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首页 Lady or Tiger

Lady or Tiger.pdf

Lady or Tiger

xubniz
2010-01-20 0人阅读 举报 0 0 暂无简介

简介:本文档为《Lady or Tigerpdf》,可适用于人文社科领域

ALSOBYRAYMONDSMULLYANTheoryofFormalSystemsFirstOrderLogicTheTaoIsSilentWhatIstheNameofThisBookThisBookNeedsNoTitleTheChessMysteriesofSherlockHolmesTheChessMysteriesoftheArabianKnightsTheDevil,CantorandInfinityTHELADYORTHETIGERTHELADYORTHETIGERandOtherLogicPuzzlesINCLUDINGAMATHEMATICALNOVELTHATFEATURESGODEL'SGREATDISCOVERYBYRaymondSmullyanCopyright©byRaymondSmullyanAllrightsreservedunderInternationalandPanAmericanCopyrightConventionsPublishedintheUnitedStatesbyTimesBooks,adivisionofRandomHouse,Inc,NewYork,andsimultaneouslyinCanadabyRandomHouseofCanadaLimited,TorontoThisworkwasoriginallypublishedinhardcoverbyAlfredAKnopf,Inc,NewYorkinTheproblemsinChapter,"TheAsylumofDoctorTarrandProfessorFether,"originallyappearedinTheTwoYearCollegeMathematicsJournalLibraryofCongressCataloginginPublicationDataSmullyan,RaymondMTheladyorthetiger:andotherlogicpuzzles,includingamathematicalnovelthatfeaturesCodersgreatdiscoverybyRaymondSmullyanstedpcmOriginallypublished:NewYork:Knopf,ISBN(pbk)PuzzlesMathematicalrecreationsPhilosophicalrecreationsITitleCV·S·dcManufacturedintheUnitedStatesofAmericaFirstTimesBooksPaperbackEditionContentsPrefaceviiPARTITHELADYORTHETIGERChestnutsOldandNewLadiesorTigersTheAsylumofDoctorTauandProfessorFetherInspectorCraigVisitsTransylvaniaPARTIIPUZZLESANDMETAPU·ZZLES•TheIslandofQuestionersTheIsleofDreamsMetapuzzlesPARTIIITHEMYSTERYOFTHEMONTECARIOLOCKTheMysteryoftheMonteCarloLockACuriousNumberMachineCraig'sLawvCONTENTSFergusson'sLawsInterlude:Let'sGeneralize!TheKeyPARTIVSOLVABLEORUNSOLVABLEFergusson'sLogicMachinegProvabilityandTruthMachinesThatTalkAboutThemselvesMortalandImmortalNumbersTheMachineThatNeverGotBuiltLeibniiz'sDreamviPrefaceAmongthenumerousfascinatinglettersIhavereceivedcon­cerningmyfirstpuzzlebook(whosenameIcanneverre­member!),onewasfromthetenyearoldsonofafamousmathematicianwhowasaformerclassmateofmineThelet­tercontainedabeautifuloriginalpuzzle,inspiredbysomeofthepuzzlesinmybookwhichtheboyhadbeenavidlyread­ingIpromptlyphonedthefathertocongratulatehimonhisson'sclevernessBeforehecalledtheboytothephone,thefathersaidtomeinsoftconspiratorialtones:"Heisreadingyourbookandlovesit!Butwhenyouspeaktohim,don'tlethimknowthatwhatheisdoingismath,becausehehatesmath!Ifhehadanyideathatthisisreallymath,hewouldstopreadingthebookimmediately!"Imentionthisbecauseitillustratesamostcurious,yetcommon,phenomenon:SomanypeopleIhavemetclaimtohatemath,andyetareenormouslyintriguedbyanylogicormathproblemIgivethem,providedIpresentitintheformofapuzzleIwouldnotbeatallsurprisedifgoodpuzzlebooksprovetobeoneofthebestcuresforsocalled"mathanxiety"Moreover,anymathtreatisecouldbewrittenintheformatofapuzzlebook!IhavesometimeswonderedwhatwouldhavehappenedifEuclidhadwrittenhisclassicEleviiPREFACEmentsinsuchaformatForexample,insteadofstatingasatheoremthatthebaseanglesofanisoscelestriangleareequal,andthengivingtheproof,hecouldhavewritten:"Problem:Givenatrianglewithtwoequalsides,aretwooftheanglesnecessarilyequalWhy,orwhynot(Forthesolu­tion,seepage)"Andsimilarlywithalltherestofhistheo­remsSuchabookmightwellhavebecomeoneofthemostpopularpuzzlebooksinhistorylIngeneral,myownpuzzlebookstendtobedifferentfromothersinthatIamprimarilyconcernedwithpuzzlesthatbearasignificantrelationtodeepandimportantresultsinlogicandmathematicsThus,therealaimofmyfirstlogicbookwastogivethegeneralpublicaninklingofwhatGodeI'sgreattheoremwasaboutThevolumeyouarenowholdinggoesstillfurtherinthisdirectionIusedthemanu­scriptofitinacourseentitled"PuzzlesandParadoxes,"whereoneofthestudentsremarkedtome:"Youknow,thiswholebookparticularlypartsThreeandFourhasmuchtheflavorofamathematicalnovelIhaveneverbeforeseenanythinglikeit!"Ithinkthephrase"mathematicalnovel"isparticularlyaptMostofthisbookisindeedintheformofanarrative,andagoodalternativetitleforitwouldbe"TheMysteryoftheMonteCarloLock,"sincethelasthalfconcernsacaseinwhichInspectorCraigofScotlandYardmustdiscoveracom­binationthatwillopenthelockofasafeinMonteCarlotopreventadisasterWhenhisinitialeffortstocrackthecaseproveunsuccessful,theinspectorreturnstoLondon,whereheserendipitouslyrenewsacquaintancewithabrilliantandeccentricinventorofnumbermachinesTogethertheyteamupwithamathematicallogician,andsoonthethreefindthemselvesineverdeepeningwatersthatflowintotheveryheartofGodel'sgreatdiscoveryTheMonteCarlolock,ofcourse,turnsouttobea"Godelian"lockindisguise,itsviiiPREFACEmodusoperandibeautifullyreflectingafundamentalideaofCodel'sthathasbasicramificationsinmanyscientifictheoriesdealingwiththeremarkablephenomenonofselfre­productionAsanoteworthydividend,theinvestigationsofCraigandhisfriendsleadtosomestartlingmathematicaldiscoverieshithertounknowntoeitherthegeneralpublicorthescien­tificcommunityTheseare"Craig'slaws"and"Fergusson'slaws,"whicharepublishedherefortheveryfirsttimeTheyshouldproveofequalinteresttothelayman,thelogician,thelinguist,andthecomputerscientistThiswholebookhasbeengreatfuntowrite,andshouldbeequalfuntoreadIamplanningseveralsequelsAgainIwishtothankmyeditor,AnnClose,andtheproductioneditor,MelvinRosenthal,forthewonderfulhelptheyhavegivenmeElkaPark,NYFebruaryixRAYMONDSMULLYANPARTONETHELADYORTHETIGER•IIIIIII'I'I'I'I,I,I,I,I,I,I,IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIChestnuts­OldandNewIwouldliketobeginwithaseriesofmiscellaneousarithmeti­calandlogicalpuzzlessomenew,someold•HowMuchSupposeyouandIhavethesameamountofmoneyHowmuchmustIgiveyousothatyouhavetendollarsmorethanI(Solutionscomeattheendofeachchapter)•ThePoliticianPuzzleAcertainconventionnumberedonehundredpoliticiansEachpoliticianwaseithercrookedorhonestWearegiventhefollowingtwofacts:()Atleastoneofthepoliticianswashonest()Givenanytwoofthepoliticians,atleastoneofthetwowascrookedCanitbedeterminedfromthesetwofactshowmanyofthepoliticianswerehonestandhowmanywerecrookedTHELADYORTHETIGER•OldWineinaNotsonewBottleAbottleofwinecosttendollarsThewinewasworthninedollarsmorethanthebottleHowmuchwasthebottleworth•HowMuchProfitTheamazingthingaboutthispuzzleisthatpeoplealwaysseemtofightovertheanswer!Yes,differentpeopleworkitoutindifferentwaysandcomeupwithdifferentanswers,andeachinsistshisansweriscorrectThepuzzleisthis:Adealerboughtanarticlefor$,solditfor$,boughtitbackfor$,andsolditfor$Howmuchprofitdidhemake•ProblemoftheTenPetsTheinstructivethingaboutthispuzzleisthatalthoughitcaneasilybesolvedbyusingelementaryalgebra,itcanalsobesolvedwithoutanyalgebraatalljustbyplaincommonsenseMoreover,thecommonsensesolutionis,inmyjudg­ment,farmoreinterestingandinformativeandcertainlymorecreativethanthealgebraicsolutionFiftysixbiscuitsaretobefedtotenpetseachpetiseitheracatoradogEachdogistogetsixbiscuits,andeachcatistogetfiveHowmanydogsandhowmanycatsarethereAnyreaderfamiliarwithalgebracangetthisimmediatelyAlso,theproblemcanbesolvedroutinelybytrialanderror:thereareelevenpossibilitiesforthenumberofcats(any­wherefromzerototen),soeachpossibilitycanbetrieduntilthecorrectanswerisfoundButifyoulookatthisprobleminCHESTNUTSOLDANDNEWjusttherightlight,thereisasurprisinglysimplesolutionthatinvolvesneitheralgebranortrialanderrorSo,IurgeeventhoseofyouwhohavegottentheanswerbyyourownmethodtoconsultthesolutionIgive•LargeBirdsandSmallBirdsHereisanotherpuzzlethatcanbesolvedeitherbyalgebraorbycommonsense,andagainIpreferthecommonsensesolutionAcertainpetshopsellslargebirdsandsmallbirdseachlargebirdfetchestwicethepriceofasmalloneAladycameinandpurchasedfivelargebirdsandthreesmallonesIf,in­stead,shehadboughtthreelargebirdsandfivesmallbirds,shewouldhavespent$lessWhatisthepriceofeachbird•TheDisadvantagesofBeingAbsentmindedThefollowingstoryhappenstobetrue:Itiswellknownthatinanygroupofatleastpeople,theoddsaregreaterthanpercentthatatleasttwoofthemwillhavethesamebirthdayNow,Iwasonceteachinganun­dergraduatemathematicsclassatPrinceton,andweweredoingalittleelementaryprobabilitytheoryIexplainedtotheclassthatwithpeopleinsteadof,theoddswouldbecomeenormouslyhighthatatleasttwoofthemhadthesamebirthday"Now,"Icontinued,"sincethereareonlynineteenstu­dentsinthisclass,theoddsaremuchlessthanfiftypercentthatanytwoofyouhavethesamebirthday"Atthispointoneofthestudentsraisedhishandandsaid,THELADYORTHETIGER''I'llbetyouthatatleasttwoofusherehavethesamebirth­day""Itwouldn'tberightformetotakethebet,"Ireplied,"becausetheprobabilitieswouldbehighlyinmyfavor""Idon'tcare,"saidthestudent,''I'llbetyouanyhow!""Allright,"Isaid,thinkingtoteachthestudentagoodles­sonIthenproceededtocallonthestudentsonebyonetoannouncetheirbirthdaysuntil,abouthalfwaythrough,bothIandtheclassburstoutlaughingatmystupidityTheboywhohadsoconfidentlymadethebetdidnotknowthebirthdayofanyonepresentexcepthisownCanyouguesswhyhewassoconfident•RepublicansandDemocratsInacertainlodge,eachmemberwaseitheraRepublicanoraDemocratOnedayoneoftheDemocratsdecidedtobecomeaRepublican,andafterthishappened,therewasthesamenumberofRepublicansasDemocratsAfewweekslater,thenewRepublicandecidedtobecomeaDemocratagain,andsoIthingswerebacktonormalThenanotherRepublicande­cidedtobecomeaDemocrat,atwhichpointthereweretwiceasmanyDemocratsasRepublicansHowmanymembersdidthelodgecontain•ANew"ColoredHats"ProblemThreesubjectsA,B,andCwereallperfectlogiciansEachcouldinstantlydeduceallconsequencesofanysetofpremisesAlso,eachwasawarethateachoftheotherswasaperfectlogicianThethreewereshownsevenstamps:twoCHESTNUTSOLDANDNEWredones,twoyellowones,andthreegreenonesTheywerethenblindfolded,andastampwaspastedoneachoftheirforeheadstheremainingfourstampswereplacedinadrawerWhentheblindfoldswereremoved,Awasasked,"Doyouknowonecolorthatyoudefinitelydonothave"Areplied,"No"ThenBwasaskedthesamequestionandre­plied,"No"Isitpossible,fromthisinformation,todeducethecolorofA'sstamp,orofB's,orofC's•ForThoseWhoKnowtheRulesofChessIwouldliketocallyourattentiontoafascinatingvarietyofchessproblemwhich,unliketheconventionalchessprob­lemWhitetoplayandmateinsomanymovesinvolvesananalysisofthepasthistoryofthegame:howthepositionaroseInspectorCraigofScotlandYard,�whoseinterestinthistypeofproblemwasequaltothatofSherlockHolmes,toncewalkedwithafriendintoachessclub,wheretheycameacrossanabandonedchessboard"Whoeverplayedthisgame,"saidthefriend,"certainlydoesn'tknowtherulesofchessThepositionisquiteimpos­sible!""Why"askedCraig"Because,"repliedthefriend,"BlackisnowincheckfromboththeWhiterookandtheWhitebishopHowcouldWhitepossiblyhaveadministeredthischeckIfhehasjust•InspectorCraigisacharacterfrommypreviousbookoflogicpuzzles,WhatIstheNameofThisBooktMybookTheChessMysteriesofSherlockHolmescontainsmanypuzzlesofthisgenreTHELADYORTHETIGERmovedtherook,theBlackkingwouldalreadybeincheckfromthebishop,andifhehasjustmovedthebishop,thekingwouldalreadybeincheckfromtherookSo,yousee,thepo­sitionisimpossible"Craigstudiedthepositionforawhile"Notso,"hesaid,"theposition,thoughexceedinglybizarre,iswellwithintheboundsoflegalpossibilities"Craigwasabsolutelyright!Despiteallappearancestothecontrary,thepositionreallyispossible,anditcanbededucedwhat"White'slastmovewas"Whatwasit•SOLUTIONS•Acommonwrongansweris$Now,supposeweeachhad,say,$IfIgaveyou$,youwouldthenhave$andIwouldhaveonly$henceyouwouldhave·$morethanI,ratherthan$Thecorrectansweris$•Afairlycommonansweris"honestandcrooked"Anotherratherfrequentoneis"honestandcrooked"CHESTNUTSOLDANDNEWBothanswersarewrong!Nowletusseehowtofindthecor­rectsolutionWearegiventhatatleastonepersonishonestLetuspickoutanyonehonestperson,whosename,say,isFrankNowpickanyoftheremainingcallhimJohnBythesecondgivencondition,atleastoneofthetwomenFrank,JoOO­iscrookedSinceFrankisnotcrooked,itmustbeJohnSinceJohnarbitrarilyrepresentsanyoftheremainingmen,theneachofthosemenmustbecrookedSotheansweristhatoneishonestandarecrookedAnotherwayofprovingitisthis:Thestatementthatgivenanytwo,atleastoneiscrooked,saysnothingmorenorlessthanthatgivenanytwo,theyarenotbothhonestinotherwords,notwoarehonestThismeansthatatmostoneishonestAlso(bythefirstcondition),atleastoneishonestHenceexactlyoneishonestDoyoupreferoneprooftotheother•Acommonwrongansweris$Now,ifthebottlewerereallyworthadollar,thenthewine,beingworth$morethanthebottle,wouldbeworth$Hencethewineandbottletogetherwouldbeworth$Thecorrectansweristhatthebottleisworth¢andthewineisworth$Thenthetwoaddupto$•Someargueasfollows:Afterhavingboughtthearticlefor$andsolditfor$,hehasmadeadollarprofitThen,bybuyingthearticlebackfor$afterhavingsolditfor$,helosesadollarhenceatthispointheisevenButthen(thear­gumentcontinues)bysellingfor$whathehasboughtfor$,hehasmadeadollaragaintherefore,histotalprofitis$Anotherargumentleadstotheconclusionthatthedealerbrokeeven:Whenhesoldthearticlefor$afterhavingTHELADYORTHETIGERboughtitfor$,hemade$profitButthenheloses$bybuyingbackfor$theitemforwhichheoriginallypaid$,andsoatthispointheis$intheholeThenhegetsbackthedollarbysellingfor$thearticleforwhichhelastpaid$,andsonowheisevenBothargumentsarewrongthecorrectansweristhatthedealermade$Thereareseveralwaystoarriveatthisonesuchgoesasfollows:First,aftersellingfor$thearticleforwhichhehaspaid$,hehasclearlymade$Now,supposethatinsteadofbuyingbackthesamearticlefor$and,thensellingitfor$,heweretobuyadifferentarticlefor$andsellitfor$WouldthisreallybeanydifferentfromapurelyeconomicpointofviewOfcoursenot!Hewouldob­viously,then,bemakinganotherdollaronthebuyingandsellingofthissecondarticleThus,hehasmade$Anotherandverysimpleproof:Thedealer'stotaloutlayis$$=$,andhistotalreturnis$$=$,givingaprofitof$Forthosenotconvincedbythesearguments,letussupposethatthedealerhasacertainamountofmoneysay,$­attheopeningofthedayandthathemakesjustthesefourdealsHowmuchwillhehaveatthecloseofthedayWell,firsthepays$forthearticle,leavinghimwith$Thenhesellsthearticlefor$,givinghim$Nexthebuysthear­ticlebackfor$,bringinghimdownto$Finally,hesellsthearticlefor$andthuswindsupwith$Sohehasstartedthedaywith$andendeditwith$How,then,couldhisprofitbeanythingotherthan$•ThesolutionIhaveinmindisthis:FirstfeedfivebiscuitstoeachofthetenpetsthisleavessixbiscuitsNow,thecatshavealreadyhadtheirportion!Therefore,thesixremain­ingbiscuitsareforthedogs,andsinceeachdogistogetCHESTNUTSOLDANDNEWonemorebiscuit,theremustbesixdogs,andthusfourcatsOfcourse,wecancheckoutthesolution:SixdogseachgettingsixbiscuitsaccountsforthirtysixbiscuitsFourcatseachgettingfivebiscuitsaccountsfortwentybiscuitsThetotal()is,asitshouldbe•Sinceeachlargebirdisworthtwosmallbirds,thenfivelargebirdsareworthtensmallbirdsHencefivelargebirdsplusthreesmallbirdsareworththirteensmallbirdsOntheotherhand,threelargebirdsplusfivesmallbirdsareworthelevensmallbirdsSothedifferencebetweenbuyingfivelargeandthreesmallbirdsorbuyingthreelargeandfivesmanbirdsisthesameasthedifferencebetweenbuyingthir­teensmallbirdsandbuyingelevensmallbirds,whichistwosmallbirdsWeknowthatthedifferenceis$Sotwosmallbirdsareworth$,whichmeansonesmallbirdisworth$Letuscheck:Asmallbirdisworth$,andalargebird$Therefore,thelady'sbillforfivelargeandthreesmallbirdswas$Hadsheboughtthreelargeandfivesmallbirds,shewouldhavespent$,whichisindeed$less•AtthetimeIacceptedthebetfromthestudent,Ihadtotallyforgottenthattwooftheotherstudents,whoalwayssatnexttoeachother,wereidenticaltwins•Thereweretwelvemembers:sevenDemocratsandfiveRepublicans•TheonlyonewhosecolorcanbedeterminedisCIfC'sstampwerered,thenBwouldhaveknownthathisstampwasnotredbyreasoning:"Ifmystampwerealsored,thenA,

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