How to Solve It: A New Aspect of Mathematical Method
A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight.\ In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out—from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft—indeed, brilliant—instructions on stripping away irrelevancies and going...
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A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight.In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out--from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft--indeed, brilliant--instructions on stripping away irrelevancies and going straight to the heart of the problem.In this best-selling classic, George Pólya revealed how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out--from building a bridge to winning a game of anagrams. Generations of readers have relished Pólya's deft instructions on stripping away irrelevancies and going straight to the heart of a problem. How to Solve It popularized heuristics, the art and science of discovery and invention. It has been in print continuously since 1945 and has been translated into twenty-three different languages.Pólya was one of the most influential mathematicians of the twentieth century. He made important contributions to a great variety of mathematical research: from complex analysis to mathematical physics, number theory, probability, geometry, astronomy, and combinatorics. He was also an extraordinary teacher--he taught until he was ninety--and maintained a strong interest in pedagogical matters throughout his long career. In addition to How to Solve It, he published a two-volume work on the topic of problem solving, Mathematics of Plausible Reasoning, also with Princeton.Pólya is one of the most frequently quoted mathematicians, and the following statements from How to Solve It make clear why: "My method to overcome a difficulty is to go around it." "Geometry is the science of correct reasoning on incorrect figures." "In order to solve this differential equation you look at it till a solution occurs to you." E. T. Bell - Mathematical Monthly Every prospective teacher should read it. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.'
From the preface to the first printingFrom the preface to the seventh printingPreface to the second edition"How to solve it" listForewordIntroductionPt. 1In the classroom1Helping the student12Questions, recommendations, mental operations13Generality24Common sense35Teacher and student, imitation and practice36Four phases57Understanding the problem68Example79Devising a plan810Example1011Carrying out the plan1212Example1313Looking back1414Example1615Various approaches1916The teacher's method of questioning2017Good questions and bad questions2218A problem of construction2319A problem to prove2520A rate problem29Pt. IIHow to solve itA dialogue33Pt. IIIShort dictionary of heuristicAnalogy37Auxiliary elements46Auxiliary problem50Bolzano57Bright idea58Can you check the result?59Can you derive the result differently?61Can you use the result?64Carrying out68Condition72Contradictory73Corollary73Could you derive something useful from the data?73Could you restate the problems?75Decomposing and recombining75Definition85Descartes92Determination, hope, success93Diagnosis94Did you use all the data?95Do you know a related problem?98Draw a figure99Examine your guess99Figures103Generalization108Have you seen it before?110Here is a problem related to yours and solved before110Heuristic112Heuristic reasoning113If you cannot solve the proposed problem114Induction and mathematical induction114Inventor's paradox121Is it possible to satisfy the condition?122Leibnitz123Lemma123Look at the unknown123Modern heuristic129Notation134Pappus141Pedantry and mastery148Practical problems149Problems to find, problems to prove154Progress and achievement157Puzzles160Reductio ad absurdum and indirect proof162Redundant171Routine problem171Rules of discovery172Rules of style172Rules of teaching173Separate the various parts of the condition173Setting up equations174Signs of progress178Specialization190Subconscious work197Symmetry199Terms, old and new200Test by dimension202The future mathematician205The intelligent problem-solver206The intelligent reader207The traditional mathematics professor208Variation of the problem209What is the unknown?214Why proofs?215Wisdom of proverbs221Working backwards225Pt. IVProblems, hints, solutionsProblems234Hints238Solutions242
\ Mathematical MonthlyEvery prospective teacher should read it. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.'\ — E. T. Bell\ \ \ \ \ Mathematical Review[This] elementary textbook on heuristic reasoning, shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected.\ — Herman Weyl\ \ \ Scientific MonthlyI recommend it highly to any person who is seriously interested in finding out methods of solving problems, and who does not object to being entertained while he does it.\ \ \ \ \ American Journal of PsychologyAny young person seeking a career in the sciences would do well to ponder this important contribution to the teacher's art.\ — A. C. Schaeffer\ \ \ \ \ Mathematics MagazineEvery mathematics student should experience and live this book\ \ \ \ \ Mathematical Monthly\ - E.T. Bell\ Every prospective teacher should read it. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.'\ \ \ \ \ Mathematical Review\ - Herman Weyl\ [This] elementary textbook on heuristic reasoning, shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected.\ \ \ \ \ American Journal of Psychology\ - A.C. Schaeffer\ Any young person seeking a career in the sciences would do well to ponder this important contribution to the teacher's art.\ \ \ \ \ Mathematical Monthly\ - E. T. Bell\ \ Every prospective teacher should read it. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.'\ \ \ \ \ American Journal of Psychology\ - A. C. Schaeffer\ \ Any young person seeking a career in the sciences would do well to ponder this important contribution to the teacher's art.\ \ \ \ \ Mathematical MonthlyEvery prospective teacher should read it. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.'\ — E. T. Bell\ \
