Jason Rosenhouse

The mathematics of various entertaining subjects : research in recreational math

Foreword by Raymond Smullyan vii Preface and Acknowledgments x PART I VIGNETTES 1 Should You Be Happy? 3 Peter Winkler 2 One-Move Puzzles with Mathematical Content 11 Anany Levitin 3 Minimalist Approaches to Figurative Maze Design 29 Robert Bosch, Tim Chartier, and Michael Rowan 4 Some ABCs of Graphs and Games 43 Jennifer Beineke and Lowell Beineke PART II PROBLEMS INSPIRED BY CLASSIC PUZZLES 5 Solving the Tower of Hanoi with Random Moves 65 Max A. Alekseyev and Toby Berger 6 Groups Associated to Tetraflexagons 81 Julie Beier and Carolyn Yackel 7 Parallel Weighings of Coins 95 Tanya Khovanova 8 Analysis of Crossword Puzzle Difficulty Using a Random Graph Process 105 John K. McSweeney 9 From the Outside In: Solving Generalizations of the Slothouber-Graatsma-Conway Puzzle 127 Derek Smith PART III PLAYING CARDS 10 Gallia Est Omnis Divisa in Partes Quattuor 139 Neil Calkin and Colm Mulcahy 11 Heartless Poker 149 Dominic Lanphier and Laura Taalman 12 An Introduction to Gilbreath Numbers 163 Robert W. Vallin PART IV GAMES 13 Tic-tac-toe on Affine Planes 175 Maureen T. Carroll and Steven T. Dougherty 14 Error Detection and Correction Using SET 199 Gary Gordon and Elizabeth McMahon 15 Connection Games and Sperners Lemma 213 David Molnar PART V FIBONACCI NUMBERS 16 The Cookie Monster Problem 231 Leigh Marie Braswell and Tanya Khovanova 17 Representing Numbers Using Fibonacci Variants 245 Stephen K. Lucas About the Editors 261 About the Contributors 263 Index 269

Fuzzy Knights and Knaves

Summary Puzzles about knights, who only make true statements, and knaves, who only make false statements, have long been a mainstay of classical logic. They are valuable not just as recreational puzzles, but as a pedagogical device for exploring fundamental issues in logic. However, the possibilities for puzzles based on nonclassical logics have been mostly unexplored. In this paper we consider knight/knave dialogs based on fuzzy logic, in which a truth value can be any real number between zero and one inclusive. Following the example of classical logic, our purpose is both recreational and educational.

The Deal or No Deal Problem

Monty Small Problem: In this variant, the host is only somewhat tired. If he has a choice of doors to open, then he has a small probability p of opening the largest number available door, otherwise (with probability 1-p) he opens the smallest number available door. What is the probability that you will win the car if you then switch to the third door? (The case p = 112 is the original problem, while p = 0 is Monty Crawl.) In this case, if you select Door #1, the probabilities that the host will open Door #3 are respectively p, 1, and 0. Hence, in this case, the probability of winning if you switch is 11 (l + p). [Exercise: If the host had instead opened Door #2, this probability would instead be 11 (2p).]