魔术方块的泛化(数字逻辑)3×3及其倍数(3×3)×(3×3)(IJISA-V5-N1-9)

I.J.IntelligentSystemsandApplications,2013,01,90-97PublishedOnlineDecember2012inMECS()DOI:10.5815/ijisa.2013.01.09Copyright©2013MECSI.J.IntelligentSystemsandApplications,2013,01,90-97GeneralizationofMagicSquare(NumericalLogic)3×3anditsMultiples(3×3)×(3×3)BLKaulDept.ofMechanicalEngineering,R.K.G.I.T,Ghaziabad,U.P.,[e-mail]Professor,Dept.ofComputerScience&Engineering,GreaterNoidaInstituteofTechnology,GreaterNoida,U.P.,[e-mail];[e-mail]—Amagicsquareof3×3anditsmultiplesi.e.(9×9)squaresandsoon,oforderNarecomposedof(n×n)matrixhavingfilledwithnumbersinsuchawaythatthetotalssumalongtherows,columnsandmaindiagonalsaddsupthesame.Byusingaspecialgeometricalfiguredeveloped.IndexTerms—MagicSquare,SquareMatrix,Integer,RequiredSumI.IntroductionThesumofthreeoritsmultiplesquarenumbersinalldirectioncanbeachievedthesame(equal)byusingafigurestartinganumberfromoneendtotheotherendeitherclockwiseoranticlockwiseseriallyusinganumberonceonly,wecanachievethiseasilybyplacingthefiguresinfourdifferentposition.Amagicsquareisann×nmatrixfilledwiththeintegersinsuchawaythatthesumofthenumbersineachrow,eachcolumnordiagonallyremainthesame,inwhichoneintegerisusedonceonly.Magicsquareanditsrelatedclassesofintegermatriceshavebeenstudiedextensivelyandstillitisgoingon.Asweknow,ancientIndiaistheoriginofnumbersanditsproperties,thenafteritspreadsallovertheworld.AsARYABHATTgavezero(0)totheWorld.Theinterestedreadermayconsultthereferencestherein[1,2,3,4,5].But,thereadercaneasilyextractthatthereisrequirementofknowledgeofalgebra,numberanditspropertiesandmanydifferentbranchesofmathematicsformagicsquares.Here,westartwith3×3matrixandwiththehelpofspecialgeometryabletoobtainamagicsquareonlyhavingtheknowledgeofsumandmultiplicationofintegers.II.TheoryFor(3×3),anynumberfifteen(15)oraboveuptotheinfinitydivisibleby3andfor(3×3)x(3x3)Threehundredsixtynine(369)lowestanduptotheinfinity,whichisdivisibleby9respectivelycanbeachievedasasumofthenumbersof3×3squaresand9×9squaresinalldirectionthesame.Byusing9or81numbersseriallyandusinganeachnumberonceonlyasbelow.83415159156721515151515Fig.1:A6510211173214982121212121Fig.1:B30274299453321992439369999999999Fig.1:CGeneralizationofMagicSquare(NumericalLogic)3×3anditsmultiples(3×3)×(3×3)91Copyright©2013MECSI.J.IntelligentSystemsandApplications,2013,01,90-97III.OurProcessFortheninesquaresafigurehavebeendevlopedwhichcanbeputintheninesquarecubeandstartingbythenumbergotbydevudingthesumbythree(3)andsubtractingacontestoffour(4)ormunlipleoffour(4)fromitsuchaswewantthesumofthenumbersof3×3squareshouldbe36inallthedirection.Devide36by3weget36/3=12.Wecanstartbythenumber12-4=8or12-(4+4)=4.Thedifferencebeetweenthenumbersusewillbeoneforsubtractingfour(4)aconstrintanddifferenceoftwoforthesubtratingof(4+4)twoconstraintsandsoon.i.e.:-Sr=36,a=8,d=1Sr=36,a=4,d=211101536161283691413363636363610818362012436616143636363636Fig.2:AFig.2:BWecanhavefourcombinationsmorebyplacingthefigureinfourdiffentdirectionclockwiseoranticlockwiseFig3.AFig3.BFig3.CFig3.DWealsocanhavemorecombinetionpossibleforahighersumrequired(SR)bysubtractingmultiplesofconstraints4(four)fromthenumberobtainedbyafterdevidingSRby3.Figureshavebeendevlopedandifthesefiguresareplacedon3×3squaresormultiplesof3×3i.e.9×9andsoon.Thesumofallthelinesofanyblockinalldirectioncanbeachievedthesamebyusinganumberonceonly.Thefigureusedin3×3oritsmultiplesisgivenbelow:-Fig4:A.92GeneralizationofMagicSquare(NumericalLogic)3×3anditsmultiples(3×3)×(3×3)Copyright©2013MECSI.J.IntelligentSystemsandApplications,2013,01,90-97Fig4:BAnynumber15oraboveuptotheinfinity,whichisdevisibleby3canbegotinallthedirectionbyputtingtheabovefigureindifferentposition.Forexamplein3x3blockwewantsumrequired(SR)147.Thesumofallthenumbersofeachlinesaresameinallthedirection,wemustfirstdevideitby3.As:-147÷3=49ThensubtractFour(4)aconstraintnumberfromiti.e.49–4=45Wecanstartwith45inarthimaticalprogressionthatis45,46,47andsoontoget147inalldirection.484752534945147465150147Fig5:Thesamemethodcanbeusedwithothernumbershavingdifferentequaldifferencebeetweenthemlikenumbers1,2,3,4,5,……………ArthimaticalProgression1,3,5,7,9,……………GeomatricalProgression1,4,7,10,13,……………GeomatricalProgression1,5,9,13,17,……………GeomatricalProgressionAsabove147Onehundredfortysevencanbeachivedwithusingothernumbersalso,havingdifferentequaldifferencebeetweenthem,butthesubtractingfactorwillbemultipleofconstrintsthatisfour(4).Like:-SR=SumRequireda=StartingNumberd=DifferencebeetweennumbersTable:1SRNumbersad147147÷3=49–(4×1)451147147÷3=49–(4×2)412147147÷3=49–(4×3)373147147÷3=49–(4×4)334147147÷3=49–(4×5)295147147÷3=49–(4×6)256147147÷3=49–(4×7)217147147÷3=49–(4×8)178147147÷3=49–(4×9)139147147÷3=49–(4×10)0910147147÷3=49–(4×11)0511147147÷3=49–(4×12)0112Togetthesumonehundredandfortyseven(147)thefollowingstartingnumbersanddifferencewillbethemwillbeas:-GeneralizationofMagicSquare(NumericalLogic)3×3anditsmultiples(3×3)×(3×3)93Copyright©2013MECSI.J.IntelligentSystemsandApplications,2013,01,90-97