Solving Math Problems:

Some Practical Remarks

Overview

This paper describes a method for solving math problems.The basic idea is to combine two things:

• First, a simple method for making handwritten notes while thinking about a problem.This method is aimed at supporting

– a step-by-step approach to problem solving and

– reflective thinking: Better understand and control what you do while solving a problem.

• Second, a densely packed cheat sheet with broad advice on math problem solving.At present, this sheet focuses on general methods for problem solving.Later versions may contain material on specific domains like calculus or algebra.

You can download this paper

• as a .pdf file: http://dl.dropbox.com/u/4884231/MathProbSolv.pdf,

• as a .tex file: http://dl.dropbox.com/u/4884231/MathProbSolv.tex.This is useful if you want to adapt the cheat sheet.

Making Notes

Layout for the notes

Here are some ideas on how to use the notemaking process.

• Use a two column layout.(Read on! This is not the two-column proof you may know from geometry lessons.)The paper (in portrait format) is divided by a vertical line. The left part is about two thirdsof the the page width and is used for the main body of notes. The right part is used forreflection about what you are doing.You can use the problem name as a headline.

• Organize your notes by process stages.As we will see a few paragraphs later, the cheat sheet suggests several process stages, like”Getting Started”, ”Make a plan” and so on. For each process stage a collection of usefulproblem solving tools is given - for the ”Getting Started” stage you can use tools like ”makea diagram”, ”introduce useful notation” or ”make a table of special cases for small numbers”.

• Use abbreviations for important stages and tools.For example, use ”signposts” like ”gs” for ”Getting Started”, ”rep” for ”Representations” andso on.

• Use the reflection column.Here you can add what you think about the material in the main left column. E.g., are youstuck? What can you do about it? What are alternatives to your current approach?The cheat sheet contains a number of useful tools for reflection. You can use tool abbreviationsagain - a question mark for obstacles or an exclamation mark for insights.Don’t worry whether your notes really belong in the left or right column - remember these areyour work notes and not a final presentation of your results. The separate reflection columnis merely a vehicle to give you better control over what you are doing in your problem solving.

• Use a hierarchic layout, linking ideas by short lines.This may be a matter of liking, but again it works well for me. With these lines, I can bettersee connections between ideas, especially if I add an idea later.

Some details

Here are some remarks about special aspects of notemaking.

• Use reflections at least at the end of each stage, and whenever you feel confused.Writing up is often a great help.

• Dealing with parallel approaches.Often enough a problem can be tackled via different roads - e.g. using induction or contra-diction. You could note both approaches and examine them one after the other, starting withthe most promising one, or you can try a parallel strategy and start a separate sheet for eachapproach. For complex problems, this may be the better process.

• Referencing.Sometimes you have to make a reference to another part of your notes. A simple way to dothis is via numbers. Number your pages to manage references between sheets, e.g. 5-3: page5, remark number 3.

Finally, here are two trivial things that work well for me.

• Write small.Getting large amounts of information on one sheet is very useful.Perhaps you’ll find that larger sheets of paper and finer pencils work well for you.

• Write lightly.This makes erasing much easier.

An example

The picture on the next page shows a number of layout essentials.

Remarks on the cheat sheet

• A focus on heuristics.The cheat sheet in its present state contains only heuristic advice. You’ll find almost nodomain specific information, e.g. about primes or about calculus.

• Use of endnotes.The cheat sheet is meant to contain as much information as reasonable and legible (hrmpf?),so it consists mainly of short descriptions of concepts that are by no means self-explanatory.More (but arguably still insufficient) information about some of the cheat sheet items can befound in the endnotes.

• Redundant information.You’ll find that some items appear more than once. This is to increase chances that you findit quickly in different contexts.

• References.A simple ”>” points to other paragraphs of the cheat sheet.

• Adapt.This is by far the most important point: Adapt this cheat sheet for your own purposes.(Sharing and discussing the result with others might be a good idea.)

• Document structure.This document was designed for easy adaptation, so I tried to keep the LATEX-aspects of thedocument easy. There are a few parameters you can change to manipulate the cheat sheet,especially the font size and the number of columns.

• Download.You can download the .tex file for this document athttp://dl.dropbox.com/u/4884231/MathProbSolv.tex.

• More Cheat Sheets.You find more cheat sheets onhttp://math-blog.com/2008/09/20/13-useful-math-cheat-sheets/ orhttp://www.cheat-sheets.org/#Math orhttp://prairiestate.edu/skifowit/sheets/sheets.htm orhttp://physics.eou.edu/opensource/math/index.html.

On the following page you find the cheat sheet.

Math Tool Box

Use Layout Elementsuse two columns

main columnreflection column

—use ”signposts” forimportant elements of problem solving:

U: unknownD: dataC: condition1

—pr: problem2

an: analysisdr: desired resultpl: plan

op: options3

so?!: what to do next?4

si: spontaneous ideasgs: getting started

rep: representations5

hyp: hypothesis6

...7

Use A General Processgo through the following stages8

leave out or repeat stages if necessary—get started >make a plan >carry out the plan >reflect >

Get Startedlook at special cases

collect and organize datalook at extreme casesmake things simpler9

—find representations >—break down the problem into

unknowndatacondition

examine these partssimplify these partsmodify these parts—check definitions

Make a Planconstruct a proof hierarchy:10

proof strategy11

proof tacticsproof details

—construction tools:

ask repeatedly:

”How can I do this?”12

work top-downwork bottom-upconstruct intermediate elementsof the proof hierarchy

construct a penultimate step13

use wishful thinking / make it easier14

—use forward searchuse backward search

—generate seminal ideas for a proof

look at related problems >use your knowledge >use important principles >find new approaches >

Carry Out the Planwork with carecorrectness proved?correctness evident?be critical

Reflectuse the reflection column—reflect on the way:15

ask ”So what?”collect questionswhat’s the problem / obstacle?what’s the conflict?what’s your aim?what’s your plan?what can you do?can you do something better?

—reflect at the end:

what worked?what didn’t work? and why?use results elsewhereuse methods elsewhere

—check list of Common Errors>

Look at Related Problemslook at more special problemslook at similar / analog problems

look at more general problems16

Use Your Knowledgeuse relevant theoremsuse solved problems

use their resultsuse their methods

—how to find such material:

from memoryvia the unknown:which material has the same unknown?

via explicit search processgather information >

—utilize the material:

make material applicable to problemmodify your problemmodify your material

Find Representationsintroduce notationdraw diagrams

develop mental images17

—use representation types:

geometriccoordinate systems

cartesian, polar, cylindrical...choose the origin

—algebraic

b-adicfactorisations

—algorithmicmiscellaneous

—organize data:

in tablesin diagrams

figures, hierarchical trees...—exploit symmetries, invariants...

Types of Proofdirect proof18

proof by mathematical inductionproof by transpositionproof by contradictionproof by constructionproof by exhaustionprobabilistic proofcombinatorial proofnonconstructive proofvisual proofcomputer-assisted proofs

Important Principlesanalogy1920

Fubini principle21

parity22

Dirichlet principle23

inclusion/exclusion24

opposites25

induction26

generalisation27

specialisation28

variation29

invariance30

monovariance31

infinite descent32

symmetry33

extremes34

recursion35

stepwise approximation

colouring36

randomisation37

change of perspective38

modularisation39

brute force40

—use a computer...

for computationfor simulation

—use a ”greedy algorithm”41

build a model42

guess and check43

Find New Approachesgeneral idea:

choose objects of problem solvingmodify objectsobserve the results

—what to modify:

problem objects—problem elements

unknowndatacondition

—representations >points of viewcontext of referencediagrams

—how to modify:

substitute, replacecombine with other elementsreverse, rearrangeeliminateexchangeadapt, alteraddminimize, maximizebreak down into partsapproximate

—what to look at:

symmetrypatternsextremeslimitsdatainvariantsdetails - more or less of themparity...

Useful Math Conceptscomplex numbersgraphsgenerating functions

...—constant vs. variable parameters - change—estimates and approximations

use inequalities

Change Heuristic Approachgeneral idea:

take heuristic objectsmodify them

—heuristic objects:

proof hierarchyproof strategyproof tacticsproof details

representationsproblem ingredients

unknowndataconditon

search directionforward / backward

—modifications:

try alternativestry opposites

Gather Informationtalk to others

directly, via email—use the internet:

math encyclopedias

Wolfram MathWorld44

PlanetMath45

Springer Encyclopedia46

Wikipedia...

—math communities47

AoPS48

PlanetMath49

mathoverflow50

...—

accessing literature

MathSciNet51

Zentralblatt MATH52

...—use books and libraries:

scripts, textbooks, formularies...

Common Errorsthinking that is...

hastynarrowfuzzy

sprawling53

—working without aimworking without planerrors in carrying out a planlack of reflection—check lists of errors54

Advice from tricki.orgDon’t start from scratch55

Hunt for analogies56

Mathematicians need to be metamathemati-cians57

Think about the converse58

Try to prove the opposite59

Look for related problems60

Work on clusters of problems61

Look at small cases62

Try to prove a stronger result63

Prove a consequence first64

Think axiomatically even about concrete ob-

jects65

Temporarily suspend rigor66

Turn off all but one of the difficulties67

Simplify your problem by generalizing it68

If you don’t know how to make a decision, then

don’t make it69

If an argument looks promising but needs sometechnical hypothesis, try assuming that hy-

pothesis for now, but aim to remove it later.70

Your Own Tools

Stay Functionaltalk to people—eat / drink...exercise, physical activitybreathe deeply and calmlytake a break; sleepwork in a new setting—music - make or listennonmath activitymath activity outside your domainflood yourself with new ideas

Persistwork on for just 15 minutes

(and repeat this)use coping self talkimagine the work doneremember previous successes

Notes1What is unknown? What are the data? What is the condition? - These fundamental questions are of course due

to George Polya’s ”How to Solve It”, Princeton 1988.2Use ”pr” not only for the initial problem, but for later sub-problems that arise in the course of your work.3Use this for collecting the options you have in a given situation.4For me, this is an especially useful tool.5Cf. the section on representations.6Use this to formulate a hypothesis. Then try to proof or disproof it.7You can highlight these signposts by drawing a little circle around them.8The General Process and much of the rest of this sheet is of course masssively influenced by Polya’s ”How to

Solve It”.9E.g., replace nasty things with nice ones. Cf. the ”wishful thinking tool” below.

10Build a hierarchy of steps to prove something.Example - Induction: On the top level, we have the induction principle. On the level below, we have the base case(often for n = 0 or n = 1) and the induction step. Then we have arguments for proving the base case and argumentsfor the induction step.In most cases however, a proof cannot be constructed in such a neat top-down manner - we have to assemble it fromseveral building blocks, using a combination of top-down and bottom-up strategies.To get a better impression of the proof hierarchy idea - Leslie Lamport’s article on writing structured proofs is aworthwhile read:http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf.

11The three hierarchy levels strategy, tactics and details are not quite sharp, but nevertheless useful.12This and the next tool are of course closely releated to the ”working backwards” tool.13Ask yourself what might be the last step in your argument, the one that will yield the conclusion. (From: Paul

Zeitz, Art and Craft of Problem Solving, New York 1999, chapter 2.2).14Try to make difficulties in your problem disappear - ”[...] if the problem involves big, ugly numbers, make them

small and pretty” - again taken from Paul Zeitz’ book.15Use the reflection column for applying these items.16Looking at more general problems is sometimes useful. In induction for example, you have more to prove - but

also more to build on.17There are countless websites that illustrate math concepts. For starters, have a look at

http://www.cinderella.de/files/HTMLDemos/,http://www.artofproblemsolving.com/Resources/videos.php?type=other orhttp://demonstrations.wolfram.com/.

18This list is largely taken fromhttp://en.wikipedia.org/wiki/Mathematical_proof.

19Major source for this list of ”Important Principles”: Christian Hesse: Das kleine Einmaleineins des klarenDenkens, Munich 2009

20Try to reduce the problem to another problem that is already solved.21Count something by counting something else. Little Gauss’ summation of 1 + ... + 100 is a famous example.22Consider even and odd numbers, or more abstract: Try to establish two nonoverlapping classes and extract

information from this.23If n + 1 items are put into n boxes, there must be at least one box with two or more items.24|A ∪B| = |A|+ |B| − |A ∩B| and the general case for this.25Assume the opposite of what is to be shown and develop a contradiction.26Besides classical induction, think of more elaborate versions like downward induction.27Try to solve a more general problem.28Consider special cases.29Get insight into the problem by varying several aspects of it.30Construct something that remains unchanged under certain transformations.31Construct something that can only increase under certain transformations.32Tool for contradiction proofs: Show that if a certain solution exists, there should be a smaller one, and then

another even smaller, but that such an infinite descent isn’t possible for the given problem.33Look for symmetries in a given system.34Look at extremal elements.35Can you reduce a problem to a simpler version of itself?36Colour your problem and derive information from this. Example: Missing corners in a checkerboard.

37Introduce elements of chance into your problem to make it simpler.38Work backwards.39Divide the problem into smaller parts, solve them and combine this for a solution of the initial problem.40Check all possible solutions.41If you have to construct something via an algorithm, take the most you can get in every step, or more abstract:

Use locally optimal solutions to construct a global optimum. Cf. Arthur Engel’s ”Problem Solving Strategies”, NewYork 1998.

42For example, build a physical model of a spatial construct.43Try to guess a solution and check it.44http://mathworld.wolfram.com/.45http://planetmath.org/encyclopedia.46http://eom.springer.de/.47Purpose and scope of the sites differ. Read the introductions and FAQs carefully.48http://www.artofproblemsolving.com/Forum/index.php.49http://planetmath.org/?op=forums.50http://mathoverflow.net/ - for research level math questions.51http://www.ams.org/mathscinet/.52http://www.zentralblatt-math.org/zbmath/advanced/.53This diagnosis is taken from David Perkins, Outsmarting IQ54For a list of common erros in undergraduate mathematics, have a look at

http://www.math.vanderbilt.edu/~schectex/commerrs/.55A common mistake people make when trying to answer a mathematical question is to work from first principles:

it is almost always easier to modify something you already know. This article illustrates the point with examplesthat range from simple arithmetic to problems from the forefront of research.

56It is surprising how often the following general approach to problem-solving is successful: you have a problemyou don’t yet know how to solve; you think of a somewhat similar context where you can formulate an analogousproblem that you do know how to solve; you then work out what the corresponding solution ought to be in thecontext you started with. Even quite loose analogies can do a wonderful job of guiding you in the right direction.

57If you want to prove a theorem, then one way of looking at your task is to regard it as a search, amongst thehuge space of potential arguments, for one that will actually work. One can often considerably narrow down thisformidable search by thinking hard about properties that a successful argument would have to have. In other words,it is a good idea to focus not just on the mathematical ideas associated with your hoped-for theorem, but also onthe properties of different kinds of proofs. This very important principle is best illustrated with some examples.

58If you are trying to prove a mathematical statement, it is often a good idea to think about its converse, especiallyif the converse is not obvious. This is particularly useful if the statement you are trying to prove is a lemma thatyou would like to use to prove something else. Some examples will help to explain why. More obviously, it is usefulif you are trying to prove a result that would, if true, be best possible.

59If you want to prove a mathematical statement, try proving the negation of that statement. Very often it givesyou an insight into why the original statement is true, and sometimes you discover that it is not true. This tip canbe iterated.

60When you are trying to solve a problem, it can be very helpful to formulate similar-looking problems and thinkabout those too. Sometimes they turn out to be interesting in themselves, and sometimes they lead you to ideasthat are useful for the original problem.

61It is not usually a good research strategy to think about one isolated problem, unless you are already some wayto solving it. Solving a problem involves a certain degree of luck, so your chances of success are much greater if youlook at a cluster of related problems.

62Can’t see how to solve a problem? Then see if you can solve it in special cases. Some special cases will be tooeasy to give you a good idea of how to approach the main problem, and some will be more or less as difficult as themain problem, but if you search for the boundary between these two extremes, you will often discover where the truedifficulty lies and what it is. And that is progress.

63As a student, one is asked to prove many statements that have been carefully designed so that their hypothesesare exactly the appropriate ones for deducing the conclusion. This makes it possible to design ”trick” questions withunnecessarily strong hypotheses: these questions can be hard if one tries to use the hypotheses as they stand, sincetheir full strength is irrelevant. The problems that arise when one is doing research are often of the ”trick” variety:one has not been set them by a benign professor in the sky. So it is a good idea to investigate whether weakerhypotheses will suffice. As with looking at small cases, this can help one to locate the true point of difficulty of aproblem. Similarly, if you are trying to find an example of a mathematical structure X that has a certain property P ,

it may be easier to look for an X that has a stronger property Q. And even if you fail, you are likely to understandmuch better what is required to find an X that satisfies P .

64This is the flip side of ”Try to prove a stronger result”; if one wants to prove some statement, it can be useful tofirst prove a weaker consequence of that statement, and then use that weaker result as a stepping stone to the fullresult.

65If you are trying to prove a fact about the exponential function, it may be easier not to use any of the commondefinitions of this function, but to use instead a few properties that it has, of which the most important is thatexp(x + y) = exp(x) exp(y). In general, it is often possible to turn concrete problems into abstract ones in this way,and doing so can considerably clarify the problems and their solutions.

66The final proof of a result should of course be fully rigorous. But this certainly does not prevent one fromsuspending rigor in order to locate the right proof strategy to pursue. For instance, if one needs to compute somecomplicated integral expression, one can temporarily suspend concerns about whether operations such as interchangeof integrals is actually justified, and go ahead and perform these operations anyway in order to find a plausible answer.One can always go back later and try to make the argument more rigorous.

67A problem may have several independent difficulties plaguing it; for instance one may need to establish anestimate which is uniform both with respect to a large parameter N , and an independent small parameter ε. In thatcase, one can often proceed by passing to a special case in which only one of the difficulties is ”active”, solving eachof these basic special cases, and then try to merge the arguments together. For instance, one could set N = 1 andget an argument which is uniform as ε→ 0, then set ε = 1 and get an argument which is uniform as N →∞, thentry to put them together.

68Sometimes if you generalize a statement, the result is easier to prove. There are several reasons for this, discussedin separate articles linked to from this page.

69Very often a proof requires one to choose some object that will make the rest of the proof work. And very oftenit is far from obvious how to make the choice. Often a good way of getting round the problem is to take an arbitraryobject of the given type, give it a name X, and continue with the proof as if you had chosen X. Along the way,you will find that you need to assume certain properties P1, . . . , Pk of X. Then your original problem is reduced tothe question ”Does there exist an object of the given type with properties P1, . . . , Pk?” Often, this is a much morestraightforward question than the main problem you were trying to solve.

70This entire section is an unchanged quotation from:http://www.tricki.org/article/General_problem-solving_tips.If there are any copyright problems, please let me know.

Thomas TeepeDecember 15, 2011