what is the name of this book (gnv64)

7/30/2019 What is the Name of This Book (Gnv64)

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now inroduc Profssor Smullyan whowill prov you ha ihr h dosnxist or you dont xis, ut you ontnw which

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Wich is bettr, etrnal happiness or aham sandwich? I would apar thaetrnal happiness s betr, but his isrealy not so! Aftr all nothing is btrhan etrnal happiness, and a hamsandwich is cerinly betr than noth-

ing. Therefor a ham sandwich sbetr tan etrn happiness

A train eaves rom Boston to NewYork An hour later a trn leaves fromew York Bston. Te two trains aregoing at exactly the same speed Wichtrain be nearer to Boston when

they meet?

man was looking at a porrait Wenasked whose potrait he replied:"Brothers and sistrs have I none butis man's fathr s my faers sonWhos cre ws h oong at?

f olms ch as hs inrigu youn flx your an wh his nrainng mnagr of ovr indndingogal ur aradox and divrons Pnd in a livy cnraionas wh ancdos an whimsca llusraion inrrs hroghou his fasinaing gro of riddl hallng yor

rasoning ow ad omo nginning wi fnfild mony

ri and casi rain asrs whdvish nw ws aymond Smulyanhigy rscd Prfssor of Mhmaa gi a prosional magian

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Also y Raymond M Smullyan

THEORY OF FORMAL SYSTEMSFIRST ORDER LOGICTHE TAO IS SENT


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Wat Is the Name ofThi Book?-The Riddle of Draculaand Other Logical Puzzles, y Raymon M SmllyanCopyght y Raymon M Smllyan

All ghts eseve o pat of ths oo may eepoe n any fom o y any means eeptfo the nlson of ef qotatons n a evewwthot pemsson n wtng fom the plshePnte n the Unte States of Amea

PenteHall nteatonal n ononPenteHall of Astala Pty t SyneyPenteHall of Canaa t ToontoPenteHall of na Pvate t ew DelhPenteHall of Japan n ToyoPenteHall of Sotheast Asa Pte t SngapoeWhtehall Boos mte Wellngton ew Zealan

L f C C Smllyan Raymon M

What s the name of ths Pzzles . Ttle

GS SB


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Dedicated toLinda Wetze and Joseph Bevandowhose wise counses have been invauabe


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ott

art One LOGICAL RECREATIONS

Fooled? 32 uzzles nd Monkey Tricks 7

Solutons 1 4 Knights nd Knves 20

Solutons 26 lice i n the Forest of Forgetfulness 36

Solutons 46

art o _ ORTIAS CASETS AND

OTHER MYSTERIES The Myster of ortis Cskets 5 5

Solutons 62 From the Files ofnspector Crig 67

Solutons 74 How to void Werewolvesnd Other

rcticl Bits of dvice 2Solutons 90

ogic uzzles 99Solutons 1 1 0

9 Bellini o r Cellini? 1 1 Solutons 124

art Three EIRD TALES

The slnd of Bl

Solutons The slnd of Zombies

Solutons2 s Drcul Still live?

Solutons

135

1 42149

153

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art Four LOGIC IS A SLENDOREDTHING

ogic nd ife How to rove nything From rdox to Truth

Solutons Gdels Discovery

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My Ths to First I wish to thank my friends Robert and Ise Cowen andtheir tenyearoddaughter Lenore who went through thismanuscript together and provided many hepfu sugges-tions (Lenore incidentay suspected a aong the trueanswer to the key question of Chapter 4 Does Tweededooreay exist or is he merey a fabrication of Humpty

Dumpty?)I am gratefu to Greer and Mevin Fitting (authors ofthe charming and usefu booknse of Smple Thngs) fortheir kindy interest in my work and for having caed it to theattention of Oscar Coier of PrenticeHa I aso thinkMevin shoud be thanked for actuay appearing in thisbook (thereby refuting my proof that he coudnt appear!)

It was a peasure working with Oscar Coier and others

at PrenticeHa Mrs Iene McGrath who first copyeditedthe text made many suggestions which I have gratefuyadopted I thank Dorothy Lachmann for her expert handingof production detais

I wsh to again mention my two dedicatees JosephBevando and Linda Wetze who have been heart and souwith this book from its very inception

My dear wife Banche has heped me with many a

query It is my hope that this book wi enabe her to decidewhether she is married to a knight or a knave


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10Foo?

1 . Was I Fooled? My introduction to ogic was at the age of six It happenedthis way On Apri 1 1925 1 was sick in bed with grippe oru or something the moing my brother Emie (tenyears my senior) came into my bedroom and said "WeRaymond today is Apri Foo's Day and 1 wi foo you asyou have never been fooed before! 1waited a day ong forhim to foo me but he didn't Late that night my motherasked me "Why don't you go to seep? 1 repied "I'mwaiting for Emie to foo me My mother turned to Emieand said "Emie wi you pease foo the chid! Emie thentued to me and the foowing diaogue ensued

Emle / o you expected me to foo you didn't you?Raymond / Yes

Emle / But 1 didn't did I?Raymond / NoEmle / But you expected me to didn't you?ymond / YesEmle / o 1 fooed you didn't I !

We 1 reca ying in bed ong after the ights were tuedout wondering whether or not 1 had reay been fooed On

the one hand if 1 wasn't fooed then 1 did not get what 1

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epected hece I was fooed (This was Emie s argumentBut ith equa reason it can be said that l was fooed thenI dd get what I epected so then in what sense was Ifooed o was I fooed or wasnt I?

I sha not answer this puzze now we sha return to it

in one form or another severa times in the course of thisbook It embodies a subte principe which sha be one ofour maor themes

Ws i A reated incident occurred many years ater whe I was agraduate student at the University of Chicago I was a pro-

fessiona magician at the time but my magic business wassow for a brief period and I had to suppement my incomesomehow I decided to try getting a ob as a saesman Iappied to a vacuum ceaner company and hd to take anaptitude test One of the questions was "Do you obect toteing a itte ie every now and again? Now at the time Idefinitey dd obectI particuary obect to saesmenying and misrepresenting their products But I thought to

mysef that I truthfuy voiced my obection then Iwoudnt get the ob Hence I ied and said "NoRiing back home after the interview I had the fo-

owing thoughts I asked mysef whether I obected to the ieI had given to the saes company My answer was "NoWe now since I didn't obect to that particuar ie then itfoows that I don't obect to a ies hence my "No answeron the test was not a ie but the truth!

To this day it is not quite cear to me whether I wasying or not I guess ogic might require me to say that I wasteing the truth since the assumption that I was ying eadsto a contradiction o ogic requires me to beieve I wasteing the truth But at the time I sure felt as though I wasying!

peaking of ying I must te you the incident of Bertrand

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Russe and the phiosopher G E Moore Russe des-cribed Mooreas one of the most truthfu peope he had evermet He once asked Moore "Have you ever ied? oorerepied "Yes In describing this ncident Russe wroteI think this is the ony ie Moore ever tod!

e incident of my eperience with the saes companyraises the question of whether it is possibe for a person toie without knowing it I woud answer "No To me, ngmeans making a statement, not which s fase, but which oneeleves to be fase Indeed a person makes a statementwhich happens to be true, but which he beieves to be fase,then I woud say he is teing a ie

I read of the foowing incident in a tetbook onabnorma psychoogy The doctors in a menta instittionwere thinking of reeasing a certain schizophrenic patientThey decided to give him a iedetector tst One of thequestions they asked him was, Are you Napoeon? Herepied, No The machine showed he was ying

I aso read somewhere the foowing incident showing how

animas can sometimes dissimuate An eperiment wasconducted with a chimpanzee in a room in which a bananawas suspended by a string from the center of the ceiingThe banana was too high to reach The room was emptyecept for the chimp, the eperimenter, the banana andstring, and severa wooden boes of various sizes Thepurpose of the eperiment was to determine whether thechimp was cever enough to make a scaffoding of the boes,

cimb up, and reach the banana What reay happened wasths The eperimenter stood in the corner of the room towatch the proceedings The chimp came over to the coerand aniousy tugged the eperimenter by the seeve indi-cating that he wanted him to move owy the eperimenterfoowed the chimp When they came to about the center ofthe room, the chimp suddeny umped on his shouders andgot the banana

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3 The Joke Ws on MeA feow graduate student of mne at the Unversty ofChcago ha two brothers, aged s and eght I was afrequent stor to ther house and often dd trcks for the

chdren One dayI came and sad, " I have a trck n whch Icoud t you both nto ons To my surprse, one ofthem sid, "Okay, tu us into ions I repied, "We, uh,reay, uh I shoudnt do that, because there is no way Icoud tu you back agan The itte one said, "I dontcare I want you to tu us into ions anyway I repied,"No, reay, theres no way I can tu you back The oderone shouted, "wantyou to turn us nto lons The tte onethen asked, "How do you tu us into ions? I repied, "Bysaying the magic words One of them asked, "What are themagic words? I repied, "If I tod you the magic words, Iwoud be saying them, and so you woud tu into ionsThey thought about this for a whie, and then one of themasked, Arent there any magic words which woud bring usback? I repied Yes, there are, bu t the trou be is this Isaid the first magic words, then not ony you two but every-body in the wordincuding mysefwoud tu into aion And ions cant tak so there woud be no one eft tosay the other magic words to bring us back The oder onethen said, "Write them down! The itte one said, "But Icant read! I repied, "No, no, writing them down is out ofthe question even if they were wrtten down rather thansaid, everyone in the word woud sti turn into a ionThey said, "Oh

Aut a week ater I met the eightyearod, and hesaid, "muyan, there s something Ive been wanting to askyou someing which has been puzzing me I repied," Yes? He said " How dd you ever learn the magc words?

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u/ ' Ps ndMonky ck

SOM GOOD OD-TMRS

We wi start with some good odtime puzzes which haveamused many a generation ome of these, many of youaready know, but even for those in the know, I have a fewnew wrinkes

ose Pcture m ookg t? is puzze was etremey popuar during my chidhood,but today it seems ess widey known The remarkabe thingaut this probem is that most peope get the wronganswer but insist (despite a argument) that they are rightI reca one occasion about 50 years ago when we had somecompany and had an argument about this probem whichseemed to ast hours, and in which those who had the rightanswer ust coud not convince the others that they were

right The probem is thisA man was ooking at a portrait omeone asked him,"Whose picture are you ooking at? He repied "Brothersand sisters have I none, but this man's father is my father sson ("This mans father means, of course, the father ofthe man in the picture)

Whose picture was the man ooking at?

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5 uppose, the above situation, the man had insteadanswere " Brothers and sisters have I none, but this mansson my fathers son Now whose icture the man

ooking at?

t Hppes f esstble Cobll Hts mmovble Post? This another probem from my chidhood which I ikevery much By an irresistibe cannonba we sha mean acannonba which knocks over everything in its way By an

immovabe post we sha mean a post which cannot beknocked over by anything o what happens if an irresisble cannonball hits an immable post?

7. ____________The following is a very simple problem which many of youknow Twentyfour red socks and 24 ble socks are lying ina drawer in a dark room What is the minimum number ofsocks I must take out of the drawer which will guaranteethat I have at least two socks of the same color?

8A new twst on the above problem uppose some bluesocks and the same number of red socks are in a draweruppose it tu out that the minimum number of socks Imust pick in order to be sure of getting at least one pair ofthe same color is the same as the minimum number I mustpick in order to be sure of getting at least two socks ofdifferent colors How many socks are in the drawer?

9.Here a wellknown logic puzzle Given that there are more

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inhabitants of New York City than there are hairs on thehead of any inhabitant and that no inhabitant is totallybald does it necessarily follow that there must be at leasttwo inhabitants with eactly the same number of hairs?

Here is a little variant of this problem: In the town of

Podunk the following facts are true:

(1) No two inhabitants have eactly the same number ohairs

(2 No inhabitant has eactly 518 hairs.() There are more imhabitants than there are hairs on the

head o any one inhabitant

What is the largest possible number of inhabitants ofPodunk?

0 Who Ws the Murderer? This story concerns a caravan going through the aharadesert One night they pitched tents Our three principlecharacters are A B and C A hated C and decided tomurder him by putting poison in the water of his canteen

(is would be Cs only water supply) uite independentlyof this B also decided to murder C so (without realizingthat C s water was already poisoned) he drilled a tiny holein C s canteen so that the water would slowly leak out As aresult several days later C died of thirst The question iswho was the murderer A or B? According to one argumentB was the murderer since C never did drink the poison putin by A hence he would have died even if A hadnt poisoned

the water According to the opposite argument A was thereal murderer since Bs actions had absolutely no effect onthe otcome; once A poisned the water C was doomedhence A would have died even if B had not drilled the hole

Which argument is correct?

At this point Ill tell you the oke of a woodchopper from theMiddle East who came looking for a ob at a lumber camp

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The foreman said I dont know f this the kind of ob youwant; here we chop trees The woodchopper said "Thatsprecisely the srt of work I do The foreman replied"Okay heres an aelets see how long it takes you tochop down this tree here The woodchopper went over to

the tree and felled it with one blow The foreman amazedsaid " Okay try that big one over there The woodchopperwent over to the treebff bamin wo strokes the treewas down " Fantastic ! cried the foreman " Of course youare hired but how did you ever lea to chop like that?" Oh he replied "Ive had plenty of practice in the aharaForest The foreman thought for a moment "You meanhe said " the ahara Desert "Oh yes replied the wood-

chopper "it is now!

other egl Puzzle Two men were being tried for a urder The ury found oneof them guilty and the other one not guilty The udgeturned to the guilty one and said: "This is the strangest caseI have ever come across! Though your guilt has been estab-

lished beyond any reasonable doubts the law compels meto set you free How do you eplain this?

Two ds Two American Indians were sitting on a loga big Indianand a little Indian The little Indian was the son of the big

Indian but the big Indian was not the father of the littleIndianHow do you eplain that?

3 The Clock Tht Stopped Here a cute simple oldtime puzle A man owned nowatch but he had an accurate clock which however he

sometimes forgot to wind Once when this happened he

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went to the house of a friend passed the evening with himwent back home and set his clock How could he do thiswithout knowing beforehand the length of the trip?

Problem of the Ber

The interesting thing about this problem is that manypeople have heard it and know the answer but their reasonsfor the answer are insufficient o even if you think youknow the answer be sure and consult the solution

A man is 100 yards due south of a bear He walks 100yards due east then faces due north fires his gun duenorth and hits the bear

What color was the bear?

MONKEY TRICKS

At first I was undecided what title to give this book; Ithought of "Recreational Logic "Logical Recreations andDiversions and others but I was not too satisfied TheIdecided to consult Rogets Thesaurus I looked in the indeunder "Re creations and was directed to section40 entit-led "Amusement There I came across such choice itemsas "fun "frolic "merriment ollity "heyday "ocos-i "drollery "buffoonery "tomfoolery "mummery the net paragraph I came across "play "play at"romps "gambols "pranks "antic "lark "gam-ade monkey trck. Well when I saw "monkey trick Iughed ad said to my wife "Hey maybe I should call thisook "Monkey Tricks D elightful that title is howevert would have been misleading for this book a whole sinceost portions can hardly be described as monkey tricksBut the title serves perfectly for the items of this section ashe reader will soon realize

ls mne

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5 Problem o the Two CoTwo Amercan coins add up to thirty cents yet one of themis not a nickel What coins are they?

Those of you who know anything about Catholicism do youhappe to know f the Catholic Church allows a man tomarr his wdows sister?

17. A man lived on the twentyffth oor of a thirtystory apartment building Every moing (ecept aturdays and undays he got into the elevator got off at the ground oorand went to work In the evening he came home got intothe elevator got off at the twentyfourth oor and walkedp one ight

Why did he get off at the twentyfourth oor instead ofthe twentyfifth?

18. A QuestionofGrammar._Those of you who are interested in uesons of good grammacal usage is it more coect to say the yolk s white othe yolk are white?

19. Rte-Tme Problem A train leaves from Boston to New York An hour later atrain leaves from New York to Boston The two trains aregoing at eactly the same speed Which train will be nearerto Boston when they meet?

20. A Queston ofSlopen a certain house the two halves of the roof are uneually

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pitched; one half slopes downward at an angle of 60 andthe other half at an angle of 70 uppose a rooster lays anegg right on the peak On which side of the roof would theegg fall?

How My 9s? A certain street contains 100 buildings A signmaker iscalled to number the houses from 1 to 100 He has to ordernumerals to do the ob Without using pencil and paper canyou figure out in your head how many 9 s he will need?

The Rcetrck

A certain snail takes an hour and a half to crawl clockwisearound a certain racetrack yet when he crawls counter-clockwise around that same racetrack it takes him onlyninety minutes Why this discrepancy?

3. Questo of tetol w an airplane crashes right on the border of the Unitedtates and Canada in which country would you bury thesurvivors?

How Do You xpl Ths? certain Mr mith and his son Arthur were driving in a carThe car crashed; the father was killed outright and the son

Arthur was critically inured and rushed to a hospital Theold surgeon took a look at him and said "I cant operate onhim; he is my son Arthur!

How do you eplain this?

25. And Now! _And now what is the name of this book?

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SOLUTIONS

A remarkably large umber of people arrive at the wrongnswer that the man is looking at his own picture They putthemselves in the place of the man looking at the pictureand reason as follows: "ince I have no brothers or sistersthen my fathers son must be me Therefore I am looking ata picture of myself

The first statement of this reasoning absolutely corret; I have neither brothers nor sisters then my fathersson indeed myself But it doesnt follow that "myself is

the answer to the problem the second clause of theproblem had been "this man my fathers son then theanswer to the problem would have been "myself But theproblem didnt say that; it said "this mans father is myfather s son From which it follows that this mans father ismyself (since my fathers son is mysel ince this mansfather myself then I am this mans father hence this manmust be my son Thus the correct answer to the problem is

that the man is looking at a picture of his son the skeptical reader is still not convinced (and Im

sure many of you are not!) it might help if you look at thematter a bit more graphically as follows:

(1 ) This mans father is my fathers son

ubstituting the word "myself for the more cumbersome

phrase "my fathers son we get

(2) This mans father is myself

Now are you convinced?

5

The answer to the second problem "Brothers and sisters

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have I none but this mans son is my fathers son is thatthe man is looking at a picture of his father

6.

The given conditions of the problem are logically contradictory It is logicaly impossible that there can eist both aniesistible canonba ad an immovable post an ie-sistible canonball should eist then by defnition it wouldknock over any post in its way hence there couldn t eist animmovable post Alteatively there eisted an immov-able post then by definition no cannonball could knock itover hence there could not eist an iesistible canonball

Thus the eistence of an irresistible cannonball is in itselfnot logicaly contradictory nor is the eistence of an immovable post in itself contradictory; but to assert they botheist is to assert a contradiction

The situation is not really very different than had Iasked you "There are two people John and Jack John istaller than Jack and Jack is taller than John Now how doyou eplain that? Your best answer would be "Either you

are lying or you are mistaken

7.The most cmmon wrong answer is 25 the problemhad been "What is the smallest number I must pick inorder to be sure of getting at least two socks of derentcolors then the correct answer would have been 2 5 Butthe problem calls for at least two socks of the same color sothe correct answer is "ee I pick three socks theneither they are all of the same color (in which case I cer-tainy have at least two of the same color) or else two are ofone color and the third is of the other color so I then havetwo of the same color

8

The answer is four

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9 the first problem the answer is "ye. For definitenessasume thee are eactly 8 million people in New York. each inhabitant had a different number of hairs then ther

would be 8 million dfferent positive whole numbers eachless than 8 million. This is impossible!For the second problem the answer is 518! To see

this suppose there were more than 518 inhabitantssay520. Then there would have to be 5 20 distinct numbers allless than 520 and none of them equal to 518. This is im-possible; there are eactly 520 distinct numbers (includingzero) less than 520 hence there are only 51 9 numbers other

than 518 which are less than 520.Incidentally one of the inhabitants ofPodunk must bebald. Why?

0I doubt that either argument can precisely called " cor-rect or " ncorrect. Im afraid that in a problem of thistype one mans opinion is as good as anothers. My per-sonal belief is that if anybody should be regarded as thecause of Cs death it was A. Indeed I were the defenseattorney of B I would point out to the court two things:(1) removing poisoned water from a man is in no sensekilling him; (2) anything B s actions probably served onlyto prolong As lfe (even though this was not his intention) since death by poisoning is likely to quicker than deathby thirst.

But then As attoey could counter "How can any-one in right mind convict A of murder by poisoning when fact C never drank any of the poison? So this problem isa real puzzler! It is complicated by the fact that it can belooked at from a moral angle a legal angle and a purlyscientific angle involving the notion of causation. From amoral angle obviously both men were guilty of intent tomurder but the sentence for actual murder is far more

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drastic egarding the legal angle I do not know how thelaw would decideperhaps different juries would decidedifferently As for the scientfic aspects of the problem thewhole notion of causaon presents many problems I thinka whole book could be written on this puzzle

The two defendants were Siamese twins

The big Indian was the mother of the little Indian

3. When the man left his house he started the clock and jotteddown the time it then showed When he got to his friendshouse he noted the time when he arrived and the time whenhe left Thus he knew how long he was at his friends house.When he got back home he looked at the clock so he knew

how long he had been away from home Subtracting fromthis the time he had spent at his friends house he knewhow long the walk back and forth had been. Adding half ofthis to the time he left his friend s house he then knew whatme it really was now

The bear must be white; it must be a polar bear The usualreason given is that the bear must have been standing at theNorth Pole Well this indeed is one possibility but not theonly one rom the North Pole all directions are south so fthe bear is standing at the North Pole and the man is 100yards south of him and waks 100 yards east then when hefaces north he will be facing the North Pole again But as Isaid this is not the only solution Indeed there is an infinite

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number of solutions It could be for eample that the manis very close to the South Pole on a spot where the Polarcircle passng through that spot has a circumference ofeactly 100 yards and the bear is standing 1 00 yards northof him Then if the man walks east 100 yards he would walk

right around that cirle and be right bak at the point hestarted from So that is a second solution But again theman could be still a little closer to the South Pole at a pointwhere the polar circle has a circumference of eactly 50yards so if he walked east 100 yards he would walk aroundthat little circle twice and be back where he started Or hecould be still a little closer t the South Pole at a pointwhere the circumference of the polar circle is onethird of

100 yards and walk east around the circle three times andbe back where he started And so frth for any positiveinteger n. Thus there is really an infinite number of placeson the earth where the given conditions could be met

Of course in ay of these solutions the bear is suffi-ciently close to either the North Pole or the South Pole toqualify a polar bear There is of course the remote pos-sibility that some mischievous human being deliberately

transported a brown bear to the North Pole just to spite theauthor of this problem

5 The answer a quarter and a nickel One of them (namelythe quarter) is not a nickel

16. How can a dead man marry anybody?

17. He was a midget and couldnt reach the elevator button fothe twentyffth oor

Someone I know (who is obviously not very good attelling okes) once told this joke at a party at which I was

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present He began thus: "On the twentyfifth oor of anapartment building lived a midget "

8

Actually the yolk is yellow

19 Obviously the two trains will at the same distance fromBoston when they meet

Roosters dont lay eggs Twenty

There is no discrepancy an hour and a half is the same asninety minutes

3 One would hardly wish to bury the survivors!

e surgeon was Arthur Smiths mother

Unfortunately I cannot right now remember the name ofthis book but dont worry m sure it will come to mesooner or later

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Kn n naes

A THE ISND OF KNIGHTS ND AVES

There is a wide variety of puzzles about an island in whichcertain inhabitants called "knights always tell the truthd others called "knaves always lie It is assumed thatevery inhabitant of the island is either a knight or a knave I

shall start with a wellknown puzzle of this type and thenfollow it with a variety of puzzles of my own

6According to this old problem three of the inhabitantsAB and Cwere standing together in a garden A strangerpassed by and asked A "Are you a knight or a knave? A

answered but rather indistinctly the stranger could notmake out what he said The stranger than asked B "Whatdid A say? B replied "A said that he is a knave At thispoint the third man C said "Dont believe ; he is lying!

The question is what are B and C?

When I came upon the above problem it immediately

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struck me that C did not really function in any essentialway; he was sort of an appendage. That is to say themoment B spoke one could tell without C s testimony thatB was lying (see solution). The following variant of theproblem eliminates that feature.

Suppose the stranger instead of asking A what he isasked A "How many knights are among you? Again Aanswers indistinctly. S the stranger asks B "What did Asay? B replies "A said that there is one knight among u.Then C says "Dont believe B; he is lying!

Now what are B and C?

28 this problem there are only two people A and B each ofwhom is either a knight or a knave. A makes the followingstatement: "At least one of us is a knave.

What are A nd B?

290 Suppose A says "Either I am a knave or B is a knight.What are A and B?

3. Suppose A says "Either I am a knave or else two plus twoequals five. What would you conclude?

3 1 Again we have three people A B C each of whom is eithera knight or a knave. A and B make the following statements:

A All of us are knaves. Eactly one of us is a knight.

What are A B C?

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2 .Suppose instead A and B say the following:

A All of us are knaves

Eactly one f us is a knaveCan it be determined what B is? Can it be determned whatC is?

3 Suppose A says "I am a knave but B isnt

What are A and B?

34 ._______We again have three inhabitants A B and C each of whomis a knight or a knave Two people are said to be of the samepe if they are both knights or both knaves A and B makethe following statements:

A B is a knave A and C are of the same type

What is C?

Again three people A B and C A says B and C are of thesame type Someone then asks C "Are A and B of thesame type?

What does C answer?

36 dveture of Mne This an unusual puzzle; moreover it taken from reallife Once when I visited the island of knights and knaves I

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came across two of the inhabitants resting under a tree. Iasked one of them Is eiher of you a knight? He re-sponded and I knew the answer to my question.

What is the person to whom I addressed the questionis he a knight or a knave; And what is the other one? I can

assure you I have given you enough information to solvethis problem.

3 Suppose you visit the island of knights and knaves. Youcome across two of the inhabitants lazily lying in the sun.You ask one of them whether the other one is a knight and

you get a (yesorno) answer. Then you ask the second onewhether the first one is a knight. You get a (yesorno)answer.

Are the two answers necessarily the same?

38 dwd dw? time you come across just one inhabitant lazily lying inthe sun. You remember that his first name is either Edwinor Edward but you cannot remember which. So you askhim his first name and he answers "Edward.

What is his first name?

KNIGHTS KNAVES NORMALS

equally fascinating type of problem deals with threetypes of people: knights who always tell the truth; knaveswho always lie; and normal people who sometimes lie andsometimes tell the truth. Here are some puzzles of mineabout knights knaves and normals.

39 We are given three people ABC one of whom is a kniht

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one a knave and one normal (but not necessarly thatorde). They make the followng statements

I am normal. That is true

C I am not nomalhat are A, B, and C?

ere is an unusual one Two people A and B, each of whomis either a knight or knave or a normal make the followng

satement B is a knight. A is not a knight.

rove that at least one of them is telling the truth ut is nota knight.

his time A and B say the ollowng

B is a knight. A is a knave

ove that either one of them is teling the truth but is no aknight or one of them is lying but is not a knave.

tter of Rn n this island of knights knaves and normals knaves aresaid to be of the lowest rank normals of middle rank andknights of highest rank.

I am particularly partial to the followng problemGiven two people A,B, each of whom is a knight a knave ora normal they make the followng statements

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A: am of lower rank than B.B That's not true!

Can the ranks of either A or B be determined? Can it bedetermined of either of these statements whether it is true

or false? Given three people A,B, C, one of whom a knight, one aknave, and one normal A,B, make the following statements

A: B is of higher rank than C.B C is of higher rank than A

Then C is asked "Who has higher rank, A or B? What doesC answer?

c THE ISLD OF AHAVA

The island of Bahava a female liberationist island; hence

the women are also called knighs knaves or noals Anancient empress of Bahava once, a whimsical moment,passed a curious decree that a knight could marry only aknave and a knave could marry only a knight. (Hence anormal can may only a normal) Thus, given any marriedcouple, either they are both normal, or one of them is aknight and the other a knave

The net three stories all ake place on the island of

Bahava

44We first consider a married couple, Mr and Mrs A Theymake the following statements

Mr A / My wife is not normalMrs A / My husband is not normal.

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What are Mr. and Mrs. A?

45 . Suppose, instead, tey had said

Mr A / My wfe is normaMr A / My husband is norma

Would the answer have been different?

6 This problem concerns two married couples on the island ofBahava, Mr. and Mrs. A, and Mr. and Mrs. B. They arebeing interviewed, and three of the four people give thefollowing testimony

Mr A / Mr. B is a knight.M A / My husband is right; Mr. B is a knight.M B / That's right. My husband is indeed a knight.

What are each of the four people, and which of the threestatements are true?

SOLUTIONS

6 It is impossible for either a knight or a knave to say, I'm aknave, because a knight wouldn't make the false state-ment that he is a knave, and a knave wouldn't make the truestatement that he is a knave. Therefore A never did say thathe was a knave. So B lied when he said that A said that ewas a knave. Hence B is a knave. Since C said that B waslying and B was indeed lying, then C spoke the truth, hence

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is a knight. Thus B is a knave and C is a knight. (It is impos-sible to know what A is.)

The answer is the same that of the preceding problem,ough the reasoning is a bit dfferent.

The first thing to observe is that B and C must be ofopposite types, snce B contradicts C. So of these two, oneis a knight and the other a knave. Now, if A were a knight,en there would be two knights present, hence A wouldnot have lied and said there was only one. On the otherhand, if A were a knave, then it would be true that there was

eactly one knight present; but then A, being a knave,couldn't have made that true statement. Therefore A couldnot have said that there was one knight among them. So Bfalsely reported A's statement, and thus B is a knave and Cis a knight.

28Suppose A were a knave. Then the statement " At least oneof us is a knave would be false (since knaves make falsestatements) ; hence they would both be knights. Thus, if Awere a knave he would also have to a knight, which is im-possible. Therefore A is not a knave; he is a knight. There-fore his statement must true, so at least one of themreally is a knave. Since A is a knight, then B must be theknave. So A is a knight and B is a knave.

9 This problem is a good introduction to the logic of disjunc-on. Given any two statementsq the statement "eitheror q means that at least one (and possibly both) of thestatements q are true. the statement "either or q

should false, then both the statements qare false. For

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eample, I should say, "Either it rag or it issnowing, then if my statement is incorrect, i t is both falsethat it s raining and false that it is sowing.

This is the way "either/or is used in logic, and is theway it will be used thoughout this book. In daily life, it is

sometimes used this way (allowing the possibility that bothalternatives hold) and sometimes in the socalled "eclu-ve sensethat one and only one of the conditions holds.As an eample of the eclusive use, I say; "I will marryBetty or I wll marry Jane, it understood that the twopossibilities are mutually eclusivethat is, that I will notmarry both girls. On the other hand, a college cataloguestates that an entering student is required to have had

either a year of mathematics or a year of a foreign language,the college is certainly not going to eclude you you hadboth! This is the inclusive use of either/or and is theone we will constantly employ.

Another important property of the disjunction rela-on either this or that is this. Consider the statement or q (which short for either or q) Suppose the state-ment happens to true. Then if false, q must be true

(because at least one of them true, so is false, q mustbe the true one). For eample, suppose it is true that it iseither raining or snowing, but it is false that it is raining.Then it must be true that it is snowing.

We apply these two principles as follows. A made astatement of the disjunctive type "Either I am a knave or B a knight. Suppose A a knave. Then the above state-ment must be false. This means that it is neither true that A

a knave nor that B a knight. So A were a knave, then itwould follow that he not a knavewhich would be acontradiction. Therefore A must be a knight.

We have thus established that A a knight. Thereforehis statement is true that at least one of the ossibilitiesholds (1 A a knave; (2) B a knight. Since possibility (1 false (since A is a knight) then possibility (2) must be thecoect one, i. e., B is a knight. Hence A,B, are both knights.

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30.The only valid conclusion that the author of this problemis not a knight. The fact is that neither a knight nor a knavecould possibly make such a statement. If A were a knight,

en the statement that either A is a knave or that two plustwo equals five would be false, since it is neither the casethat A is a knave nor that two p us two equals five. Thus A, aknight, would have made a false statement, which impos-sible. On the other hand, A were a knave, then thestatement that either A is a knave or that two plus two equalsfive would be true, since the first clause that A is a knave istrue. Thus A, a knave, would have made a true statement,which is equally impossible.

Therefore the conditions of the problem are contra-dictory (just like the problem of the iesistible cannonballand the immovable post). Therefore, I, the author of theproblem, was either mistaken or lying. I can assure you Iwasn't mistaken. Hence it follows that I am not a knight.

For the sake of the records, I would like to testify that Ihave told the truth at least once in my life, hence I am not aknave either.

0 To begin with, A must be a knave, for he were a knight,en it would be true that all three are knaves and hencethat A too is a knave. A were a knight he would have to bea knave, which is impossible. So A is a knave. Hence hisstatement was false, so in fact there is at least one knight

among them.Now, suppose B were a knave. Then A and B would

both be knaves, so C would be a knight (since there is atleast one knight among them). This would mean that therewas exactly one knight among them, hence B's statementwould be true. We would thus have the impossibility of aknave making a true statement. Therefore B must be aknight.

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We now know that A s a knave and that B s a knight.Since B s a knight, hs statement s true, so there s exactlyone knight amog them. This knight must be B, hence Cmust a knave. Thus the answer s that A s a knave, B s aknight, and C s a knave.

t cannot be determined what B is, but it can be proved thatC s a knight.

To begin with, A must be a knave for the same reasonsas in the preceding problem; hence also there is at least oneknight among them. Now, either B is a knight or a knave.

Suppose he s a knight. Then it s true that exactly one ofthem s a knave. Ths only knave must be A, so C would aknight. So B s a knight, so s C. On the other hand, if B isa knave, then C must be a knight, snce all three can't beknaves (as we have seen). So in either case, C must be aknight.

3 To begin with, A can't a knight or hs statement would betrue, in which case he would have to be a knave. ThereforeA is a knave. Hence also his statement is false. If B were aknight, then A's statement would be true. Hence B s also aknave. So A,B are both knaves.

34.Suppose A s a knight. Then hs statement that B s a knavemust be true, so B is then a knave. Hence B's statementthat A and C are of the same type is false, o A and C are ofdfferent types. Hence C must be a knave (since A is aknight) . Thus if A s a knight, then C s a knave.

On the other hand, suppose A is a knave. Then his

statement that B s a knave s false, hence B s a knight.

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Hence B's statement is true that A and C are of the sametype This means that C must be a knave (since A is).

We have shown that regardless of whether A is aknight or a knave, C must be a knave. Hence C is a knave.

35.m afraid we can solve this problem only by analysis intocases.

Case One A is a knight. Then B,C really are of the sametype C is a knight, then B is also a knight, hence is of thesame type as A, so C being truthful must answer "Yes. If C

is a knave, then B is also a knave (since he is the same typeas C), hence is of a different type than A. So C, being aknave, must lie and say "Yes.

Case Two A is a knave. Then B,C are of different types. C is a knight, then B is a knave, hence he is of the same typeas A. So C, being a knight, must answer "Yes. C is aknave, then B, being of a dfferent type than C, is a knight,

hence is of a dfferent type than A. Then C, being a knave,must lie about A and C being of fferent types, so he willanswer "Yes.

Thus in both cases, C answers "Yes.

6 To solve this problem, you must use the information I gave

you that after the speaker's response, I knew the trueanswer to my question.Suppose the speakecall him Ahad answered

"Yes Could I have then known whether at least one ofthem was a knight? Certainly not. For it could that A wasa knight and truthfully answered "Yes (which would betruthful, since at least onenamely Awas a knight), or itcould be that both of them were knaves, in which case A

would have falsely answered "Yes (which would indeed be

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false since neither was a knight). So if A had answered"Yes would have had no way of knowng. But told youthat dd know after A's answer. Therefore A must haveanswered "No.

The reader can now easily see what A and the other

call him Bust be A were a knight, he couldnt havetruthfully answered "No, so A is a knave. Since his answer"No is false, then there is at least one knight present.Hence A is a knave and B is a knight.

3Yes, they are. they are both knights, then they will both

nswer "Yes. they are both knaves, then again they wllboth answer "Yes. one is a knight and the other a knave,en the knight will answer "No, and the knave wll alsonswer "No.

38 feel entitled, occasionlly, to a little horseplay. The vitalclue gave you was that the man was lazily lying the sun.From this it follows that he was lying in the sun. From this itfollows that he was , hence he is knave. So his name isEwin.

39 To begin wth, A cannot be a knight, because a knight wouldnever say that he is normal. So A is a knave or is normal.

Suppose A were normal. Then B' s statement would be true,hence B is a knight or a normal, but B can't be normal (sinceA is), so B is a knight. This leaves C a knave. But a knavecannot say that he is not normal (because a knave reallyisn't normal) , so we have contradiction. Therefore Acannot be normal. Hence A is a knave. Then B' s statementis false, so B must be normal (he can't be a knave since A is) .Thus A is the knave, B is the normal one, hence C is the

knight.

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0The interesting thing about this problem is that it is im-possible to know whether it is A who is telling the truth butisn't a knight or whether it is B who is telling the truth but

isn't a knight; all we can prove is that at least one of themhas that property.Either A is telling the truth or he isn't. We shall prove

(1) he is, then A is telling the truth but isn't a knight; (2) he isn't then B is telling the truth but isn't a knight.

(1) Suppose A is telling the truth. Then B really is aknight. Hence B is telling the truth, so A is t a knight. Thus A is telling the truth then A is a person who is telling the

truth but isn't a knight.(2) Suppose A is not telling the truth. Then B isn't aknight. But B must be telling the truth, since A can't be aknight (because A is not telling the truth). So in this case Bis telling the truth but isn't a knight.

We shall show that B is telling the truth then he isn't akight and he isn't telling the truth then A is lying butisn't a ave.

(1 ) Suppose B is telling the truth. Then A is a knave,hence A is certainly not telling the truth, hence B is not aknight. So in this case B is telling the truth but isn't aknight.

(2) Suppose B is not telling the truth. Then A is notreally a knave. But A is certainly lying about B, because Bcan't a knight if he isn't telling the truth. So in this case,A is lying but isn't a knave.

2 To begin with, A can't a knight, because it can't be truethat a knight is of lower rank than anyone else. Now,suppose A is a knave. Then his statement is false, hence he

is not of lower rank than B. Then B must also be a knave (for

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he weren' then A would be of lower rank than B) . S if Ais a knave, is B But this is impossible bcause B iscontradicting A, and two contradictory claims can't both befalse. Therefore the assumption that A is a knave leads to acontradiction. Therefore A is not a knave. Hence A must be

normaNow, what about B? Well, he were a knight, then A

(being normal) actually would be of lower rank than B,hence A's statement would be true, hence B's statementfalse, and we would have the impossibility of a knightmaking a false statement Thus B is no a knight. SupposeB were a knave. Then A's statement would be false, henceB's would be true, and we would have a knave making a tre

statement. Therefore B can't be a knave either. Hence B isnormaThus A and B are both normal. So also, A's statement

is false and B's statement is true. So the problem admits ofa complete solution.

3 Step : We first show that from A's statement follows thatC cannot be normal. Well, if A is a knight then B really is ofhigher rank than C, hence B must be normal and C must bea knave. So in this case, C is not norma Suppose A is aknave. Then B is not really of higher rank than C, hence B isof lower rank, so B must be normal and C must be a knight.So in this case, C again is not norma The third possiblecase is that A is normal, in which case C certainly isn't

(since only one of A, B, C is normal). Thus C is not normaStep By similar reasoning, it follows from B's state-ment that A is not normal. Thus neither A nor C is normaTherefore B is norma

Step Since C is not normal, then he is a knight or aknave. Suppose he is a knight Then A is a knave (since B isnormal) hence B is of higher rank than A. So C, being aknight, would truthfully answer, "B is of higher rank. On

the other hand, suppose C is a knave. Then A must be a

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knight, so B not of higher rank than Then C, being aknave, would lie and say, "B is of higher rank than A. Soregardless of whether C is a knight or a knave, he answersthat B is of higher rank than A.

A cannot be a knave, because then h wfe would be aknight and hence not normal, so Mr. A's statement wouldhave been true Similarly Mrs A cannot be a knave There-fore neither is a knight either {or the souse would then be aknave), so they are both normal (and both lying).

For the second problem, the answer the same. Why?

46.It turns out that all four are noral, and all three state-ments are lies

First of all, Mrs B must be normal, for f she were aknight her husband would be a knave, hence she wouldnthave lied and said he was a knight. she were a knave, herhusband would be a knight, but then she wouldn't have toldthe truth about this Therefore Mrs B is normal Hencealso Mr B is normal This means that Mr. and Mrs. A wereboth lying. Thereore neither one is a knight, and they can'tth knaves, so they are both norma

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h L Alice inth ForesO Foene

A THE LION AD THE UNICORN

e Ace etere the Forest of Forgeuless, she otorget even oly certa thgs. She ofte forgot herame a the one thg she was most lkely to forget wasthe ay of he week ow the Lo an the Ucorn werefrequent vstors to the forest These two are sgecreatures The Lon les on Moays Tuesays a We-

esays a tells the truth o the other ays of the weeke Uco o the other ha les o Thursays F-ays a Saturays but tells the truth o he other ays ofthe week

One ay Alce met the Lo a the Unco restg uner a

tree They mae the followg statements:

on / Yesteray was oe of my lyg aysUnco / Yesteray was oe of my lyg ays too

From these two statements Alce (who was a ver brght s able to euce the ay of the week What ay was t?

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48 . On another occason Alce met the Lon alone He made thefollowing two statements:

(1) led yesterday(2) le agan two days after tomorrow

What day of the week was t?

49 On what days of the week t possble for the Lon to makethe followng two statements:

(1) led yesterday(2) le agan tomorow

On what days of the week s t possble for the Lon to makethe following sngle statement: "I led yesterday and 1 willle agan tomorrow Wan! The answer s not th sameas that of the preceding problem

. TWEEDLEDUM D TWEEDLDEE

During one month the Lon and the Unco were absentfrom the Forest of Forgetfulness They were elsewhere,

busly fghtng for the crownHowever, Tweedledum and Tweedledee were fre-

quent vstors to the forest Now, one of the two s lke theLon, lying on Mondays, Tuesdays, and Wednesdays andtellng the truth on the other days of the week The otherone s lke the Unco he lies on Thursdays, Frdays, andSaturdays but tells the truth the other days of the weekAlce ddnt know whch one was lke the Lon an whch

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one was lke the Unco To make matters worse, thebrothers looked so much alke, that Alce could not even tellthem apart (except when they wore ther embroderedcollars whch tey seldom dd) Thus poor Alce found thestuaton most confusng ndeed! Now, here are some of

Alces adventures wth Tweedledum and Tweedledee

1 . One day Alce met the brothers together and they made thefollowng statements:

Frst One / Im TweedledumSecond One / Im Tweedledee

Whch one was really Tweedledum and whch one wasTweedledee?

2 On another day of that same week, the two brothers madethe followng statements:

Frst One / Im TweedledumSecond One / I that s really true, then Im Tweedle-

dee!

Whch was whch?

3 On another occason, Alce met the two brothers, and askedone of them, "Do you e on Sundays? He repled "YesThen she asked the other one the same queston What ddhe answer?

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5. On another occasion, the brothers made the followngstatements:

Frst One / (1) I lie on Saturdays(2) I lie on Sundays

Second One / I will lie tomorrow

What day of the week was it?

"5 5 . ____One day Alice came across ust one of the brothers Hemade the following statement: "I am lying today and I amTweedledee

Who was speaking?

56

Suppose, instead, he had said: "I am lying today or I amTweedledee Would it have been possible to determinewho it was?

57 .______One day Alice came across both brothers They made thefollowing statements:

Frst One / I m Tweedledum then he s TweedledeeSecond One / I hes Tweedledee then I m

weedledum

Is it possible to determine who is who? Is it possible todetermine the day of the week?

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58. A Mystery Resolved! On ts geat occason Alce resolved three grand mys-teres She cae across the two broters grnnng under atree Se hoped that on ths encounter she would find out

three thngs: (1) the day of the week (2) whch of the twowas Tweedledum 3 whether Tweedledum was lke theLon or the Uncorn n hs lyng habts (a fact she had longdesred to know!)

Well, the two brothers made the followng statements:

rst One / Today not SundaySecond One / fact, today s MondayFrst One / Tomorrow s one of Tweedledees lyng

daysSecond One / The Lon led yesterday

Alce clapped her hands n oy The problem was now com-pletely solved What s the soluton?

C. WHO OWNS THE RATTLE?

Tweedledum and TweedledeeAgreed to have a battle;

For Tweedledum sad TweedledeeHa spoled hs nce new rate

Just then flew down a monstrous crowAs black as a tarbarrel

Whch frghtened both the heroes so

They qute forgot ther quarrelOld Nursery Rhyme

"Well, well, trumphantly exclamed the Whte Kng toAlce one day, "Ive found the rattle, and Ive had it re-stored Doesnt t look as good as new?

"Yes, indeed, repled Alce, admrngly, "t looks asgood as the day t was made Even a baby couldnt tell the

dfference

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"What do you mean even a baby? cred the hteKng sternly "That not very logcal you know Of coursea baby couldnt tell the dfferenceone would hardly ex-pect a baby to do that!

"What you should have sad contnued the Kng,

somewhat more gently, "s that even a grownup couldnttell the dfferencenot even the worlds greatest rattleexpert

"Anyway contnued the Kng "well magne t sadThe mportant thng s to restore the rattle to ts rghtfulowner Wll you please do ths for me?

"Who s the rghtful owner? asked Alce"I shouldnt have to tell you that! cred the Kng

impatently"Why not? inqured Alce"Because it says qute explicitly n the rhymewhch

I'm sure you knowthat Tweedledum sad that Tweele-dee had spoled his nce new rattle, so the rattle belongs toTweedledum, of course!

"Nt necessarly replied Alce, who was n a moofor a lttle argument, "I know the rhyme well, and I beleve

it "Then whats the problem? cre the Kng, morepuzzled than ever

"Very smple, really explaned Alce "I grant thatwhat the rhyme says is true Therefore Tweedledum didndeed say that Tweedledee had spoled hs rattle Butbecause Tweedledum sad t, it does not mean that t isnecessarly true Perhaps Tweedledum sad i t on one of hs

lying days Indeed for all I know, t may be the other wayaroundmaybe t was Tweedledum who spoiled Tweedle-dees new rattle

"Oh, dear, repled the Kng dsconsolately, "I neverthought of that Now all my good intentons are waste

The poor king looked so deected Alve thought hewould cry "Never mnd, sad Alce as cheerfully as shecould "Gve me the rattle and I wll try to fnd out who is

the true owner I've had some experence wth liars and

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truthtellers around here, and 1 have gotten a lttle of theknack of how to handle them

"I hope so! repled the Kng mourfullyNow 1 sha tell you of Alces actual adventures wth

the ratte

59She took the rattle and went nto the Forest of Forget-fulness, hopng to fnd at least one of the brothers To hergreat delght, she suddenly came across both of themgrnnng under a tree She went to the frst one and sternlysad: "I want the truth now! Who really owns the rattle? He

replied, "Tweedledee owns the rattle She thought for awhle, and asked the second one, "Who are you? Herepled, "Tweedledee

Now, Alce dd not remember the day of the week, butshe was sure t was not Sunday

T whom should Alce gve the rattle?

60Alce restored the rattle to ts rghtful owner Several dayslater, the other brother broke the rattle agan Ths tme, noblack crow came to frghten the brothers, so they beganslammng and bangng away at each other Alce pcked upthe broken rattle and ran out of the forest as fast as shecould

Some tme later, she agan came acrss the Whte

Kng She thoroughly explaned the stuaton to hm"Very nterestng, repled the Kng "The most re-markable part s that although you knew to whom to gve t,we stll do not know t s Tweedledee or Tweedledumwho owns the rattle

"Very true, repled Alce, "but what do 1 do now?"No problem, repled the Kng, 1 can easly have the

rattle fxed agan

True to hs word, the Whte Kng had the rattle

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perfectly restored and gave it to Alice some days laterce went trepidly into the forest fearing that the battlemght still be on As a matter offact, the brothers had calleda temporary truce, and Alice came across ust one of themrestng wearily under a tree Alice went over to hm and

asked, "Who really owns this rattle? He quzzically re-pled, "The true owner of this rattle is lying today

What are the chances that the speaker owns the rattle?

6 _Several days later Alice again came upon ust one of thebrothers lyng under a tree She asked the same question,

and the reply was, "The owner of this rattle is telling thetruth todayAlce pondered over ths she wondered ust what were

the chances that the speaker owned the rattle"I know what you are thinking, said Humpty Dumpty,

who happened to be standing nearby, "and the chances areexactly thirteen out of fourteen!

How did Humpty Dumpty ever arrive at those numbers?

62 This tme Alice came across both brothers together Aliceasked the first one, "Do you own this rattle? He replied"Yes Then Alice asked the second one, "Do you own thsrattle? The second one answered, and Alice gave one ofthem the rattle

Did Alice gve the rattle to the first or the second one?

D. FROM THE MOUTH OF THEJAERWOCKY

Of all the adventures Alice had with the Tweedle brothersn the Forest of Forgetfulness, the one 1 am about to relatewas the most eerie, and the one Alce remembered most

vvidly



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It started ths way: One day Humpty Dumpty metAlce and said Chld I wish to tell you a great secret Mostpeople dont know it but Tweeedee and Tweedledumactually have a thrd brotherhs name is Tweedled Helives n a faroff land but occasonally comes around to

these parts He loks as much like Tweedledee and Twee-edum as Tweedledee and Tweedledum look like eachother

Ths informaton dsturbed Alce dreadfully! For onethng the possibility that there really was a third one wouldmean that all her past inferences were invalidated and thatshe really may not have figured out the day of the weekwhen she thought she had Of even greater practcal m-

portance she may not have restored the rattle to ts rightfulowner after allAlce pondered deeply overthese troublesome thoughts

Finally she asked Humpty Dumpty a sensible question"On what days ds Tweedledoo lie?"Tweedledoo always les replied Humpty umptyAlce walked away in troubled silence "Perhaps the

whole thing s only a fabricaton of Humpty Dumpty Alice

thought to hersef "It certanly sounds a most unlikely taleto me Stl Ace was haunted by the thought that it mhtbe true

There are four dfferent accounts of ust what hap-pened net and I shall tell you all of them I ask the readerto assume two thngs () f there really s an ndivdualother than Tweedledee or Tweedledum who looks indistin-guisha ble from them then his name really s Tweedledoo

(2) f such an ndivdual ests, then he really dslie all thetme I might remark that the second assumpton s notnecessary for the soluton of the next mystery but t s forthe two which follow after that

63 The Frst Verson Alice came across ust one brother alone n the forest At

least he looked lke he was Tweedledee or Tweedledum

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Alce told hm Humpty Dumptys story, and then askedhm "Who are you really? He gave the enigmatc reply, " ether Tweedledee or Tweedledum and today s one ofmy lyng days

The queston s, does Tweedledoo really exst, or s he

ust a fabrcaton of Humpty Dumpty?

6 The Second Verson Accordng to this verson Alce came across (what seemedto be) both brothers She asked the frst one: "Who reallyare you? She got the followng reples:

Frst One / "m Tweedledooecond One / "Yes, he s!

What do you make of this verson?

65 The Thrd Verson Accordng to this verson, Alce came across ust one ofthem He made the followng statement: "Today s one of

my lyng days What do you make of ths verson?

66 The Fourth Verson Accordng to ths verson Alce met (what seemed to be)th brothers on a weekday She asked, "Does Tweedledooeally exst? She got the followng reples:

Frst One / Tweedledoo exstsSecond One / exst

What do you make of ths verson?

poue

Now, what is the real truth of the matter does Tweedledoo

really exst or not? Well, have gven you four conctng



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versions of what really happened How come four versons?Well, to tell you the truth, 1 ddnt nvent these storesmysel 1 heard them all from the mouth of the Jabber-wocky Now, th conversaton between Alce and HumptyDumpty really happened Alce told me this herself, and

Alice is always truthful But the four versons of what hap-pened after that were all told to me by the JabberwockyNow, 1 know that the Jabberwocky lies on the same days asthe Lion Monday, Tuesday, Wednesday and he told methese stories on four consecutve weekdays ( know theywere weekdays, because 1 am lazy and sleep all day Sat-urdays and Sundays They were told to me n the sameorder as 1 recounted them

From ths nformaton, the reader should have no df-culty n ascertainng whether Tweedledoo really exsts orwhether Humpty Dumpty was lyng Does Alce knowwhether Tweedledoo exists?

SOLUTIONS

4 . The only days the Lion can say " led yesterday areMondays and Thursdays The only days the Unicorn cansay lied yesterday are Thursdays and Sundays There-fore the only day they can both say that s on Thursday

48 The lions frst statement imples that t is Monday orThursday The second statement imples that it s notThursday Hence t is Monday

4 On no day of the week is this possble Only on Mondaysand Thursdays could he make the frst statement only on

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Wednesdays and Sundays ould he make the second Sothere s no day he could say both

50Ths s a very dfferent stuaton! It well llustrates tedfference between makng two statements separately andmakng one statement whch s the conuncton of thetwo Indeed gven any two statements X f the snglestatement "X and s true then t of course followsthat X are true separately but f the conuncton Xand s false t only follows that at least one of them s

falseNow the only day of the week t could be true that theLon led yesterday and wll le agan tomorrow s Tuesday(ths s the one and only day whch occurs between two ofthe Lons lyng days) So the day the Lion sad thatcouldnt be Tuesday, for on Tuesdays that statement istrue but the Lon doesnt make true statements on Tuesdays Therefore it is not Tuesday hence the Lons state

ment s false so the Lon s lying Therefore the day must beether Monday or Wednesday

I the frst statement s true, then the frst one really sTweedledum, hence the second one s Tweedledee and thesecond statement is also true If the frst statement s false

then the frst one is actually Tweedledee and the secondone s Tweedledum and hence he second statement s alsofalse Therefore ether both statements are true or bothstatements are false They cant both false snce thebrothers never le on the same day Therefore both statements must be true o the frst one s Tweedledum and thesecond one s Tweedledee Also the day of the encountermust be Sunday

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5 This is a horse of a ver dfferent color! The secon one'sstatement is certainly true Now we are given that the ayof the week is fferent from that of the lat problem so it is

a weekay Therefore it cannot be that both statements aretrue so the frst one must be fase Therefore the first one isTweedledee and the second is Tweedledum

5 3 . The frst answer was clearly a lie, hence the event musthave taken place on a weekday Therefore the other onemust have answered truthfully and said "No

5 Statement (2) of the first one is clearly fase, hece statement (1) is fase too (nce it is uttered on the same day) Therefore the first one ds not lie on Saturdays, so thesecond one lies on Saturdays The second one is telling thetruth on this day (since the first one is lyin so it is now

onday Tuesday or Wednesday The nly one of thesedays n which it is true that he lie tomorrow is Wednesday So the day is Wednesday

55.is statement is certainly fase (for i t were ue then hewould be lyng tay, which is a contradicton) Therefore

at least one of the two clauses " lying tay, " amTweedledee must be fase The first clause (" lyngtoday) is true therefore e second clause must be faseSo he is Tweedledum

56 es it wod he were lying tay then e first clause of

the disunction would be true hence the whole statement

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would be true, which is a contradicon Therefore he itellng the truth tay So his statement is true: either he islying today or he is Tweedledee Since he is not lying todayen he s Tweedledee

57.Bo statements are obviously true, so i t is a Sunday I t isnot possible to determine who is who

58To begin wi, it is impossible on a Sunday for either

brother to lie and say that it is not Sunday Therefore todaycannot be Sunday So the frst one is tellng the truth, and(since it is not Sunday), the second one is therefore lyingtoday The second one says today is Monday, but he islying, so t is not Monday either

Now, the second one has also told the lie that the Lioed yesterday, hence yesterday was really one of the Liontruthful days This means that yesterday was Thursday

Frday, Saturday, or Sunday, so today s Frday, SaturdaySunday, or Monday We have aready ruled out Sunday andMonday, so today must be Friday or Saturday

Next we observe that tomorrow is one of Tweedledees lying days (since the frst one, wh is speaking thetruth, said so) Therefore today cannot be Saturday Hencetoday is Frday

From ths it further follows that Tweedledee lies o

Saturdays, hence he s lke the Unicorn Also, the frst oneis tellng the truth today, whch is a Frday, hence he sTwedledum This proves everything

59 . __Suppose the frst one told the truth Then the rattle belongsto Tweedledee The second speaker must be lying (since it

is not Sunday) , hence his name is not really Tweedledee it

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s Tweedledum Hence the frst speaker s Tweedledee andshould get the rattle

Suppose the frst one led Then the rattle belongs toTweededum Then also the second one told the truth so sreally Tweeledee Then agan the frst one owns the rattle

S n ether case, th rattle belongs to the frst speaker

6 The chances are zero! Suppose hs statement s true Thenthe owner of the rattle s lyng today, hence cannot be thespeaker Suppose on the other hand that hs statement sfalse Then the owner of the rattle s tellng the truth today

hence agan cannot be the speaker

61 Humpty Dumpty was rght! Suppose the speaker s lyngThen the owner of the rattle s not tellng the truth today hes lyng today hence must be the speaker But suppose thespeaker s tellng the truth Then the owner of the rattle s

ndeed tellng the truth today If t s a weekday then hemust be the owner but f t s a Sunday then both brothersare tellng the truth today so ether could be the owner

In summary f t s a weekday then the speaker s definately the owner If t s Sunday then the chances are eventhat he s the owner Therefore the chances are 6% out of7or 3 out of 4that he s the owner

62The clue here s that Alce dd know who to gve t to Hadthe second one answered "Yes then one of them wouldhave been tellng the truth and the other lyng hence Alcewould have no way of knowng who owned the rattle But Itold you she dd know hence the second one ddnt answer"Yes Therefore they were both lyng or both tellng thetruth Ths means they were both tellng the truth, and t

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must have been Sunda So Alce gave t to the frst one

63 Yes Tweedledoo must est Ace was ust talking to him

The speaker claimed that the following statements areoth true

(1) He s ether Tweedledee or Tweedledum(2) He s lyng today

I his claim were true then (1) and (2) would both be truehence (2) would be true, whch would be a contradicton

Therefore hs claim is false (1) and (2) cannot bo betrue Now, (2) is true (nce his claim on this day is false) , sot must be (1) that s not true Therefore he s neitherTweedledee nor Tweedledum, so he must be Tweedledoo

64The first one cant really be Tweedledoo (since Tweedledoo

always les) so he s Tweedledee or Tweedledu but he slng Then the second one s also lng I the second onewere Tweedledee or Tweedled u then Tweedledee andTweedledum woud be lng on the same day, whch is impossible Therefore the second one must be Tweedledoo

65 Ths veron ust simply false66 hoever the second one is, his statement is certainly true( thnk Descartes pointed out that anyone who says heests is makng a true statement certainly 1 have never metanyone who didn t exst) Snce the second statement s trueand t s not Sunday then the frst statement must be fal

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S f ths verson of the story correct Tweedledoo doesntexst

Soluton to the ploue The thrd verson of the story s defntely false Also noneof the stores was told on a Saturday or Sunday The onlyway these four stores can be ftted nto four consecutvedays satsfyng these condtons s that the thrd versonwas told on a Wednesday So the last verson was told on aThursday hence must be the true one S Tweedledoodoesnt really exst! (Im qute sure ncdentally that hadTweedledoo really ested Lews Carroll wouldhave known

about t) for Alce snce the fourth verson the only one

whch really took place then Alce should have no dfficultyn realzng that all these "Tweedledoo fears were ground-less

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& @Prt'sCskesd Othstes

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he yser or:oPortia'sCasketsA. THE FIRST TAE

67a. Shakespeare's Merchant o Vence Portia had threecasketsgold, silver, and leadinside one of which wasPortia's portrait The suitor was to choose one of thecaskets, and he was lucky enough (or wise enough) tochoose the one with the portrait, then he could claim Portiaas his brde On the lid of each casket was an inscrption to

help the suitor choose wiselyNow, suppose Portia wshed to choose her husband

not on the basis of virtue, but simply on the basis of in-telligence She had the following inscriptions put on thecaskets

Gold

THE PORTRTS THSCSKET

Slver

THE PORTRTS NOT THS CSKET

Lead

THE PORTRTS NOT THEGOLD CSKET

Portia eplained to the suitor that of the three statements,at most one was true

Which casket should the suitor choose?

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67bo Poria's suitor chose correctly, so they married and livedquite happilyat least for a while Then, one day, Portiahad the following thoughts "Though my husband showed

some intelligence in choosing the right casket, the problemwasn't really that difficult Surely, I could have made theproblem harder and gotten a reallyclever husband So sheforthwith divorced her husband and decided to get a clev-erer one

This time she had the following inscriptions put on thecaskets

dTHE PORTRATS NOT THE

SLVER CASKET

Slver

THE PORTRATS NOT

THS CASKET

Lead

THE PORTRATS THSCASKET

ortia eplained to the suitor that at least one of the threestatements was true and that at least one of them was false

Which casket contains the portrait?

po As fate would have it, the first suitor tued out to beorta's ehusband He was really quite right enough tofigure out this problem too So they were remarried Thehusband took Portia home, turned her over his knee, gaveher a goo sound spanking, and Portia never had any

foolish ideas again

B. THE SECOND TALE

Portia and her husband did, as a matter of fact, live happilyever after They had a daughter Portia IIhenceforth to becalled "Portia When the young Portia grew to young

womanhood, she was both clever and beautiful, just like her

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mommy She also decided to select her husband by thecasket method The suitor had to pass two tests in order towin her

68a The Frst Test

this test each lid contained two statements, and Portiaeplained that no lid contained more than one falsestatementd Silv.e

THE PORTRAT THE PORTRAIT

S NOT HERE S NOT THEGOLD CASKET

( ) THE ARTST OFTHE PORTRAT S

FROM VENCE

( ) THE ARTIST OFTHE PORTRAIT S

REALLY FROMFLORENCE


68b The Second Test

La( ) THE PORTRAT

S REALLY THE SLVER

CASKET

I the suitor passed the first test, he was taken into anotherroom in which there were three more caskets Again eachcasket had two sentences inscribed on the lid Portia e-plaine that on one of the lids, both statements were true onanother, both statements were false and on the third, one

statement was true and one was false Lad THE PORTRAIT

S NOT THSCASKET

THE MYSTERY OF PORTAS CASKETS 57


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C. INTRODUCING BELLINI AND CELLII

The suitor of the last tale passed both tests and happilyclaimed Portia II as his bride They lived happily ever afterand had a lovely daughter Portia IIIhenceforth to becalled "Portia When she grew up to young womanhood,she was born smart and beautifuljust like her mommyand grandmommy She also decided to choose her husbandby the casket method The suitor had to pass three tests inorder to wn her! The tests were quite ingenious She went

ack to her grandmother's idea of having only one statement inscribed on each casket rather than two But sheintroduced the following new wrinkle She eplained to thesuitor that each casket was fashioned by one of two famousFlorentine craftsmenCellini or Bellini Whenever Cellinifashioned a casket, he always put a false inscription on it,whereas Bellini put only true inscriptions on his caskets

69a The Frst Test

this unusual test the suitor (if he guessed blindly) wouldhave a two out of three rather than a one out of threechance Instead of using a portrait, Portia used a daggerwhich was placed in one of the three caskets the other twocaskets were empty If the suitor could avod the casket wththe dagger then he could take the net test The inscrip-tions on the caskets were as follows

Gold Silver Lead-AT MOST ONE OF THESE

THREE CASKETS WASFASHIONED BY BELL

Which casket should the suitor choose?

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69b The Second Test this test, the suitor's chaces ( he guessed blindly) wereone out of two Portia used only two caskets, gold andsilver, and one of them contained her portrait (no dagger

was used in this test) Again each casket was fashionedeither by Cellini or Bellini The caskets read

Gold

EXACTLY ONE OF THESETWO CASKETS WAS

FASHONED BY BELL

Which casket should the suitor choose in order to find theportrait

69c The Thrd Test I the suitor passed these two tests, he was led into anotherroom containing a gold, silver, and lead casket Again, eachcasket was fashioned by either Cellini or Bellini Now in thistest, the suitor's chances were one out of three (if heguessed blindly) Portia used a portrait of herself, and theportrait was in one of the caskets To pass the test, thesuitor had to (1) select the casket containing the portrait(2) tell the maker of each casket

The three inscriptions read

Gold Silver

What is the solution

Lead AT LEAS TWO OF

THESE CASTS WEEFASHIONED BY CELLI

THE MYSTERY OF PORTAS CASKETS 59


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D. THE MYSTERY: WHAT WENT WRONG?

The fourth and final tale is the most baffling of all, and itillustrates a logical principle of basic importance

The suitor of the last story passed all three tests andhappily claimed Portia II as his bride They had manychildren, greatgrandchildren, etc

Several generations later a descendant was born inAmerica who looked so much like the ancestral portraitsthat she was named Portia Nthhenceforth to be referredto as "Portia When is Pora grew to young womanhood

she was both clever and beautfuljust like all the otherPoras In addition, she was highly vivacious and a bit onthe mischievous side She also decided to select her hus-band by the casket method (which was somewhat of annomaly in modern New York, b ut let that pass)

The test she used appeared simple enough she hadonly two caskets, silver nd gold, in one of which wasPort's portrait The lids bore the following inscripons

Gold SilverEXALY ONE OF

THESE TWOST ATEMENTS S TRUE

Which casket would you choose? Well, the suitor reasoneds follows I e statement on e silver casket is true, thenit is e case that eactly one of e two sttements is trueThis means that the statement on the gold casket must befalse On the other hand, suppose the statement on thesilver casket is false Then it is not the case that ectly oneof e statements is true this means that the statements areeither both true or both false They can't both be true(under the assumption that the second is false) , hence theyre both false Therefore gin, the statement on e gold

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casket false So regardless of whether the statement othe silver casket is true or false, the statement on the goldcasket must be false Thereor the portrait must be in thegold casket

So the suitor triumphantly eclaimed, "The portrait

must be in the gold casket and opened the lid To his utterhorror the gold casket was empty! The suitor was stunnedand claimed that Portia had deceived him "I don't stoop todeceptions, laughed Portia, and with a haughty, trium-phant, and disdaiful air opened the silver casket Sureenough, the portrait was there

Now, what on earth went wrong with the suitor'sreasoning?

"Well, well! said Portia, evidently enjoying the situationenormously, "so your reason didn't do you much good,did it? However, you seem like a very attractive young man,so I think I'll give you another chance I really shouldn't dothis, but I will! In fact, I'll forget the last test and give you asimpler one in which your chances of winning me will be twoout of three rather than one out of two It resembles one of

the tests given by my ancestor Portia III Now surely youshould be able to pass this one!So saying, she led the suitor into another room in

which there were three casketsgold, silver, and leadPortia eplained that one of them contained a dagger andthe other two were empty To win her, the suitor merelyneed choose one of the empty ones The inscriptions on thecaskets read follows

Gold Silver

THE DAGGER THSS THS CASKETCASKE T S EMPTY

LeadAT MOST ONE

OF THES THREESTATEMENTS S TRUE

(Compare this problem with the first test of Portia III!Doesn't it seem to be eactly the same problem?)

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Well, the suitor reasoned ve carefully this time asfollows Suppose statement (3) is true Then both otherstateents must be falsein particular is false, so thedagger is then in the silver casket On the other hand, if (3)is false, then there must be at least two true statements

present, hence (1) must be one of them, so in this case thedagger is in the gold casket In either case the lead casket isempty

So the suitor chose the lead casket, opened the lid,and to his horror, there was the dagger! Laughingly, Portiaopened the other two caskets and they were empty!

I'm sure the reader will be happy to hear that Portiamarried her suitor anyhow (She had decided this long

before the tests, and merely used the tests to tease him alittle) But this still leaves unanswered the queson Whatwas wrong wth the sutors reasonng

SOLUTIONS

67eThe statements on the gold and lead caskets say theopposite, hence one of them must be true Since at mostone of the three statements is true, then the statement onthe silver casket is false, so the portrait is actually in thesilver casket

This problem could be alternatively solved by the fol-lowing method If the portrait were in the gold casket, wewould have two true statements (namely on the gold andlead caskets), which is contrary to what is given If theportrait were in the lead casket, we would again have twotrue statements (this time on the lead and silver caskets) Therefore the portrait must be in the silver casket

Both methods are correct, and this illustrates the factthat in many problems there can be several correct ways ofarriving at the same conclusion

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6I the portrait were in the lead asket, then all hree state-ments would be true, which s contrary to what given Ithe portrait were in the silver casket, then all three state-

ments would be false, which again contrary to what isgiven Therefore the portrait must be in the gold casket(and we have the frst two statements true and the thrd onefalse, whch s consstent wth what s gven)

68a We can mmedately rule out the lead casket, for the

portrat were there, then both statements on the leadcasket would be false So the portrat s n the gold or theslver casket Now, the frst statements on the gold andslver caskets agree, so they are both true or both false Ithey are both false, the second statements are both truebut they cannot be both true snce they are contradct

what is the name of this book (gnv64)

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