The Mathematics of Various Entertaining Subjects: Research in Recreational Math

Description

The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books exploring puzzles and brainteasers, research in recreational mathematics has often been neglected. The Mathematics of Various Entertaining Subjects brings together authors from a variety of specialties to present fascinating problems and solutions in recreational mathematics. Contributors to the book show how sophisticated mathematics can help construct mazes that look like famous people, how the analysis of crossword puzzles has much in common with understanding epidemics, and how the theory of electrical circuits is useful in understanding the classic Towers of Hanoi puzzle. The card game SET is related to the theory of error-correcting codes, and simple tic-tac-toe takes on a new life when played on an affine plane. Inspirations for the book's wealth of problems include board games, card tricks, fake coins, flexagons, pencil puzzles, poker, and so much more. Looking at a plethora of eclectic games and puzzles, The Mathematics of Various Entertaining Subjects is sure to entertain, challenge, and inspire academic mathematicians and avid math enthusiasts alike. Cover Page Title Page Copyright Page Table of Contents Foreword Preface and Acknowledgements PART I Vignettes 1 Should You Be Happy? 1 Comparing Probabilities 2 A Chess Problem, of Sorts 3 Back to Baseball 4 Coin-Flipping and Dishwashing 5 Application to Squash Strategy Acknowledgement 2 One-Move Puzzles with Mathematical Content 1 Divination Puzzles 2 Weighing Puzzles 3 Rearrangement Puzzles 4 Dissection Puzzles 5 Folding Puzzles 6 Conclusion 7 Solutions References 3 Minimalist Approaches to Figurative Maze Design 1 TSP Art Mazes 2 MST Art Mazes 3 Maze Design via Phyllotaxis 4 Seeded Stippling 5 Vortex Tiles 6 Conclusion References 4 Some ABCs of Graphs and Games 1 Amazing Asteroid 2 Bernstein’s Bijection 3 Chromatic Combat 4 Devious Dice 5 Eluding Execution 6 Flipping Fun 7 Get the Giraffe 8 Hands On References Part II Problems Inspired by Classic Puzzles 5 Solving the Tower of Hanoi with RandomMoves 1 Puzzle Variations and Preliminary Results 2 Lemmas and Proofs 3 Analysis of Puzzle 1 → 3 via Networks of Electrical Resistors 4 Discussion References 6 Groups Associated to Tetraflexagons 1 Flexagon Basics 2 A Simple Case 3 Sub-Squares Matter 4 Not All Flexes Are Equal 5 A Hamiltonian Cycle Is Not Necessary 6 Not All Flips Commute 7 Conclusion References 7 Parallel Weighings of Coins 1 Warm-up: The Multiple Pans Problem 2 A Coin’s Potential 3 Unlimited Supply of Real Coins 4 The Solution 5 More Scales 6 Find and Label 7 Lazy Coin Acknowledgments References 8 Analysis of Crossword Puzzle Difficulty Using a Random Graph Process 1 Preliminaries 2 The Puzzle as a Network 3 Difficulty Thresholds and Solution Process 3.1 Cryptic Crosswords 3.2 Relation to Network-Based Epidemic Modeling 3.3 Random Thresholds 3.4 Limitations of the Model 3.5 Structure of the Chapter 4 Completely White Grid 5 Network Properties and Simulations on Actual Puzzles 5.1 Summary Statistics of Actual Crossword Networks 5.1.1 Clustering 5.1.2 Modularity 5.1.3 Degree distribution 5.2 Simulations on Modified Networks 5.3 Simulation Results 5.4 Solution Examples 5.5 Interpretation of Results 6 Conclusion Acknowledgments References 9 From the Outside In: Solving Generalizations of the Slothouber-Graatsma-Conway Puzzle 1 Placing Pieces 2 Puzzling On Acknowledgments References Part III Playing Cards 10 Gallia Est Omnis Divisa in Partes Quattuor 1 The Ice Cream Trick and What’s Behind It 2 Reversing More, Cutting Less 3 What Have the Romans Ever Done for Us? 4 Dualism—It’s All in How You Group It References 11 Heartless Poker 1 Straight, Flush, or Full House? 2 More or Less: Frequencies of Generalized Poker Hands 3 Are Straight, Flush, and Full House Ever Tied? 4 Never a Tie: Applications of Diophantine Equations to Poker References 12 An Introduction to Gilbreath Numbers 1 Continued Fractions 2 Gilbreath Continued Fractions 3 An Analytic Perspective on Gilbreath Continued Fractions References Part IV Games 13 Tic-tac-toe on Affine Planes 1 Describing the Game 2 Affine Planes and Latin Squares 3 Winning and Drawing Strategies 4 The 2 × 2 and 3 × 3 Boards 5 Weight Functions and Larger Boards 6 Further Exploration References 14 Error Detection and Correction Using SET 1 The EndGame 2 Error Detection (and Correction?) 2.1 Error Detection 2.2 Error Correction? 3 Perfect Codes 4 Further Exploration References 15 Connection Games and Sperner’s Lemma 1 Hex and Y 2 Someone Must Win 3 Star Y 4 What is a Polygon? 5 Atoll 6 General Forms of Atoll and Begird 7 Extending Sperner’s Lemma 8 Once Again—Someone Must Win 9 Other Games Acknowledgments References Part V Fibonacci Numbers 16 The Cookie Monster Problem 1 The Problem 2 General Algorithms 3 Established Bounds 4 Naccis 5 Super Naccis 6 Beyond Naccis 6.1 The Sequence 6.2 The Theorem Acknowledgements References 17 Representing Numbers Using Fibonacci Variants 1 Zeckendorf Proofs 2 Efficiency of Number Representations 2.1 Lists of Natural Numbers 2.2 Arbitrary Reals and Continued Fractions 3 Generalizing Fibonacci Coding 4 Arithmetic 4.1 Addition 4.2 Subtraction 4.3 Multiplication 5 Conclusion References About the Editors About the Contributors Index