Magic Square of Squares

Transcription

1 Intrnatinal Jurnal f Enginring and Tchnical Rsarch (IJETR) ISSN: , Vlum-, Issu-8, Octbr 23 Magic Squar f Squars Shubhankar Paul Abstract A n n array f intgrs is calld a magic squar whn all th rws, all th clumns, acrss diagnals numbrs add up t sam sum. Hr w will cnsidr a 3 3 array. Magic squar f squars is a 3 3 array whs all rws, all clumns, acrss diagnals numbrs ar squar numbr and als add up t sam sum. Indx Trms n n array, intgrs, magic squar I. INTRODUCTION A squar is magic if ach f th rws, clumns, and diagnals add up t th sam ttal. S, fr xampl, th squar Dfinitin : A32² + A33² Hr w will dfin th acrnyms usd thrugh th ntir papr. Nw any f Aij² whr i =,2,3 and j =,2,3 can b ithr dd r vn. W will writ fr vn and fr dd. If any Aij is vn it will b rplacd by and if dd it will b rplacd by Assumptin will b takn n A, A2, A3, A23 and A33. Rst ar dpndnt accrding t th abv quatins. is magic, sinc vry rw, clumn, and diagnal adds up t 435. Of th nin ntris, fiv (49, 69, 289, 225, and 24) ar prfct squars. W will put a black r if it is assumd and w will put a rd r if it is drivd frm quatins. W will fllw this cnvntin thrughut this papr. Th prblm is t find a 3 by 3 magic squar all f whs ntris ar distinct prfct squars, r prv that such a squar cannt xist. W will dfin th squar. Lt th squar ntry b : A² A2² A3² A2² A22² A23² A3² A32² A33² Equatins : ) A² + A2² = A23² + A33² 2) A2² + A23² = A2² + A32² 3) A² + A33² = A3² + A3² 4) A² + A2² + A3² = A2² + A22² + A32² = A3² + A23² + A33² 5) A² + A2² + A3² = A2² + A22² + A23² = A3² + Manuscript rcivd Sptmbr 29, 23. Shubhankar Paul Passd BE in Elctrical Enginring frm Jadavpur Univrsity in 27. Wrkd at IBM fr 3 yars and 4 mnths as manual tstr with dsignatin Applicatin Cnsultant. Wrkd in IIT Bmbay fr 3 mnths as JRF. Cas : II. SOLUTION All Aij cannt b vn thrwis th whl family will b dividd by 4 until w gt sm dd. S, thr is n qustin f cnsidring this cas. Cas 2 : (ANY TWO OF FIRST ROW IS ODD AND ANOTHER EVEN) Lt s say A23 = and A33 = Cas 2a : A = ; A2 = and A3 = Explanatin f drivd vn r dds : Odd² (md 4) and vn² (md 4) Accrding t quatin 3, A3² = vn Accrding t quatin 4, A32² = vn 8

2 Magic Squar f Squars Nw, clumn 3 is 2 (md 4) Whras, clumn r 2 is r Cntradictin. S, this arrangmnt is nt pssibl. Cas 2b : A =, A2 =, A3 = Explanatins f drivd vn r dds : Accrding t quatin 3, A3² = Accrding t quatin 4, A2² = Nw, first clumn is (md 4) Hr it is clar that third clumn is 2 (md 4) whras first rw is 3 (md 4) Cntradictin. This arrangmnt is nt pssibl. Cas 3 : Fix A23 = vn and A33 = dd. Cas 3a : A = ; A2 = and A3 = Hr it is clar that first rw is (md 4) whras third clumn is 2 (md 4) Third clumn (md 4) Cntradictin. This Cas 2c : A =, A2 =, A3 = Explanatins f drivd vn r dds : Accrding t Equatin 3, A3² = dd Accrding t quatin 5, A32² = Nw, whatvr b A22 scnd clumn is 2 r 3 but third clumn is Cntradictin. This arrangmnt is nt pssibl. Cas 2d : A =, A2 = =. A3 = arrangmnt is nt pssibl. Cas 3b : A = ; A2 =, A3 = Explanatins f th drivd vn and dds : Frm quatin 3, A3 = Frm quatin 5, A32 = Nw, first rw is 2 (md 4) whras third rw is 3 (md 4) Cas 3c : A=, A2 =. A3 = Nw whatvr b th valu f A22 scnd clumn is r (md 4) whras third clumn is 2 (md 4) Cas 3d : A = ; A2 = ; A3 = 9

3 Intrnatinal Jurnal f Enginring and Tchnical Rsarch (IJETR) ISSN: , Vlum-, Issu-8, Octbr 23 It is clar that first rw is 3 (md 4) whras third clumn is 2 (md 4) Cas 4 : A3 = vn Cas 4a : A =, A2 =, A23 =, A33 = It is clar that first rw is (md 4) whras third clumn is (md 4) Cas 4b : A = ; A2 =. A23 =, A33 = Similar cnclusin as Cas 4a. Cas 4c : A =, A2 =, A23 =, A33 = Explanatin f drivd vn and dds : Frm quatin 3, A3 = Whatvr b th valu f A32 third rw is 2 r 3 (md 4) whras first rw is (md 4) Cntradictin. S, this arrangmnt is nt pssibl. Cas 4d : A= ; A2 =, A23 =, A33 = Explanatins f th drivd vn and dds : Frm quatin 3, A3 = Frm quatin 5, A32 = Nw, whatvr b th valu f A22 scnd clumn is 2 r 3 (md 4) whras third clumn is (md 4). Cas 4 : A =, A2 =, A23 =, A33 = It is clar that first rw is (md 4) whras third clumn is 2 (md 4) Cntradictin. This cas is nt pssibl. Cas 4f : A =, A2 =, A23 =, A33 = It is clar that first rw is (md 4) whras third clumn is (md 4) Cas 4g : A =, A2 =, A3 =, A33 = O O O It is clar that first rw is (md 4) whras third clumn is 2 (md 4) Cas 4h : A =, A2 =, A23 =, A33 = It is clar that first rw is (md 4) whras third clumn is (md 4) Cntradictin. S, this arrangmnt is nt pssibl. Frm th abv cass w cnclud that fr any cmbinatin f indpndnt variabls A, A2, A23, A33 th variabl A3 dsn t xist. Nw, w will cnsidr th last cas if all ar dd. Nw, any dd squar numbr whn dividd by 3 givs rmaindr ithr r. W will writ and in plac f Aij whr i =,2,3 and j =,2,3. W will us rd ink t shw drivd r. W will fllw this cnvntin hr. Cas A : St A3² (md 3) 2

4 Magic Squar f Squars Cas A : St A², A2², A23², A33² (md 3) It is clar that first rw is (md 3) whras third clumn is (md 3) Cas A2 : A², A2², A23², A33² (md 3) Similar argumnt as Cas A. Cas A3 : A², A2², A23², A33² (md 3) Nw A3² r. Whatvr b th cas Equatin 3 dsn t satisfy. Cas A4 : A², A2², A23², A33² (md 3) Accrding t quatin 3, A3² (md 3) Nw whatvr b th valu f A32² quatin 5 dsn t satisfy. Cas A5 : A², A2², A23², A33² (md 3) Frm quatin 3, A3² (md 3) Nw, whatvr b th valu f A2² quatin 4 dsn t satisfy. Cas A6 : A², A2², A23², A33² (md 3) It is clar that first rw is 2 (md 3) whras third clumn is (md 3) Cas A7 : A², A2², A23², A33² (md 3) It is clar that first rw is 2 (md 3) whras third clumn is (md 3) Cas A8 : A², A2², A23², A33² (md 3) It is clar that first rw is (md 3) whras third clumn is 2 (md 3) Cas B : Fix A3² Cas B : A², A2², A23², A33² (md 3) It is clar that first rw is (md 3) whras third clumn is 2 (md 3) Cas B2 : A², A2², A23², A33² (md 3) 2

5 Intrnatinal Jurnal f Enginring and Tchnical Rsarch (IJETR) ISSN: , Vlum-, Issu-8, Octbr 23 It is clar that first rw is (md 3) whras third clumn is 2 (md 3) Cas B3 : A², A2², A23², A33² (md 3) Accrding t quatin 3, A3² (md 3) Accrding t quatin 5, A32² (md 3) Nw, whatvr b th valu f A22², scnd clumn will b r (md 3) whras third clumn is 2 (md 3) Cas B4 : A², A2², A23², A33² (md 3) Accrding t quatin 3, A3² (md 3) Nw whatvr b th valu f A2² First clumn will b r (md 3) whras first rw is 2 (md 3). Cas B5 : A², A2², A23², A33² (md 3) Accrding t quatin 3, A3² (md 3) Nw, whatvr b th valu f A32², third rw will b r (md 3) whras first rw is 2 (md 3). Cas B6 : A², A2², A23², A33² (md 3) It is clar that first rw is (md 3) whras third clumn is 2 (md 3). Cas B7 : A², A2², A23², A33² (md 3) It is clar that first rw is (md 3) whras third clumn is (md 3) Cas B8 : A², A2², A23², A33² (md 3) It is clar that first rw is (md 3) whras third clumn is (md 3) Nw, cms th intrsting stry. What if all numbrs ar (md 3). Thn all numbrs ar (md 4) S, th numbrs ar f th frm 2n+. All numbrs must b r ± (md 5) S, th last digit f th numbrs can b 9, r 5. S, sum f any rw r any clumn r acrss th diagnals shuld b divisibl by 5. Lt s adrn th numbr in th 3 3 array with last digit. Bst cmbinatin can b Nw, A3² + A3² 5 (md ) S, (A² + A33²) & (A2² + A32²) & (A2² + A23²) must b 5 (md ) 22

6 Magic Squar f Squars Th tnth digit cmbinatin can b (9,5) ; (8,6) ; (7,7) ; (2,2) and (3,) Frm ths nly (8,6) and (2,2) is pssibl cmbinatin. Othr cass ar nt applicabl. S, th last tw digits f th numbr is f th frm 89, 8, 69, 6, 29, 2. Th pairs can b (29, 2); (89,6); (8,69) If w g n squaring th dd numbrs frm 9 t, w gt as last tw digits as fllwing : 2 -> ², 39², 6², 89² 29 -> 23², 27², 73², 77² 89 -> 7²,33², 83², 33² 6 -> 9², 3², 8², 3² 8 -> 9², 4², 59², 9² 69 -> 3², 37², 63², 87² W s a pattrn that 2 and 29 cms with a diffrnc f 2 whras 89 and 6 cms as a diffrnc f 2 and 8 and 6 cms as a diffrnc f 4. III. CONCLUSION W hav fund th last tw digits f th squar numbrs f Magic squar. But w can t cnclud that n dsn t xist. REFERENCES [] Matthias Bck, Msh Chn, Jssica Cum, and Paul Griblyuk, Th Numbr f Magic Squars, Cubs, and Hyprcubs, Appard in Amrican Mathmatical Mnthly, n. 8 (23), [2] M. Ahmd, J. DLra, and R. Hmmck, Plyhdral cns f magic cubs and squars,prprint (arxiv:math.co/28) (22); t appar in: J. Pach, S. Basu, M. Sharir, ds., [3] Discrt and Cmputatinal Gmtry Th Gdman-Pllack Fstschrift. [4] H. Anand, V. C. Dumir, and H. Gupta, A cmbinatrial distributin prblm, Duk Math. J.33 (966) [5] W. S. Andrws, Magic Squars and Cubs, 2nd d., Dvr, Nw Yrk, 96. [6] M. Bck and D. Pixtn, Th Ehrhart plynmial f th Birkhff plytp, prprint (arxiv:math.co/22267) (22); t appar in Discrt Cmp. Gm. [7] M. B na, Sur l num ratin ds cubs magiqus, C. R. Acad. Sci. Paris S r. I Math. 36 (993) [8] E. Ehrhart, Sur un prbl`m d g m tri diphantinn lin air. II. Syst`ms diphantinslin airs, J. Rin Angw. Math. 227 (967) Shubhankar Paul Passd BE in Elctrical Enginring frm Jadavpur Univrsity in 27. Wrkd at IBM fr 3 yars and 4 mnths as manual tstr with dsignatin Applicatin Cnsultant. Wrkd in IIT Bmbay fr 3 mnths as JRF. W als can nt that 2 numbrs ccur in th diffrnc f 28,22, sris ccur n th diffrnc f 4, 46,4. 89 sris ccur in th diffrnc f 6, 5, 5. 6 sris ccur in th diffrnc f 2, 5, 5. 8 sris ccur in th diffrnc f 32,8, sris ccur in th diffrnc f 24,26,24. S, w s that th sris sum f vry tw cnscutiv diffrnc is 5 xcpt fr 89 and 6. Nw, w s that 6 and 89 cms in a sris f diffrnc 5. S, w can say any numbr f th sris ar (3+5n)² and (33+5m)² Nw, if w tak numbrs frm 2 and 29 sris n aftr anthr thn it will b, (+5p)² and (23+5q)² Nw, (3+5n)² + (33+5m)² = (+5p)² + (23+5q)² 4 + (3n + 33m) (p + 23q) + 25 (m² + n² - p² - q²) =...quatin () If w tak th thr sris f 2 and 29 w gt, (3+5n)² + (33+5m)² = (39+5p)² + (27+5q)² 2 + (39p + 27q) (3m + 33n) + 25(p²+q²-m²-n²) =...quatin (2) Nw, quatin () r (2) shuld hav a slutin t hld magic squar f squars. Similarly, w can frm quatin fr 8 and