PhysicsRecreations:TheDoomsdayAlgorithmDr.D.G.SimpsonDepartmentofPhysi - PDF document

PhysicsRecreations:TheDoomsdayAlgorithmDr.D.G.SimpsonDepartmentofPhysicalSciencesandEngineeringPrinceGeorge’sCommunityCollegeOctober21,2011IntroductionThedoomsdayalgorithmisaclevermethodfordeterminingthedayofweekforanycalendardate,which Example.OnwhatdayoftheweekisApril15,2011?Solution.Againwealreadyknowthatdoomsdayfor2011isMonday.Then4/4(April4)isalsoadoomsday(Monday).Adding7,wethereforeknowthatApril11mustbeonaMonday,andthereforeApril15(fourdayslater)mustbeonaFriday.Example.OnwhatdayoftheweekisChristmasDayin2011?Solution.Weknowthatdoomsdayfor2011isMonday.Therefore12/12(December12)isalsoadoomsday(Monday).Ifweadd7days,weDecember19isalsoonaMonday,andChristmasDayis6dayslater,soitmustbeonaSunday.OddMonthsSonowweknowhowtodoeven-numberedmonths.Whataboutodd-numberedmonths?Thesewouldbe1,3,5,7,9,and11(i.e.January,March,May,July,September,andNovember).Conwaycameupwithaclevermnemonicdeviceforodd-numberedmonths5through11:“Iwork9to5atthe7-Eleven.”Thisissupposedtoremindyouthatforodd-numberedmonths(forgettingJanuaryandMarchforthemoment),9/5,5/9,7/11,and11/7arealldoomsdays.(Thatis,September5,May9,July11,andNovember7.)We’veleftoutMarch.WeknowdoomsdayisthedayoftheweekofthelastdayofFebruary,whichwecanthinkofasMarch0,soofcourseMarch7willalsobeadoomsday.We’vealsoleftoutJanuary.ForJanuary,doomsdayisJanuary3,orJanuary4duringleapyear.Thisiseasytoremember:it’sJanuary3duringthe3commonyearsofafour-yearcycle,orJanuary4duringthefourth(leap)year.Example.OnwhatdayoftheweekisJuly4,2011?Solution.Wealreadyknowthatdoomsdayfor2011isMonday.Therefore7/11(July11)isalsoadoomsday(Monday).Then7daysearlier(July4)isalsoonaMonday.Example.OnwhatdayoftheweekisMay5,2011?Solution.Sincedoomsdayfor2011isMonday,weknowthat5/9(May9)isalsoadoomsday(Monday).Subtracting7days,thismeansMay2isaMonday,andsoMay5isaThursday.Example.OnwhatdayoftheweekisNewYear’sDayin2011?Solution.Sincedoomsdayfor2011isMonday,weknowthatJanuary3isalsoaMonday,andsoNewYear’sDay(twodaysearlier)isaSaturday.OtherYearsAlloftheexamplessofarhavebeenfor2011.Whataboutotheryears?First,let’sconsiderthe20thcentury.Thekeythingtorememberforthe20thcenturyis:Doomsdayfor1900isWednesday.(Ifyouwereborninthe1900s,youcanrememberthisas“1900=We-in-dis-day”.)Thenforanyyearoftheform1.Thenumberof12sin2.Theremainderfromstep1;and3.Thenumberof4sintheremainderinstep1.Addthesethreenumberstogetherandsubtractmultiplesof7togetanumberbetween0and6.Thenaddthatnumberofdaystodoomsdayfor1900(Wednesday)tonddoomsdayforthatyear,thencontinueasbefore. Example.ThearmisticeendingWorldWarIwassignedonNovember11,1918.Whatdayoftheweekwasthis?Solution.Goingthroughthecalculationjustdescribed,=18;thereisone12in18,witharemainderof6,andthereisone4intheremainder.Addingthese,weget1+6+1=8;subtracting7,weget1.Sincedoomsdayfor1900isWednesday,weknowdoomsdayfor1918isonedayafterWednesday,orThursday.Nowweknow11/7isalsoadoomsday(Thursday),andsoNovember11isfourdayslater,onaMonday.Example.OnwhatdayoftheweekwasPearlHarborattacked?(YoumaywellalreadyknowtheanswerifyouknowAmericanhistory.)Solution.TheattackonPearlHarbor,Hawai‘iwasonDecember7,1941.Inthiscase=41.Thereare312sin41,witharemainderof5,andthereisasingle4inthisremainder.Thesumofthethreenumbersis3+5+1=9,andwecansubtractasingle7toget2.Recallthatdoomsday1900isWednesday,sodoomsdayfor1941istwodayslater(Friday).Therefore12/12(December12)isalsoadoomsday(Friday),andsoaweekearlier(December5)wasalsoaFriday,andsotwodayslater,December7,wasaSunday.Forthe21stcentury(yearsoftheform),notethatdoomsdayfor2000isTuesday(“2000Tue.”),thenproceedasbefore.Example.OnwhatdayoftheweekisMay15,2075?Solution.Herethelasttwodigits=75,whichcontains612s,witharemainderof3;thereareno4sinthisremainder.Therefore6+3+0=9;subtract7andwehave2,sodoomsdayfor2075istwodaysafterTuesday,orThursday.So5/9(May9)isalsoonaThursday,andMay15is6dayslater,orWednesday.OtherCenturiesForothercenturies,simplynotethedoomsdaysforthesecenturyyears,andproceedaswithyearsoftheform•1700:Sunday•1800:Friday•1900:Wednesday•2000:TuesdayThecyclerepeatsafterthis:2100isSunday;2200isFriday,etc.TheGregoriancalendarrepeatsafter400years,soyoucanalwaysaddorsubtractmultiplesof400yearstobringthedateintotherange1700–2099tondthedayofweek.Example.OnwhatdayoftheweekwasNovember17,1858?Solution.Thelasttwodigitsoftheyear=58,whichcontains412s,witharemainderof10;thereare24sinthisremainder.Therefore4+10+2=16;subtracttwo7sandwehave2,sodoomsdayfor1858istwodaysafterdoomsdayfor1800(Friday),whichisSunday.Thenweknow11/7isalsoadoomsday(Sunday);adding7days,weknow11/14mustthereforealsobeaSunday,andsothe17th(3dayslater)mustbeaWednesday.TheJulianCalendarEverythingdiscussedupuntilnowisfortheGregoriancalendar,whichisvalidfordatesofThursday,Septem-ber14,1752andlater(intheU.S.andtheU.K.).Thedaybeforethis—thelastdaywewereontheJuliancalendar—wascalledWednesday,September2,1752.InswitchingfromtheJuliantotheGregoriancalendar, wechangedtheleap-yearruleandalsodropped11daysfromthecalendar,sotherewasnoSeptember3–13in1752.FordatesontheJuliancalendar(onorbeforeSeptember2,1752),weneedtomodifythecenturyrulefordeterminingdoomsday.FortheJuliancalendar:•1000:Thursday•1100:Wednesday•1200:Tuesday•1300:Monday•1400:Sunday•1500:Saturday•1600:FridayThecyclerepeatsafterthis,inbothdirections:1700isThursday,900isFriday,800isSaturday,etc.Noticethepattern:foreverycenturyyougobackintime,doomsdayforthatcenturyyearmovesaheadbyoneday.SincetheJuliancalendarrepeatsevery700years,youcanalwaysaddmultiplesof700yearstothedatetobringitintotherange1000–1699,thenusethetableabove.Conwaysuggestsawaytorememberastartingpoint:“1000THOUSday(Thursday).”Example.OnwhatdayoftheweekwasBattleofHastings(October14,1066)?Solution.ThisdateisontheJuliancalendar,soweusethecenturyrulesforJuliandates:doomsdayfor1000isThursday.Thelasttwodigitsoftheyear=66,whichcontains512s,witharemainderof6;thereis14inthisremainder.Therefore5+6+1=12;subtract7andwehave5,sodoomsdayfor1066mustbe5daysafterdoomsdayfor1000:Thursdayplus5daysisTuesday.Weknow10/10mustthereforealsobeadoomsday(Tuesday),andso4dayslater(October14)musthavebeenonaSaturday.B.C.DatesAllB.C.datesareontheoldJuliancalendar.Therewasnoyear0;theyearbeforeA.D.1was1B.C.TotreatB.C.datesproperlywiththesealgorithms,youwillneedtoconverttheyearintoanequivalentnon-positivenumber.Forexample,change1B.C.to0,2B.C.totheyear,etc.Ingeneral,subtract1fromtheB.C.yearandputaminussignonit;thenaddmultiplesof700yearstobringtheyearintotherange1000–1699,andusethetableshownintheprevioussection.Example.OnwhatdayoftheweekwasJuliusCaesarassassinated(March15,44B.C.)?Solution.ThisisaB.C.date,soweusetheJuliancalendar.Theyear44B.C.isequivalentto;let’sadd700yearstogetA.D.657.Thisisn’tenoughtogettotherange1000–1699,solet’sadd700againtogetA.D.1357.Sothecalendarfor44B.C.isthesameasthecalendarforA.D.1357.Nowasseenintheprevioussection,doomsdayfor1300isMonday.Thelasttwodigitsoftheyear=57,whichcontains412s,witharemainderof9;thereare24sintheremainder.Therefore4+9+2=15;subtracttwo7stoget1,sodoomsdayfor1357isonedayafterdoomsdayfor1300:Mondayplus1dayisTuesday.WeknowthatMarch7mustthereforealsobeadoomsday(Tuesday),andso1weeklater(March14)isalsoaTuesday.Thenextday,March15,wasthereforeaWednesday. Summaryndthedayofweekforagivendate(here“doomsday”meansthedayofweekofthelastdayofFebruary):1.Determinedoomsdayforthebeginningofthecenturyofinterest:1900Wednesday,2000Tuesday.(Seetextforothercenturies.)2.Determinedoomsdayfortheyearofinterest.Todothis:Addtogether:(a)thenumberof12sinthelasttwodigitsoftheyear,(b)theremainder;and(c)thenumberof4sintheremainder.Subtract7untilyouhaveanumberbetween0and6,thenaddthismanydaystothecenturydoomsdayinstep1.Theresultisthatyear’sdoomsday.3.Findadoomsdayinthemonthofinterest(leapyearvaluesinparentheses):(a)Evenmonths:2/28(29),4/4,6/6,8/8,10/10,12/12Oddmonths:1/3(4),3/7,5/9,7/11,9/5,11/7(“Iwork9to5atthe7-Eleven.”)4.Nowaddorsubtractmultiplesof7daystogetotherdoomsdaysinthatmonthuntilnearthedateofinterest,thencountoutindividualdaystondthedayofweekofthedesireddate. ExercisesForfun,youmightliketoseeifyoucannd,inyourhead,thedayoftheweekforeachofthesedates:1.Thedayyouwereborn.2.ThesigningoftheMayowerCompactonNovember11,1620(Juliancalendar).3.TheadoptionoftheDeclarationofIndependenceonJuly4,1776.4.TheattackonFortSumteronApril12,1861.5.TheassassinationofPresidentLincolnonApril14,1865.6.TherstsuccessfultestofEdison’selectriclightonOctober22,1879.7.TheshootingofPresidentGareldonJuly2,1881.8.TherstWrightBrothersightonDecember17,9.ThesinkingoftheTitaniconApril15,1912.10.ThestockmarketcrashonOctober29,1929.11.ThecrashoftheairshipHindenburgonMay6,1937.12.D-Day:June6,1944.13.ThedateMartyMcFlyarrivesinthepastinthemovieBacktotheFuture:November5,1955.14.ThelaunchoftheSovietSputniksatelliteonOctober4,1957.15.TheassassinationofPresidentKennedyonNovember22,1963.16.TherstMoonlandingonJuly20,1969.17.TheresignationofPresidentNixononAugust9,1974.18.America’sBicentennial(July4,1976).19.ThersttestofthetimemachineinthemovieBacktotheFuture:October25,1985.20.Therstdayofthe21stcentury(January1,2001).21.September11,2001.22.The“endoftheMayancalendar”onDecember20,23.MartyMcFly’sarrivalinthefutureinthemovieBacktotheFutureII:October21,2015.24.“Doomsday,”accordingtoa1960paperinthejournalScienceNovember13,2026.25.Thedateof“rstcontact”inthemovieStarTrek:FirstContact:April5,2063. Theauthorsdidacurvettohistoricalpopulationdata;theresultingequationpredictedthattheEarth’spopulationwouldbecomeniteonthisdate.(vonFoerster,Mora,andAmiot,Science(3436),1291-1295(1960).) References•“WhatDayIsDoomsday?HowtoMentallyCalculatetheDayoftheWeekforAnyDate”byCham-berlainFong.ScienticAmericanWebsite,October18,2011:http://www.scientificamerican.com/article.cfm?id=calendar-algorithm•“TheDoomsdayRule”inChapter24ofWinningWaysforYourMathematicalPlays(Vol.4)byE.R.Berlekamp,J.H.Conway,andR.K.Guy(A.K.Peters,2004).SolutionstoExercises(2)Sat.;(3)Thur.;(4)Fri.;(5)Fri.;(6)Wed.;(7)Sat.;(8)Thur.;(9)Mon.;(10)Tue.;(11)Thur.;(12)Tue.;(13)Sat.;(14)Fri.;(15)Fri;(16)Sun.;(17)Fri.;(18)Sun.;(19)Fri.;(20)Mon.;(21)Tue.;(22)Thur.;(23)Wed.;(24)Fri.;(25)Thur.