How to Solve Mathematical Problems

Description

Title Page; Copyright Page; Dedication; Table of Contents; Preface; 1 - Introduction; SCOPE OF THE BOOK; GENERAL VERSUS SPECIAL METHODS; RELATION TO ARTIFICIAL INTELLIGENCE; APPLYING METHODS TO PROBLEMS; 2 - Problem Theory; FOUR SAMPLE PROBLEMS; Instant Insanity; Chess Problem; Find Problem from Mechanics; Proof Problem from Modern Algebra; WHAT IS A PROBLEM?; Givens; Operations; Goals; Implicit Specification of Givens, Operations, and Goals; Incomplete Specification of Givens, Operations and Goals; WHAT IS A PROBLEM STATE?; WHAT IS A SOLUTION?; STATE-ACTION TREE; 3 - Inference; GIVENS.Numerical PropertiesTopological Properties; Operations; GOALS; 4 - Classification of Action Sequences; RANDOM TRIAL AND ERROR; SYSTEMATIC TRIAL AND ERROR; CLASSIFICATORY TRIAL AND ERROR; MACROACTIONS; GETTING OUT OF LOOPS; INCUBATION; 5 - State Evaluation and Hill Climbing; THEORY; APPLICATIONS; DIFFICULTIES WITH HILL CLIMBING; 6 - Subgoals; THEORY; APPLICATIONS; 7 - Contradiction; INDIRECT PROOF; MULTIPLE CHOICE-SMALL SEARCH SPACE; CLASSIFICATORY CONTRADICTION-LARGE SEARCH SPACE; ITERATIVE CONTRADICTION IN INFINITE SEARCH SPACES; 8 - Working Backward; THEORY; APPLICATIONS.9 - Relations Between ProblemsEQUIVALENT PROBLEMS; SIMILAR PROBLEMS; SPECIAL CASE; GENERALIZATION; 10 - Topics in Mathematical Representation; REPRESENTATION ON PAPER OR IN THE HEAD; DIAGRAMMATIC REPRESENTATION; SYMBOLIC REPRESENTATION; SOME IMPORTANT MATHEMATICAL CONCEPTS; 11 - Problems from Mathematics, Science, and Engineering; ALGEBRA; TRIGONOMETRY; ANALYTIC GEOMETRY; CALCULUS; DIFFERENTIAL EQUATIONS; PROBABILITY AND STATISTICS; COMBINATORIAL ANALYSIS; NUMBER THEORY; MODERN ALGEBRA; MECHANICS; HEAT; ELECTRICITY; ELECTRICAL ENGINEERING; COMPUTER PROGRAMMING; References; Index.If you've ever tried to solve mathematical problems without any idea how to go about it, this book is for you. It will improve your ability to solve all kinds of mathematical problems whether in mathematics, science, engineering, business, or purely recreational mathematical problems (puzzles, games, etc.). In the pages of this book you'll discover seven indispensable problem-solving techniques: inference, classification of action sequences, state evaluation and hill climbing, subgoals, contradiction, working backward and relations between problems. Based on modern advances in the fields of ar.