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PENGUIN BOOKSALICE IN PUZZLE-LANDBorn in New York City, Raymond Smullyan received his M.S.from the University of Chicago and his Ph.D. from PrincetonUniversity. He has taught at Dartmouth and Princeton and iscurrently on the faculty of Lehman College in New York. Inaddition to being a professor of mathematical logic, he has performed as a professional magician. His other books are Theoryof Formal Systems, First-Order Logic, The Tao Is Silent, WhatIs the Name of This Book?, This Book Needs No Title, TheChess Mysteries of Sherlock Holmes, The Chess Mysteries of theArabian Knights, and The Lady or the Tiger?. Mr. Smullyanhas also contributed to collective works on mathematics and haswritten for Scientific American and other magazines.

ALICE INPUZZLE-LANDA Carrollian Tale for Children Under EightyRAYMOND SMULLYANWith an introduction by Martin GardnerILLUSTRATED BY GREER FITTINGPENGUIN BOOKS

Penguin Books Ltd, Harmondsworth,Middlesex, EnglandPenguin Books, 40 West 23rd Street.New York, New York 10010, U.S.A.Penguin Books Australia Ltd, Ringwood,Victoria, AustraliaPenguin Books Canada Limited, 2801 John Street.Markham, Ontario, Canada L3R 1B4Penguin Books (N.Z.) Ltd, 182-190 Wairau Road,Auckland 10, New ZealandFirst published in the United States of America byWilliam M orrow and Company. Inc., 1982 Publishedin Penguin Books 1984Copyright Raymond M. Smullyan, 1982 Allrights reserved Reprinted 1984LIBRARY OF CONGRESS CATALOGING IN PUBLICATION DATASmullyan, Raymond M.Alice in puzzle-land.Originally published: New York: Morrow. 1982.Summary: A range of puzzles dealing with word playand logic, mathematics and philosophy, featuring Aliceand the creatures of Wonderland.1. Puzzles. [1. Puzzles] I. Carroll. Lewis, 1832-1898.Alice's adventures in Wonderland. II. Title.GV1493.S624 1984793.7383-25141 ISBN 0 1400.7056 7Printed in the United States of A merica byR. R. Donnelley & Sons Company, Harrisonburg, VirginiaSet in BaskervilleExcept in the United States of America, this book issold subject to the condition that it shall not, by wayof trade or otherwise, be lent, re-sold, hired out, orotherwise circulated without the publisher's priorconsent in any form of binding or cover other thanthat in which it is published and without a similarcondition including this condition being imposed onthe subsequent purchaser

PrefaceThis book, like Alice's Adventures in Wonderland and Throughthe Looking-Glass, is truly for readers of all ages. By this I do notmean that all of it is for every age, but for every age, some of it is forthat age. For example, the extremely elementary puzzles of ChapterFour are particularly for the very young reader who has not yetlearned algebra (and, as the Gryphon wisely says, "You don't needsno algebra!"). At the opposite extreme are the prize puzzles ofChapters Five and Nine; these will intrigue the expert as much asthe beginner. Chapter Ten has a special and unusual status.This year (1982) marks the one-hundred-fiftieth anniversary ofthe birth of Lewis Carroll, to whom this book is dedicated. I believeCarroll would have particularly enjoyed the Humpty chapter, whichdeals with paradoxes (one of Carroll's favorite themes), but in theinimitable manner of Humpty Dumpty. This chapter was great funto write (as were all the other chapters!). Indeed, the whole projectof re-creating the spirit of Carroll's writings has been a delight frombeginning to end.My heartfelt thanks go to Greer Fitting, for all the lovel yillustrations; to Maria Guarnaschelli, for her excellent editorship;and to Iver Kern, who carefully went through the entire manuscriptand provided a host of helpful suggestions.—RAYMOND SMULLYANElka Park, New YorkJanuary 1, 1982v

ContentsvPREFACEINTRODUCTIONviiiPart I—Wonderland Puzzles1.2.3.4.5.WHICH ALICE ?3WHO STOLE THE TARTS ?WHO IS MAD?720THE GRYPHON AND THE MOCK TURTLETHE KING' S STORY3052Part II—Looking-Glass Logic6.7.8.9.10.11.12.1THE TWELFTH QUESTION6365TWEEDLEDUM OR TWEEDI.EDEE ?74AND THAT ' S THE BEAUTIFUL PART OF IT !85THE WHITE KNIGHT COULDN'T QUITE REMEMBERLOOKING-GLASS LOGICTHE RED KING 'S THEORYWHICH ALICE ?133SOLUTIONS TO THE PUZZLES137114127101

IntroductionRaymond Smullyan is a unique set of personalities thatincludes a philosopher, logician, mathematician, musician, magician, humorist, writer, and maker of marvelous puzzles. Because heis a skillful writer and humorist, he enjoys presenting his puzzles innarrative forms that often parody great works of popular fiction.And he does this so well that his puzzle books are, incredibly, apleasure to read even if you never try to solve a single puzzle!Ray's first puzzle book (I call him Ray because we are old friends)was titled What Is the Name of This Book? It introduced his knights(who speak only truth), his knaves (who always lie), and suchcharacters as Inspector Craig, Bellini and Cellini, Count Dracula,and Lewis Carroll's Alice and the creatures of Wonderland. Ofcourse readers who actually worked on the puzzles of this book,whatever its name, found the book doubly rewarding, and at theend they were given a remarkable bonus—an insight into KurtGödel's famous proof, the greatest of modern mathematical discoveries.Ray's first collection of original chess problems, The Chess Mysteriesof Sherlock Holmes, surrounds each problem with a pastiche aboutHolmes and Watson. So faithful are these tales to the spirit of thecanon that Sherlockians who never play a game of chess can enjoythem for the dialogue alone. A second collection of chess problems,The Chess Mysteries of the Arabian Knights, embed the problems inparodies of the tales of Scheherazade.viii

IntroductionIn the volume you now hold, Alice and her friends are back againfor a puzzle romp behind the Looking-Glass that will pleaseCarrollians as much as the first book of chess puzzles pleased BakerStreet Irregulars. Ray has done it again. His characters not only talkand behave exactly like the originals, but the book also swarms withtypically Carrollian word play, logic and metalogic problems, anddark philosophical paradoxes. In Carroll's nonsense world therewere two Alices: the imaginary one and his real-life child-friendAlice Liddell. In Ray's nonsense world there are also two Alices: afriend of Ray's, and the imaginary Alice of his first book. Carrollwould have loved them both. And he would have been delighted byRay's looking-glass package that unwraps itself only when you try towrap it, and a hundred other whimsies that Carroll might havethought of himself if he had been capable of dreaming upRaymond Smullyan.As always in Ray's books, curious metaphysical questions have away of catching you by surprise. For example, when HumptyDumpty tells Alice she should think of everything, Alice sensiblydeclares this to be impossible."I never said you could," Humpty replies. "I merely said youshould.""But is it reasonable to say that I should do something that Icannot do?""That is an interesting problem in Moral Philosophy," answersHumpty, "but that would take us too far afield."It would indeed! Ray does not tell you, but Humpty has raised afamous problem known as Hintikka's paradox, after Jaako Hintikka, a leader of a fashionable new school of "possible worlds"philosophers. Is it proper to call morally wrong something a personcannot do? Hintikka has a notorious argument designed to show itis wrong to try to do something impossible. There is now a largeliterature on this strange question, which belongs to a type of modallogic called deontic logic. We learned from Carroll that Humpty isan expert on classical logic and semantics. Now we learn from Raythat the egg is also an expert on modal logic!A page or two later, Humpty bewilders Alice with an amazingone-sentence version of another famous paradox that goes underix

ALICE IN PUZZLE- LANDsuch names as the "unexpected examination" and the "unexpectedhanging." (You can read about it in the first chapter of my book,The Unexpected Hanging.) Humpty is not sure whether his elegantcompression of this puzzle is a genuine paradox or not, and youwon't be either after you understand it. As Humpty exclaims,"That's the beautiful part of it!"In his chapter about the White Knight, Carroll tells us: "Of all thestrange things that Alice saw in her journey Through the LookingGlass, this was the one that she always remembered most clearly.Years afterwards she could bring the whole scene back again, as if ithad been only yesterday. . . ."Ray has not forgotten. "Of all Alice's puzzle-adventures in theLooking-Glass," he begins Chapter Nine, "the ones that follow werethose she remembered most vividly. For years after, she kept tellingher friends these fascinating and unusual puzzles." Yes, and youwould swear it is Carroll's own White Knight who has tumbled offhis horse into Ray's pages.At the close of Carroll's second Alice book, Alice wonders if shehas dreamed about the Red King, or if she is only a sort of thing inthe Red King's dream. In his last two chapters, Ray weaves brilliantpuzzle themes around the act of dreaming. His book ends with theRed King presenting Alice with a question about dreams that is soconfusing and so deep that, as Carroll did, Ray wisely leaves itunanswered.No one can read this book, or any of Ray's books, withoutbecoming more aware of the mystery of being, of the difficulty ofdistinguishing what is true from what is false, or what is real fromwhat is unreal. That's the beautiful part of them. And you close thisbook knowing that Ray has shown you only a small part of thefantastic puzzle tricks he has up his conjuror's sleeves, that he isspeaking through the Duchess when she says, "As to confusingpuzzles, these are nothing compared to some I could tell you if Ichose!"—MARTIN GARDNERHendersonville, North Carolinax

WONDERLANDPUZZLES

Which Alice?It all started at Alice's birthday party. Not the Alice inWonderland, but my friend Alice. How the other Alice entered thestory will soon be apparent. Of course Alice's younger brother,Tony, was there, as well as her friends Michael, Lillian, and severalothers.After many games and magic tricks, the whole company wantedto hear some logic puzzles."Here's a nice one," I said. "There are two identical twins. One ofthem always lies and the other one always tells the truth.""What are their names?" asked Tony."One of them is named John," I replied."Such a common name!" exclaimed Michael. "It seems that justabout every Tom, Dick and Harry is called John!"I could not help being somewhat puzzled by this remark."What is the name of the other brother?" asked Tony."I don't remember," I replied."Why don't you remember?" asked Michael."I have no idea why I don't remember," I answered, "and thename of the other brother doesn't matter.""Is John the one who lies, or is it his brother?" asked Lillian."Good question," I answered, "but unfortunately no one knowswhether it is John or his brother who lies.""Then what is the problem?" asked Alice."The problem is this: Suppose you meet the two brothers and you3