| |
| |
| Preface | |
| |
| |
| To the Reader | |
| |
| |
| Acknowledgments | |
| |
| |
| |
| Following the Clues | |
| |
| |
| Sample problems | |
| |
| |
| Which chart or Diagram to Choose | |
| |
| |
| Presenting a Solution | |
| |
| |
| Some Steps in Problem Solving | |
| |
| |
| Tree Diagrams | |
| |
| |
| The Multiplication Principle | |
| |
| |
| Simplification | |
| |
| |
| The Chapter in Retrospect | |
| |
| |
| Exercises | |
| |
| |
| |
| Solve It With Logic | |
| |
| |
| Sample Problems | |
| |
| |
| Statements | |
| |
| |
| Variables and Connectives | |
| |
| |
| Negation; "And"—Conjunction; "Or"—Disjunction | |
| |
| |
| Conditional and Biconditional Statements | |
| |
| |
| Drawing Conclusions | |
| |
| |
| Compound Statements | |
| |
| |
| Logical Implication and Equivalence | |
| |
| |
| Arguments and Validity | |
| |
| |
| The Chapter in Retrospect | |
| |
| |
| Exercises | |
| |
| |
| |
| From Words to Equations: Algebraic Recreations | |
| |
| |
| Sample Problems | |
| |
| |
| Introducing Variables | |
| |
| |
| The Chapter in Retrospect | |
| |
| |
| Exercises | |
| |
| |
| |
| Solve It With Integers, Some Topics from Number Theory | |
| |
| |
| Sample Problems | |
| |
| |
| Diophantine Equations | |
| |
| |
| Divisibility | |
| |
| |
| Prime Numbers | |
| |
| |
| The Infinitude of Primes | |
| |
| |
| The Sieve of Eratosthenes | |
| |
| |
| More About Primes | |
| |
| |
| Linear Diophantine Equations | |
| |
| |
| Division With Remainders | |
| |
| |
| Congruence | |
| |
| |
| Casting Out Nines | |
| |
| |
| Solving Linear Congruences | |
| |
| |
| Solving Linear Diophantine Equations | |
| |
| |
| The Chapter in Retrospect | |
| |
| |
| Exercises | |
| |
| |
| |
| More About Numbers: Bases and Cryptarithmetic | |
| |
| |
| Sample Problems | |
| |
| |
| Positional Notation | |
| |
| |
| Changing Bases | |
| |
| |
| Addition and Multiplication in Other Bases | |
| |
| |
| Cryptarithmetic | |
| |
| |
| The Chapter in Retrospect | |
| |
| |
| Exercises | |
| |
| |
| |
| Solve It With Networks: An Introduction to Graph Theory | |
| |
| |
| Sample Problems | |
| |
| |
| Graphs | |
| |
| |
| Eulerian Paths and Circuits | |
| |
| |
| Odd and Even Vertices | |
| |
| |
| More Than Two Odd Vertices | |
| |
| |
| Directed Graphs | |
| |
| |
| Hamiltonian Circuits | |
| |
| |
| The Knight's Tour | |
| |
| |
| Other Applications | |
| |
| |
| Coloring Graphs and Maps | |
| |
| |
| The Chapter in Retrospect | |
| |
| |
| Exercises | |
| |
| |
| |
| Games of Strategy for Two Players | |
| |
| |
| Sample problems | |
| |
| |
| Chance-Free Decisionmaking | |
| |
| |
| Games of Perfect Information | |
| |
| |
| Finiteness | |
| |
| |
| The Existence of Winning Strategies | |
| |
| |
| Position--State of the Game | |
| |
| |
| The State Diagram of a Game | |
| |
| |
| How Do We Find a Winning Strategy? | |
| |
| |
| Finding a Winning Strategy by Working Backward | |
| |
| |
| Finding Winning Strategies by Simplifying a Game | |
| |
| |
| Finding Winning Strategies With a Frontal Assault | |
| |
| |
| How Many Possibilities Need Be Considered? | |
| |
| |
| Symmetry as a Limiting Factor | |
| |
| |
| Deja Vu—We've Seen it Before | |
| |
| |
| The Game of Nim | |
| |
| |
| Pairing Strategies | |
| |
| |
| Variations of a Game | |
| |
| |
| The Chapter in Retrospect | |
| |
| |
| Exercises | |
| |
| |
| |
| Solitaire Games and Puzzles | |
| |
| |
| Sample Problems | |
| |
| |
| The Tower of Brahma | |
| |
| |
| Dissection Problems | |
| |
| |
| Polyominoes | |
| |
| |
| Soma | |
| |
| |
| Peg Solitaire | |
| |
| |
| The Fifteen Puzzle | |
| |
| |
| Even and Odd Permutations | |
| |
| |
| Coloring and the 15 Puzzle--A Second Approach | |
| |
| |
| Colored Cubes | |
| |
| |
| Colored Cubes--A Second Approach | |
| |
| |
| The Chapter in Retrospect | |
| |
| |
| Exercises | |
| |
| |
| |
| Potpourri | |
| |
| |
| Decimation | |
| |
| |
| Coin Weighing | |
| |
| |
| Shunting | |
| |
| |
| Syllogisms | |
| |
| |
| Grab Bag | |
| |
| |
| The Book in Retrospect | |
| |
| |
| |
| Some Basic Algebraic Techniques | |
| |
| |
| |
| Mathematical Induction | |
| |
| |
| |
| Probability Bibliography | |
| |
| |
| Hints and Solutions | |
| |
| |
| Answers to Selected Problems | |
| |
| |
| Index | |