be discussed at length in 1.8. Once this has been donewe proceed to the next step.

2. Devising a Plan:

The second step is to devise a plan that will find the unknown. One knows they

have a plan when they either know completely, or know a rough outline, of thesteps they must take in order to obtain the solution. In a problem to prove , which isPolya’s name for a problem involving a mathematical proof, this may not be anexact unknown variable, but rather the main ideas of the proof.

The step of devising the plan is, in my opinion, the most overlooked by novicestudents. I find that they tend to form a general understanding of what the problemis asking them to accomplish, and from there do no more thinking on it but torigidly apply a given technique and proceed to the third step, which is carrying outthe plan (or in their case technique). This type of approach might work for some

elementary problems, but is utterly useless in any complex one. Instead when oneconfronts a complex problem, the solution to which is eluding them, they shoulddo things such as looking for a problem “related to [their]’s and solved before” inPolya’s words, or look at the problem from a number of different perspectivestaking note of the subtleties exposed by each. This will not solve all problems (forindeed not all problems can be solved), but it will certainly bring much moreclarity and method to the way one goes about attempting to solve them.



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ultimately supplies one with a problem “related to yours and solved before” whenone does encounter another similar problem.

The essence of Polya’s method is understanding; understanding of the problem,understanding of methods employed in solving the problem, understanding of howto apply those general methods to the problem at hand, and finally understandingwhy such a method was employed in the first place. All of these general steps havespecific heuristic techniques which are associated with them and which aid in theiruse. It is the most useful of these which I now present to you in the forthcomingpart of this little book.



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1.1 Condition:

“The principle part of a problem to find ” (my italics). In sum, the informationgiven by the problem constitutes the condition insofar as that information modifiesthe process taken in solving it (if not it is data ). By this logic it should seem thatany problem is merely a set of conditions, all of which are satisfied through a givenmethod, but there is a distinction between the main unknown of the problem and itssubsequent conditions. Thus the problem “find two numbers whose sum is 60 andwhose quotient is 4” is made up of an unknown and two conditions.

1. Find two number numbers (unknown)2. Their sum is 60 (condition1)

3.

Their quotient is 4 (condition 2)This is solved easily with a simple translation into algebraic notion (see 1.7),

60=x+y , x/y=4 ; so x=48 , y=12.

A condition is redundant when there is more information than required. So toadd the third condition of “their product is 80” to the equation above would createa linear system comprised of more equations than variables; therefore it would beredundant and may not have a solution. A condition is insufficient where there isless information than required. To delete one of the conditions would render theproblem above too vague, and therefore incapable of a solution. A condition iscontradictory when two or more elements are mutually opposed. To add a thirdcondition of “their sum is 43” to the problem above would be to render itcontradictory, because two numbers cannot have a sum of 60 and 43simultaneously.

1.2 Can You Derive Something Useful from the Data?

In any problem there are two places to start. The first and most prevalent is

from the unknown, but another place is from the data. Polya uses the analogy thatthese two are a separated by a gap for which a bridge must be built –where one canstart from either side. The main question of this technique is “Can you derivesomething useful from the data?”

A. Beginning the Inquiry

The inquiry begins with the usual set of questions:

1. What are the data?2. What is the unknown?



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3. What is/are the condition(s)?

The usual procedure would be to start from the unknown, i. e. set up somethinglike an x = y , where x is the unknown and y is the data and condition. At that pointone usually can obtain a solution, if it is still elusive, than one should consider thatdata more.

B. Considering the Data

In considering the data one thinks of the problem in terms of y=x or at least yimplies x. It is a deductive process, where x must be equivalent to one’s solutionbecause of the data, that is, x supports the data and not vice-versa. Use the data inisolation when possible, so that the relationship between various parts of it may beobserved. The example of the line through three points is how Polya showsworking from the data in How to solve it , see the section in the heuristic dictionarywhich bears the same title as this section for the problem itself.

1.3 Definitions:

“The definition of a term is a statement of its meaning in other terms whichare supposed to be well known”. Thus says Polya.

A. General Information

There are two types of definitions, or technical terms .

Primitive terms , or terms which are not defined, are the first. These are those suchas “point” and “line” in Euclidian geometry, and “number” and “variable” in manyalgebraic systems. These terms cannot be defined because there is nothing todefine them with.

Derived terms are terms which can be defined using primitive terms. Therefore, a

circle can be defined as “the locus of points equidistant from a single point inCartesian two-space”.

B. Going Back to Definitions

The main application of primitive terms vs. derived terms is in restating aproblem using primitive terms for derived terms, which Polya calls going back todefinitions . The point of this is to make the problem easier to understand.

One starts with a problem which uses many unfamiliar derived terms, and slowlyworks it so that the derived terms are restated in primitive terms, this may not be



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accomplished at once, but over the interval where other heuristic techniques arecoming into play as well. Drawing a figure, for example, may help one to restate

the derived terms of the problem in the primitive terms of the figure. The heuristicquestion is “Can you restate the problem?” The restatement is the deflated problem, where one has “deflated” the unfamiliar derived terms.

1.4 Did You Use All the Data?

Essential to the understanding of a problem as a whole is the understandingof its data. When one is finding themself unable to solve a problem, it may bebecause one has not taken all the data into account. One must ask themselves “didyou use all the data?”, or “did you use all of the conditions/the whole hypothesis?”,with the divide in the latter for a problem to solve and problem to proverespectively. It is a check and nothing more. Any strategy for solving the problemthat does not take into account all of the data is likely flawed, excepting thosecircumstances where some part of the data is extraneous.

One might have a flawed or incomplete notion of the problem if one doesnot take all the data into account, or does not consider in the right way some part of it. Therefore one must also ask “have you taken into account all of the essentialnotions involved in the problem?” This cannot be directly accomplished if onedoes not know what the essential notions of the problem are, of course, but rathercan be accomplished through considering the different ways all of the datafunction, both individually and as a whole.

1.5 Is it Possible to Satisfy the Condition?

Since it is pointless to work toward an end which one cannot achieve, onemust appraise the condition of the problem as to whether or not it can be satisfied.

Thus one asks themself “Is the condition sufficient to determine the unknown, or isit insufficient, redundant, or contradictory?” In any reasonable problem, thecondition can be satisfied. This is most useful at the beginning of considering theproblem, and because of that should only be a plausible guess which gives aprovisional answer.



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1.6 Separating the Condition and Setting up Equations:

“To set up equations means to express in mathematical symbols a conditionthat is stated in words; it is translation from ordinary language into the language of mathematical formulas.” He later compares it to a translation between twolanguages. The problem can either be a simple translation or a complex one. Theprocess itself is ultimately an algorithmic one, which precious few of Polya’stechniques are. Primarily this technique is used for word problems.

A. Separating the Condition

In easy cases, the word problem will split into successive parts which canthen be translated into an equation. In difficult cases, there is often some nuancethat cannot be directly translated into mathematical symbols.

First one may have to rearrange or separate the condition; to do this onemust completely understand the condition, which may require one to deflate theproblem (see 1.3). One then separates the various parts of the condition, thequestion being “Can you write [the independent parts of the condition] down?”

B. Setting Up Equations

At this point one makes the translation into the mathematical notation, if it is still

not clear as to the form the translation will take, it is likely that one has not madethe appropriate divisions. One can use a vertical line down the page to separate thestatement in words from its equivalent mathematical representation such as:

|

(Statement in words) | (Statement in Mathematics)

|

|

At this point one can proceed to solve the problem.

C. Example

This is an example taken from Polya (see pages 175-176 of How to Solve It ).

Find the breadth and height of a right prism with a square base, given it has avolume of 63 in 3 , and a surface area of 102in 2.



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What are the unknowns? The breadth and the height.

What are the Data? Same as the condition.

What is the Condition? The volume of the prism is 63 cubic inches and the surfacearea is 102 square inches.

First: Draw a figure (see 2.4).

Second: Reduce the terms to primitives; one can deflate the unfamiliar termsbreadth and height (see 1.3).

The breadth of a right prism is defined as a side of the base, one can call it x. Theheight is defined as the altitude, say y. Therefore we restate the problem:

Given a parallelogram with a square base that has a volume of 63 in 3 , and asurface area of 102in 2. Find the length of the side of the base and the altitude of the prism

Third: The problem now needs better organization, particularly of the condition.Therefore, separate the various parts of the condition .

There are two parts: first the volume, then the surface area, both of which oneneeds knowledge of Euclidian geometry to fully understand, but no more. Thus

one has successfully separated the condition so to facilitate a greater understandingof the problem at hand.

Fourth: Make the transition from words to mathematical symbols:

Find the length of the side of the base | x

Find the altitude | y

The volume is 63 in 3 | x 2 y = 63

The surface area is 102 in 2 | 2x 2 + 4xy = 102

From this one can easily solve the linear system and obtain a solution to theproblem. Notice that the equations are in terms of the main variables rather thantheir Euclidian definitions; it would be helpful in any circumstance, and evennecessary in others to define say x2 as equaling the base of the parallelogram etc.These can also be included in the “translation” with the dividing line in the middle.



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1.7 Symmetry:

Symmetry, in a general sense, is the idea that certain parts of a problem areinterchangeable with others, thus given the sum xy +xz +yz one can interchangeany two variables without changing the meaning of the expression. Any symmetryone encounters should be noted, and one should be careful not to destroy anynatural symmetry without cause, since it can help one to understand the problem.

1.8 Notation:

Notation is one of the most important aspects of problem solving. Essential

to the process of understanding the problem is to “introduce suitable notation”,using this one can better formulate a way of dealing with the problem. Most of all anotation cannot be redundant or ambiguous –one can easily hamper themselves intheir ability to work the problem if they are using a poor notation, and it makes itnear impossible to go back and check one’s work. This is all summed up by Polyawhen he says:

“A good notation should be unambiguous, pregnant, easy to remember; itshould avoid harmful second meanings and take advantage of useful secondmeanings; the order and connection of signs should suggest the order and

connection of things.”A. General Principles

The general principles of devising a notation follow Polya’s quote.

First , none of the elements of the notation can be ambiguous:

One symbol cannot denote two or more objects.

Two or more symbols can denote one object. Multiplication, for instance, can be

written × , · ,or simply . In some cases this is advantageous, but itshould never be done without cause.

Second , the notation should be easy to remember. It should remind one of the quantity it denotes when that is appropriate.

A simple devise is to use the initial of the quantity, V for volume, a foracceleration, etc. This doesn’t work when one works with two quantities whichhave the same initial, rate and radius for example, but for this there are othermethods.



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Third , the order and connection of the signs should suggest the order andconnection of the objects. These are shown many times through the alphabet used.

Letters near the beginning of the alphabet, a , b, c, usually denote constants or othergiven quantities, while letters near the end, x, y, z, usually denote variables. Thus if in a problem one is given length, width, and height, it may be more useful to writethem as a , b, c, rather than l, w, h, to show that they are given constants and notvariables.

Objects belonging to the same class are usually written within the same alphabet,thus in Euclidian geometry: A, B, C, are all points; a, b, c, are all lines; , , ,areall angles.

These are the foremost and most necessary principles of formulating anotation. There are others, but they mostly consist of nuances and so will not beincluded here (see How to Solve It 138-141).



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Devising a Plan



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2.1 Analogy:

Analogy is a very important aspect of problem solving; it allows one to makeinferences about the problem at hand through problems which are related to it.Much of the time, a good analogy makes use of an auxiliary problem (2.2) andvice-versa. One, when desiring to make use of a problem analogous to the presentone, should ask themself if there is a “simpler problem related to the present one”.

To state the obvious, simpler problems are easier to solve, and therefore onecan possibly derive some useful insight or information from them, withoutburdening oneself with yet another difficult problem and “losing sight of the goal”.Therefore, given the choice between a simple problem which is somewhat

analogous to the present one, and a another problem, just as difficult as the first,which is much more related, one should choose the simpler one because onecannot solve the difficult one; If the solution to the difficult one was obtainablethan why was the solution to the present problem evasive? There are instanceswhere this is not true (2.2 (D)) but for the most part this is a good general rule.Polya devotes pages to the subject, including numerous examples and anecdotes,but this is the heart of his argument (see How to Solve It 137-146).

2.2 Auxiliary Problems and Auxiliary Elements:

Both auxiliary problems and auxiliary elements are used as an analogy (2.1)to the present problem, usually in the format of a problem related to the presentone and solved before, but in some cases merely as a simpler problem which onecan solve easier.

1. Auxiliary Elements

An auxiliary element is defined by Polya as “an element we introduce in thehope that it will further the solution”. There are various types of auxiliary

elements, in Euclidian geometry elements such as auxiliary lines and auxiliary polygons , in algebra auxiliary unknowns and auxiliary theorems , etc.

There are also various reasons for introducing auxiliary elements, such asusing them to make the present problem similar to another problem, otherwiserelated and solved before. Thus, given a related problem whose solution has to dowith triangles, yet having no triangles in one’s figure, one can introduce a triangleas an auxiliary element to take full advantage of the related problem. And whengoing back to definitions (1.3), if one finds that the primitive term contains some



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auxiliary element, one should not hesitate to introduce it to the problem. As well asmany more reasons.

2. Auxiliary Problems

One takes up an auxiliary problem to illuminate the solution to the presentproblem.

A. Profiting from the Problem

There are two ways to profit from the problem.

When one profits from the result they use a solution obtained in the auxiliary tomake clear the solution to the present problem. Such as, given x4+x 2+32=0 , if onelets y=x 2, the solution becomes clear, and once one solves y2+y+32=0 one alsoobtains x by .

When one profits from the method however, one takes a notion involved in theauxiliary problem and applied it to the present problem. Thus one observes thecondition (1.1). Polya’s example of this is the problem: find length of the diagonalof a rectangular parallelepiped being given the lengths of the three edges drawn

from the same corner . The appropriate auxiliary problem is that of finding thediagonal of a parallelogram, which introduces the same notion that the

Pythagorean Theorem is to be used.B. Equivalence and Bilateral Reduction

Many auxiliary problems are equivalent , defined by Polya by saying “thesolution of each requires the solution of the other”. In essence, two equivalentproblems involve the same or extremely similar mathematical notions in theirsolutions. Thus the two problems

1. x2+x=0 2. 3x2+3x=0

are equivalent. They are not identical, but the same notions are involved witheach. Thus if one can realize the answer to (1) is zero, one knows the answer to (2)must be zero, and furthermore the answer to any problem of the form ax 2+ax=0must be zero. The process of proceeding from (1) to (2) is termed bilateralreduction .



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C. Chains of Equivalent Problems

Using the idea of bilateral reduction, it is possible to set up a large chain of equivalent problems which stretches to a problem which is either already solved, orone to which to solution is easily obtained. Since each problem is equivalent to theone before it, each and every problem is equivalent to the first. Thus, all algebraicmanipulation can even be termed bilateral reduction.

D. Unilateral Reduction

Given two problems, both unsolved, where the solution of the first wouldsolidify the solution of the second, but not vice-versa, thus we should solve thefirst problem first then the second. If the first problem is “more ambitious” than thesecond, there are however two ways to proceed. When one proceeds from a givenproblem to a “more ambitious” one or a “less ambitious” one, the process is termedunilateral reduction .

Of the two ways, the first is to deal with the “less ambitious” then the “moreambitious.” Even though in the scenario above this would be less advantageous,there are many circumstances where this is helpful (the second example in (A) forinstance). This is by far the easiest and most common.

The second is to proceed from the “less ambitious” to the “more ambitious.”

This was almost forbid in the analogy section (2.1), but in some special instancesthis can be helpful. The ability to solve a difficult problem before a simple one iscalled by Polya the inventor’s paradox (see How to Solve it 121-122).

2.3 Decomposing and Recombining:

Decomposing is the process by which one examines the problem by lookingat each of its elements individually. After this is accomplished, reconstructingthem, called recombining, can give one a better idea of the problem. One primarilylooks at the three principle parts of the problem, namely what is the unknown? what are the data? And what is the condition? Once each of these has beenexamined individually, they can be further decomposed by going back to thedefinitions (1.3), etc. The main point of decomposing a problem however, is toconstruct an auxiliary problem which will be able to aid in obtaining a solution tothe first. There are three primary ways of doing this: 1. Keeping the unknown andchanging the data and condition, 2. Keeping the data and changing the unknownand the condition, and 3. Changing both the unknown and the data.



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A. Keeping the Unknown and Changing the Rest:

The problem has the same unknown as the original, but other things areslightly changed. One considers “what data is appropriate to determine theunknown.” One could also keep part of the data and part of the condition, changingas little as possible, but dropping some part of either and consider the problemthen.

B. Keeping the Data and Changing the Rest:

The data is retained and a new condition and unknown are created. Theunknown should be useful and accessible, acting as a median point between thedata and the original unknown. Since it is hard however, to conceive of anunknown which is both useful and accessible, one can introduce a new unknownwhich is related to the original, but that may not be as accessible, in the hope that itwill yield to a solution more easily than the original and vice-versa.

C. Changing Both the Data and Unknown:

This type is a more radical change than those which preceded it. The newproblem however, might have a good chance of success, and thus one considers if they can “…change the data or unknown, or both if necessary, so that the new dataand unknown are closer to each other.” This is done through considering the

principles for changing the unknown and data found in (A) and (B).

2.4 Figures:

The drawing of figures is essential to one’s ability to solve the problem.Many, if not all, geometric problems have a figure which is associated with theproblem, but in other problems, it is very useful to introduce a figure which thenacts as a visual aid in one’s heuristic process.

A. Drawing Figures ExactlyExact figures are not absolutely necessary, but one should draw them as

exactly as possible. A good freehand sketch of a figure should be enough for mostproblems. A badly drawn figure will suggest a false conclusion, and therefore hurtmore than it will help. It is important to consider the problem; does the figure fitthe problem?



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B. Order of Construction

The various elements of a figure must be constructed in the correct relationsand measurements, or at least close to the correct ones. The order of theirconstruction however, is up to the problem itself. Therefore, given two angles a and b to be constructed so that a=3b , it is not possible to construct first a and thenb, one must rather construct b first and obtain a from the first. The figure in eitherwill illustrate the same concept. In many other problems, the order of constructionwill not have an effect on the figure, in which case it is optional.

C. Erroneous Conclusions

The construction of a figure should not introduce any symmetry or relationthat is not shown in the problem. Lines and angles which are not equal should notappear so; otherwise this may lead to false conclusions.

Note: the best triangle to construct for a general triangle is one with the angles 45,60, 75. This is the most remote from both an isosceles and a right triangle

D. Shading

Shading is a very important part of drawing a figure. One can shade a line orarea which has a special significance to the problem. Creating darker lines bring

out a certain area, while dotted lines hint that there is some relation between twoother things in the figure.

2.5 Look At the Unknown:

It is advantageous in any problem to consider the final goal of the problem.Without this in mind one can easily be sidetracked, thus the Latin saying “respicefinem,” or Polya’s more understandable “look at the unknown” are used todescribe the consideration of one’s goal.

A. The Unknown as an Auxiliary Problem

It is very helpful to consider an auxiliary problem (2.2) which has the sameunknown as the present one. Thus if the unknown to the present problem is thelength of a line, the unknown to the auxiliary problem should also be the length of a line. By doing this one might very well be able to profit from both the methodand the solution to the auxiliary problem.



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First one looks at a schematic of the problem, where all other parts of itexcept the unknown are omitted. The example Polya gives is:

“Given……….Find the length of the line”

This focuses our attention to the nature of the unknown, which in this case is a line

Second one can consider all of the other problems which have unknownsrelated to theirs; there is “an economy of choice”, as Polya puts it, where oneconsiders the simplest and most familiar first. In the example one sees that thelength of the line could easily be obtained if it were a side of a triangle. Thus wemust introduce the auxiliary element (2.2) of a triangle into our figure, etc…

B. General Unknowns in Auxiliary ProblemsFor any problem the process in (A) can be of use. One must consider the

typical types of problems which involve a certain unknown, thus for the unknowns:

1. “Given……….Find the angle”2. “Given……….Find the area of the cube”3. “Given……….Construct the point”

We can think of the problems for

1. To be concerned with some triangle2. To be concerned with some side or given distance3. To be concerned with some locus of points

Thus the heuristic question is “given a problem, can you think of an auxiliaryproblem having the same or a similar unknown?”

2.6 Working Backwards:

When one is confronted with a problem which is extremely perplexing, oneshould consider it from a variety of perspectives. An important one of these iscalled “working backwards” by Polya, where one considers the solution and fromthe solution derives the problem. This may seem strange, but is a good and usefulskill nonetheless, thus the saying of Pappus, “let us assume what is sought asalready found.” Many times the actual work is through a series of auxiliaryproblems (2.2) each derived from first the solution and then each other. This isillustrated best in the following example.

A. Example



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Polya’s example for this technique is so poignant and well conceived that Iwould be a fool to not include it at present. Thus it is this example I set forth.

How can you bring up from a river exactly six quarts of water having onlytwo containers to hold it, a four quart pail and a nine quart pail?

What are the unknowns? Six quarts of water, or more correctly the method for obtaining six quarts of water.

What are the Data? Same as the conditions

What are the conditions? You only have a four quart pail and a nine quartpail.

First: Draw a figure(s). I shall not include any in my explanation as Icurrently have no practical way to do this at my disposal. Refer to Polya ( How tosolve It pg 226-229) for the figures or draw your own (it’s not that difficult).

Second: The problem is much more difficult than it appears at first, andthere seems no way to obtain the solution from the problem, therefore we work backwards .

Working backwards one has 6 qts in the 9 qt container (the 4 qt containercannot hold enough water).

How could one have come by this?

If one had 1qt in the 4 qt container and had filled the 9 qt, then one couldhave poured 3 qts on top of the 1 qt in the 4 qt container (until it was full) and thenpoured all of the contents of the 4 qt into the river, leaving them with 6 qts.

How could one get the 1 qt in the 4 qt container?

If the 4 qt was empty and one had 1 qt in the 9 qt container one could havetransferred the 1 qt from the 9 qt container into the 4 qt container.

How could one have 1 qt in the 9 qt container?

Then the solution is realized. If one had filled the 9 qt container, and thenpoured its contents into the 4 qt container, emptied the 4 qt container into the river,and repeated that process one could obtain the one qt. The solution consists mostlyof simple arithmetic, 9 – 4(2) = 1 . Thus one has obtained the correct solution,which might never have occurred to them if they hadn’t thought of “workingbackwards.”



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Carrying Out the Plan



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3.1 Carrying Out:

The step of carrying out the plan is different from merely devising it. Onecan notice nuances to the problem and sees things about it that one was not awareof when the plan was first devised. Devising the plan, one makes use of manyplausible guesses and intuition. Once the plan is to be carried out however, thesemust be replaced by a more rigorous set of standards.

One should pay special attention to the order which one carries out the stepsin their plan. The major aspects to the argument should be checked before onestarts to go into its details. There should be no detail which is omitted, and therelationship between various details should be noted.

One must also verify their argument at every step of the plan if one is goingto be sure if its validity. There are two ways to do this. The first is by a directrigorous proof that the step is correct, and the second is by an intuitive notion thatone “sees” how the step is correct. Both of these must be verified when carryingout the plan, since the plan cannot possible true if one cannot prove a step, andcannot possible be practical if one cannot see in intuitively. Thus one asks themself the questions “can you prove that [the step] is correct” and “can you see clearlythat [the step] is correct.”

3.2 Generalization:

Generalization is, in sum, the technique of realizing that the problem at handis a member of a greater set of problems, all of which have similar solutions, andthen using the general solution for that set to solve the problem at hand. Polyadefines it as “passing form the consideration of a certain object to the considerationof a set that contains that object.” When one is confronted with a problem which isburdened with data, one can generalize the data so as to give a clear indication asto the solution. For given lengths one can create letters, for specific objects, onecan substitute a general class of object. Thus the problem is likely to resemblesome abstract idea which one is familiar with. Then the solution becomes apparentand the problem can be solved.



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3.3 Specialization:

Just as generalization (3.2) moves from the consideration of a specific objectto a set of objects, specialization moves from the consideration of a set of objectsto a specific object or from one member of the set to another in the set which isclosely related to it. Polya outlines a specific method by which one should createan auxiliary problem (2.2) through a specialization.

A. Specialization Though Auxiliary Problems

Given an original problem which is too difficult because of a givencondition or set of conditions, one can specialize the problem. Invent an auxiliaryproblem which ignores one or more of the conditions so that it can be solvedeasily. What is to be learned from this problem? How does it relate to the originalproblem? Once this connection has been made, one can many times go on to solvethe original problem without much trouble. Since one part of the problem is solved,solving the others, i. e. the conditions which were previously ignored, is moremanageable. Thus, the specialized problem serves as a stepping stone to theoriginal.

3.4 Induction and Mathematical Induction:

There are two types of induction in mathematics, the first is the style of formal proof using the n + 1 model, and the other is an informal way of creatingand supporting plausible guesses. In How to Solve It Polya discusses both types,especially the formal one in great detail, but since the formal type is commonknowledge, and not truly of any heuristic value, I will not include it here, only theinformal.

A. The Informal Type of Induction

In solving a problem one may chance upon some discovery without beingsure of its validity. A common way to form a plausible guess as to its validity is totest it in other scenarios, if a formula is good for one configuration of numbers, is itgood for another? How many others? If there are a substantial amount of othercases where the formula holds true, then it is most likely true (one proves nothinghowever, until one has devised a rigorous argument, and it is only a plausible guessuntil that occurs). It is a step toward a generalization (3.2), and supplying othercases of a given idea’s validity is a specialization (3.3). The best use of this is asone is carrying out a plan, if one makes some discovery, to verify the validity of it



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sec = cm mcm(sec -2)n

Distributing we get:sec = cm m+n sec -2n

To eliminate the cm terms one sets them to 0, such as:

m + n = 0

And then to complete the system, one writes the remaining equivalence:

1 = -2 n

Thus we obtain n = -½ and m = ½. The formula is now: =

One has now ascertained much more about this formula than was previouslyknown, without actually memorizing it. True, there is no way to find the constantc, but it is still better that the previous form of it.



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Looking Back



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Other Techniques



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5.1 Practical Problems:

Outside of mathematics there are many other sciences which make use of mathematical problem solving with relation a physical system. Many times theinformation in a practical problem can be staggering, with hundreds of unknowns,conditions, and thousands of data. This is because a practical problem is usuallynot as clear as purely mathematical problems.

The complexity of the unknowns, the data, and the conditions differentiate apractical problem from a purely mathematical one the most sharply. It is usuallythought that practical problems need much more experience to tackle than purelymathematical problems, but this is usually in the knowledge needed that the actual

heuristic approach to the problem. In either type, one applies their knowledge of related problems, thus the questions “have you seen the same problem in a slightlydifferent form?” and “do you know a related problem?” should be asked in anycase.

Another sharp difference is in the ability to completely understand all of thenotions related to the problem. In a mathematical problem, these are clear, and onecan fairly easily gain an understanding of them. In practical problems however,these are usually very “hazy” as Polya puts it, and the clarification of them isessential, thus the questions “have you take into account all essential notions

involved in the problem?”In pure mathematical problems one must include all data and all conditions,

but not so in a practical problem. Think of the engineer in charge of building apower plant. They must take into account things such as cost, environmentalimpact, and efficiency, but not the petty grievances of the local residents. Thus oneasks themself “did you include all the data/conditions which could influenceappreciably the solution ?” rather than just the first part of the question.

The last consideration is “can you make the problem simpler by using areasonable approximation rather than completely accurate data.” A burdensomeamount of detail in calculations may make the problem more difficult and involvedthan it should be.

5.2 Problems to Find Problems to Prove:

There are two principle types of mathematical problems. The first, problemswhich revolve around some unknown bit of information, are called “problems tofind” by Polya, and the second, problems which revolve around proving or



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disproving a given proposition, are called “problems to prove.” This entry shalltake up both separately.

A. Problems to Find

The unknown of a problem to find could be anything. In geometry, a shapeor measurement, in algebra a number, in the calculus an antiderivitive to a givenfunction, in a crossword puzzle a word of a given amount of characters, etc. Onemust know and be able to decompose (2.3) the three main parts of a problem tofind, the unknown , the data , and the condition . In simple problems there may beonly one or two of each, but in more complex problems one might have many setsof data and conditions. Either way, in solving a problem to find one asks

themselves “what is/are the unknown(s)/data/condition(s)?”, “can you separate thevarious parts of the condition?” (1.6), “can you establish a link between the dataand the unknown?” etc (there is a complete list in How to Solve it pg 155-156).

B. Problems to Prove

Unlike the unknown of a problem to find, the unknown of a problem toprove is “to show conclusively that a certain clearly stated assertion is true, or elseto show that it is false.” Thus problems to prove depend much more on the contentof the problem for their variation rather than the nature of what is sought. Usualproblems to prove can be separated into two parts, the hypothesis and theconclusion . This is not true for all problems to prove however, note the problem“prove there is infinitely many prime numbers.” In solving a problem to prove,understanding these parts is very important, and furthermore, recognizing thenature of the notions involved is essential. Thus one asks questions like “did youuse the whole hypothesis?” and “is there another problem with a similarconclusion?” etc (once again, there is a complete list of these questions in How toSolve it pg 156).

5.3 Progress and Achievement:

There are certain steps one takes as they progress toward the solution of aproblem. There are three very important ones called mobilization , organization ,and changes in the mode of conception. Each of these has a given heuristic value,and shall be discussed separately.



A. Mobilization

When first confronted with a problem one must assemble the necessaryknowledge in order to correctly form a plan to solve it. This requires eitherrecalling knowledge that one already has or gaining knowledge which one does nothave, but is conducive to the solution. This process of gathering knowledge istermed mobilization by Polya. It is a process of extraction

B. Organization

Once one has mobilized their knowledge of the problem, the knowledgemust be ordered correctly so as to find a solution. One must take the isolated factswhich are the products of the mobilization and combine them in a suitable way soas to create a path to the solution. This process is called organization by Polya, andis used to “…construct and argument containing the materials recollected to t\awell adapted whole.”

C. Changes in the Mode of Conception

Once one has mobilized and organized their knowledge, one sets off carrying out the steps to find a solution. Throughout this process one becomesaware of things which were not known before, realizes connections which one waspreviously unaware of, and recalls helpful ideas which previously didn’t occur to

them. Thus, in the process of solving a problem, one’s idea of the problem and thecontents of thee problem changes from the outset. These changes are calledchanges in the mode of conception by Polya. With several changes in the mode of conception, one becomes aware of various standpoints of the problem, thus helpingto smooth the solution to the problem by recognizing all of its parts.

the little polya

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