Some Practical Remarks On Solving Math Problems


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Combine notemaking and problem solving tools to become better at solving math problems.
The paper contains a densely packed cheat sheet with a large number of general math problem solving tools.

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Some Practical Remarks On Solving Math Problems

  1. 1. Some Practical Remarks on Solving Math Problems Thomas Teepe February 16, 2011OverviewThis 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:, • as a .tex file: This is useful if you want to adapt the cheat sheet. 1
  2. 2. Making NotesLayout for the notesHere 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 thirds of the the page width and is used for the main body of notes. The right part is used for reflection 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 useful problem solving tools is given - for the ”Getting Started” stage you can use tools like ”make a diagram”, ”introduce useful notation” or ”make a table of special cases for small numbers”. • Use abbreviations for important stages and tools. For example, use ”GS” for ”Getting Started”, ”Rep” for ”Representations” and so on. • Use the reflection column. Here you can add what you think about the material in the main left column. E.g., are you stuck? 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 abbreviations again - 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 are your work notes and not a final presentation of your results. The separate reflection column is 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 better see connections between ideas, especially if I add an idea later.Some detailsHere 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 with the most promising one, or you can try a parallel strategy and start a separate sheet for each approach. 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 do
  3. 3. this is via numbers. Number your pages to manage references between sheets, e.g. 5-3: page 5, 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 exampleThe picture on the next page shows a number of layout essentials.
  4. 4. Remarks on the cheat sheet • A focus on heuristics. The cheat sheet in its present state contains only heuristic advice. You’ll find almost no domain 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 be found in the endnotes. • Redundant information. You’ll find that some items appear more than once. This is to increase chances that you find it 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 L TEX-aspects of the A document 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 at the following page you find the cheat sheet.
  5. 5. Tool Box Find Representations Useful Math Concepts Your Own Tools introduce notation complex numbers draw diagrams graphs develop mental images11 generating functions — ...Use A General Process use representation types: geometric — constant vs. variable parameters - changego through the following stages coordinate systems —leave out or repeat stages if necessary cartesian, polar, cylindrical... estimates and approximations— choose the origin use inequalitiesget started > —use your knowledge > algebraicfind new approaches > b-adicmake a plan >carry out the plan > factorisations Change Heuristic Approach — general idea:reflect >1 algorithmic take heuristic objects miscellaneous modify them — — organize data: heuristic objects:Get Started in tables proof hierarchy in diagramslook at special cases proof strategy figures, hierarchical trees... collect and organize data proof tactics —look at extreme cases proof details exploit symmetries, invariants...make things simpler2 representations— problem ingredientsfind representations > unknown data—break down the problem into Types of Proof conditon unknown search direction direct proof12 data forward / backward proof by mathematical induction condition3 proof by transposition —examine these parts modifications: proof by contradictionsimplify these parts try alternatives proof by constructionmodify these parts try opposites proof by exhaustion— probabilistic proofcheck definitions combinatorial proof nonconstructive proof visual proof Gather Information computer-assisted proofs talk to othersUse Your Knowledge directly, via emailuse relevant theorems —use solved problems use the internet: use their results Find New Approaches math encyclopedias use their methods general idea: Wolfram MathWorld38— choose objects of problem solving PlanetMath39how to find such material: modify objects Springer Encyclopedia40 from memory observe the results Wikipedia via the unknown: — ... which material has the same unknown? what to modify: — via explicit search process problem objects math communities41 gather information > — AoPS42— PlanetMath43utilize the material: problem elements unknown mathoverflow44 Stay Functional make material applicable to problem data ... talk to people modify your problem condition — — modify your material — accessing literature eat / drink... representations > MathSciNet45 exercise, physical activity points of view Zentralblatt MATH46 breathe deeply and calmly context of reference ... take a break; sleepLook at Related Problems diagrams — work in a new setting — use books and libraries: — look at more special problems how to modify: scripts, textbooks, formularies... music - make or listen look at similar / analog problems substitute, replace nonmath activity look at more general problems4 math activity outside your domain combine with other elements reverse, rearrange flood yourself with new ideas eliminate Common Errors exchangeMake a Plan adapt, alter thinking that is...construct a proof hierarchy:5 add hasty narrow Persist minimize, maximize proof strategy6 fuzzy work on for just 15 minutes break down into parts proof tactics approximate sprawling47 (and repeat this) proof details — use coping self talk —— working without aim imagine the work done what to look at:construction tools: working without plan remember previous successes symmetry work top-down7 errors in carrying out a plan patterns work bottom-up lack of reflection extremes use important principles > — limits construct intermediate elements data check lists of errors48 of the proof hierarchy invariants construct a penultimate step8 details - more or less of them use wishful thinking / make it easier9 parity— ... Advice from use forward search use backward search Don’t start from scratch49 Hunt for analogies50 Important Principles Mathematicians need to be metamathemati- cians51Carry Out the Plan analogy13 14 Think about the converse52 Try to prove the opposite53work with care Fubini principle15 Look for related problems54correctness proved? parity16 Work on clusters of problems55correctness evident? Dirichlet principle17 Look at small cases56be critical inclusion/exclusion18 Try to prove a stronger result57 opposites19 Prove a consequence first58 induction20 Think axiomatically even about concrete ob- generalisation21Reflect specialisation22 jects59use the reflection column Temporarily suspend rigor60 variation23— Turn off all but one of the difficulties61 invariance24reflect on the way:10 Simplify your problem by generalizing it62 monovariance25 ask ”So what?” If you don’t know how to make a decision, then infinite descent26 collect questions symmetry27 don’t make it63 what’s the problem / obstacle? If an argument looks promising but needs some what’s the conflict? extremes28 technical hypothesis, try assuming that hy- recursion29 what’s your aim? stepwise approximation pothesis for now, but aim to remove it later.64 what’s your plan? what can you do? colouring30 can you do something better? randomisation31— change of perspective32reflect at the end: modularisation33 what worked? brute force34 what didn’t work? and why? — use results elsewhere use a computer... use methods elsewhere for computation— for simulationcheck list of Common Errors> — use a ”greedy algorithm”35 build a model36 guess and check37
  6. 6. Notes 1 The General Process and much of the rest of this sheet is of course masssively influenced by George Polya’s Howto Solve It, Princeton 1988. 2 E.g., replace nasty things with nice ones. Cf. the ”wishful thinking tool” below. 3 The categories ”unknown, data and condtions” are again due to Polya’s ”How to Solve It”. 4 Looking at more general problems is sometimes useful. In induction for example, you have more to prove - butalso more to build on. 5 Build 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, the proof hierarchy cannot be constructed neatly in such a top-down manner - we have to assemble itfrom several 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: 6 The three hierarchy levels strategy, tactics and details are not quite sharp, but nevertheless useful. 7 Top-down construction of the proof hierarchy is of course closely releated to the working backwards tool. 8 Ask yourself what might be the last step in your argument, the one that will yield the conclusion. (From: PaulZeitz, Art and Craft of Problem Solving, New York 1999, chapter 2.2). 9 Try to make difficulties in your problem disappear - ”[...] if the problem involves big, ugly numbers, make themsamll and pretty” - again taken from Paul Zeitz’ book. 10 Use the reflection column for applying these items. 11 There are countless websites that illustrate math concepts. For starters, have a look at, or 12 This list is largely taken from 13 Major source for this list: Christian Hesse: Das kleine Einmaleineins des klaren Denkens, Munich 2009 14 Try to reduce the problem to another problem that is already solved. 15 Count something by counting something else. Little Gauss’ summation of 1 + ... + 100 is a famous example. 16 Consider even and odd numbers, or more abstract: Try to establish two nonoverlapping classes and extractinformation from this. 17 If n + 1 items are put into n boxes, there must be at least one box with two or more items. 18 |A ∪ B| = |A| + |B| − |A ∩ B| and the general case for this. 19 Assume the opposite of what is to be shown and develop a contradiction. 20 Besides classical induction, think of more elaborate versions like downward induction. 21 Try to solve a more general problem. 22 Consider special cases. 23 Get insight into the problem by varying several aspects of it. 24 Construct something that remains unchanged under certain transformations. 25 Construct something that can only increase under certain transformations. 26 Tool for contradiction proofs: Show that if a certain solution exists, there should be a smaller one, and thenanother even smaller, but that such an infinite descent isn’t possible for the given problem. 27 Look for symmetries in a given system. 28 Look at extremal elements. 29 You you reduce a problem to a simpler version of itself? 30 Colour your problem and derive information from this. Example: Missing corner in a checkerboard. 31 Introduce elements of chance into your problem to make it simpler. 32 Work backwards. 33 Divide the problem into smaller parts, solve them and combine this for a solution of the initial problem. 34 Check all possible solutions. 35 If 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. 36 For example, build a physical model of a spatial construct.
  7. 7. 37 Try to guess a solution and check it. 38 39 40 41 Purpose and scope of the sites differ. Read the introductions and FAQs carefully. 42 43 44 - for research level math questions. 45 46 47 This diagnosis is taken from David Perkins, Outsmarting IQ 48 For a list of common erros in undergraduate mathematics, have a look at 49 A 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. 50 It 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. 51 If 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. 52 If 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. 53 If 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. 54 When 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. 55 It 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. 56 Can’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. 57 As 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 . 58 This 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. 59 If 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.
  8. 8. 60 The 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. 61 A 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. 62 Sometimes 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. 63 Very 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. 64 This entire section is an unchanged quotation from: there are any copyright problems, please let me know.