Improved Technique for Constructing Doubly-even Magic Squares using Basic Latin Squares

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1 Itertol Jourl of Scetfc d Reserch Publctos, Volume 3, Issue 6, Jue ISSN Improved Techque for Costructg Doubly-eve Mgc Squres usg Bsc Lt Squres Tomb I d Shbr N b Dept. of Mthemtcs, Mpur Uversty, Imphl, Mpur (INDIA) tombrom@gml.com b Dept. of Mthemtcs, CMJ Uversty, Shllog, Meghly(INDIA) bom4urs@gml.com Abstrct: The techque developed by Tomb (July 01) for fdg ( x ) mgc squres (whe s dvsble by 4) usg bsc Lt squres s re-dscussed. The steps clude fxg the colum ssocted wth the elemets, dcet to the pvot elemet d rrgg orderly mer, mkg symmetrc trsformtos tht geertes mgc prmetrc costt (T) d flly derved wth mor dustmet o the pr-umbers of stsfyg T [1]. Nordgre s method c be ppled so s to vod symmetrc trsformto, dustmet o the pr umbers stsfyg T d thereby shorteg 3 steps oly. The mproved techque for costructo of doublyeve mgc squres eeds three steps wth o symmetrc trsformto, mor dustmets etc. o the pr umbers stsfyg T. As such, lot of lbour d tme s sved wth the use of ths mproved techque. Key-words: Lt squre (bsc), doubly eve mgc squre (orml), mgc prmetrc costt (Tomb s costt), AMS Clssfcto No: A- 05 d A- 1. INTRODUCTION I Lt squre, Lt letters re see oce ech row d colum wheres the sums of rows d colums re equl but ot the sums of dgols. The de of Lt squres d bsc Lt squres, orml mgc squres hve studed [11] d [1]. A ( x ) bsc Lt squre s symmetrc d o-duplcted (f s odd) d symmetrc but duplcted (f s eve). () (b) Norml mgc squre hs the followg propertes; Elemets or umbers ( 0) re cosecutve d ot repeted Sums of the rows, colums d dgols (usg dgol otto s d ) re equl to the mgc sum, S b b d d, 1,, (c) Equlty property of the sum of rows, colums d dgols rem ultered for rottos d reflectos. A ( x ) rry b (wth dgol otto d ) stsfyg the followg propertes re wek mgc squres; () Elemets or umbers ( 0) re cosecutve d ot repeted (b) Sums of dgols re equl to the mgc sum, S (c) d d S ;, 1,,.. Sums of the rows, colums re equl to the mgc sum (S), except for some d b b S ;, 1,,.. (d) Equlty property of the sum of rows, colums d dgols rem ultered for rottos d reflectos. The the mtrx b s wek mgc squre. Let be mgc squre stsfyg the propertes () to (c). The lterte structures of mgc squre c be expressed (clockwse or tclockwse rotto) s (k) for ( k ; k 1,, m ) where, (k) for ll 0, 4, 8, =. TECHNIQUE (TOMBA, 01) (For costructg doubly-eve mgc squres) The costructo process s expressed the followg fve steps: Step-1: [1] Frst the cosecutve umbers (1 to ) rows d colums be rrged bsc Lt squre formt. The pvot elemet les betwee two

2 Itertol Jourl of Scetfc d Reserch Publctos, Volume 3, Issue 6, Jue 013 ISSN Step-: ( ) d 1 umbers,. Fd the mgc prmetrc costt, T= 1 Select the colum ssocted wth these two umbers, ssg ths colum s m dgol elemets d rrge other (row) elemets orderly mer to gve dgols sums equl Step-3: Mke symmetrc trsformtos of other elemets (retg the dgol elemets uchged) to costruct the extreme corer d cetrl blocks of ( x ) ech. 1 Step-4: Revertg 4 rows d colums systemtc mer, mgc prmetrc costt (T) d set of sub-mgc prmetrc costts re geerted. The reverso process wll stsfy the followg codtos; () (sgly eve), k for k = 0, 1, () (doubly eve), k for k = 0, 1, Step-5: M dustmets should be mde o the prumbers stsfyg T, wheres mor dustmets should be mde o other elemets of sub-mgc prmetrc costts to get the mgc squre for y doubly-eve. Note: The techque geertes wek mgc squres for y sgly-eve. Dfferet wek mgc squres c be derved tkg the cetrl block rbtrrly y two of the pr umbers stsfyg T. where, S for ll d S = ( 1) 1 [] Ths codto s true for ll (odd or eve) due to bsc Lt squre property. The pvot elemet s uque (for s odd) but t les betwee two umbers (for s eve). Sce the pvot elemet s ot fxed, we select the colum (row cse of LSF-), ssocted wth the umbers dcet to the pvot elemet, ssg t s the dgol elemets d rrge the other row (colum) elemets orderly mer to get ew mtrx stsfyg the property d d S for ll d [3] Reflectg odd colums of 1 st, 3 rd, ( < th) d eve colums of th, (-) th, (> th) followed by 1 1 reflecto of odd rows of 1 st, 3 rd, ( < rows of th, (-) th, (> th) d eve th), the desred mgc squre c be cheved. The process of symmetrc trsformto d dustmet o the pr stsfyg T s ot requred It gves b stsfyg, b b d d [4] b ;, 1,,.. s the doubly eve mgc squre. 4. STEPS FOR CONSTRUCTION FOR THE IMPROVED TECHNIQUE (For doubly-eve mgc squres) 3. IMPROVED TECHNIQUE (For costructg doubly-eve mgc squres) The techque of costructg doubly-eve mgc squre usg bsc Lt squre c be mproved wth the pplcto of Nordgre s method fter step-. The techque c be expressed s follows: Let the mtrx ;, 1,, wth the cosecutve umbers be rrged bsc Lt squre formt s; Improved method for costructo of mgc squre usg bsc Lt squre s expressed the followg steps: Step-1: Frst rrge the cosecutve umbers 1 to or ( 11 to ) bsc Lt squre formt. Fd T= 1. Step-: The pvot elemet les 1, 1 betwee. It s ssocted wth (or row cse of BLSF-). 1, 1 d th colum Select ths colum, ssg t s dgol elemets d rrge other row (colum) elemets orderly mer to gve the dgols sums equl. The process gves the cetrl block of (x).

3 Itertol Jourl of Scetfc d Reserch Publctos, Volume 3, Issue 6, Jue ISSN Step-3: Reflectg the frst odd colums, less th d lst eve colums greter th 1 d g reflectg the frst odd rows, less th d lst eve rows, greter th 1, the requred mgc squre s geerted. The process wll ot ffect the cetrl (x) block. 5. ADVANTAGES The mproved techque eeds o dustmets o the pr umbers stsfyg T (Tomb s costts). The process dverts from the erler method fter step-. I leu of Step-3 to step-5, Nordgre s method s ppled. The costructo of doubly eve mgc squres c be completed three steps oly leu of 5 steps eeded for the erler method, developed by Tomb. It geertes doubly eve mgc squres of y (dvsble by 4) wth dfferet vlues the cetrl (x) block. The techque does ot geerte extreme corer (x) blocks. Therefore, lot of lbour d tme s sved wth the use of ths mproved techque. 6. NUMERICAL EXAMPLES 6.1: To costruct (4 x 4) mgc squre Step-1: The umbers (1 to 16) 4 rows d 4 colums, rrged bsc Lt squre formt be: The rrgemet gves the colum totls equl, 34 for ll. Here, S = 34 d T = 17. Step-: Select the colum ssocted wth the umbers 8 d 9 (sy 3, 8, 9, 14) d ssg t s dgol elemets. Rerrgg the other elemets orderly mer gves ew mtrx stsfyg d d = Step-3: Reflectg the odd colums, less th d eve colums greter th 3 d g reflectg the sme odd rows less th d eve rows greter th 3 Reflectg the frst colum, 4 th colum d g reflectg the frst row d fourth row It represets the (4x4) mgc squre wth T = To costruct (8 x 8) mgc squre Step-1: Let the cosecutve umbers be (1,, 3, 64) 8 rows d 8 colums be rrged bsc Lt Squre formt s: Step-: Select the 5 th colum, ssocted wth the elemet 3 d 33, ssg t s dgol elemets. Rerrgg the other row elemets orderly mer gves ew mtrx stsfyg the dgol sums equl Step-3: Reflectg 1 st, 3 rd colums d 6 th d 8 th colums; d g reflectg 1 st, 3 rd rows d 6 th d 8 th rows, the 8x8 mgc squre s geerted.

4 Itertol Jourl of Scetfc d Reserch Publctos, Volume 3, Issue 6, Jue ISSN It represets the 8x8 mgc squre wth T = To costruct (1 x 1) mgc squre Step 1: Let the umbers 1 to 144 be rrged Lt Squre formt be; Step : The pvot les betwee 7 d 73. Select the 7 th.e the 1 th colum, ssocted wth these elemets d ssg t s dgol elemets. Rerrgg the other elemets orderly mer to get ew (1 x 1) rry, stsfyg colum sums equl It gves the cetrl block of (x) s Step 3: Reflectg the 1 st, 3 rd, 5 th colums (less th 6 th ) 8 th, 10 th d 1 th colums (greter th 7 th ) d g reflectg the sme odd d eve rows (1 st, 3 rd, 5 th rows d 8 th, 10 th, 1 th rows ), the 1 x 1 mgc squre s geerted The cetrl block rems uchged. 7. CONCLUSION The techque c be used for fdg mgc squres from bsc Lt squres of y order ( 1, for s doubly-eve). The costructo s doe by fxg the

5 Itertol Jourl of Scetfc d Reserch Publctos, Volume 3, Issue 6, Jue ISSN colum (row) elemets ssocted wth the pvot elemet, ssgg t s dgol elemet d rrgg other elemets orderly mer, reflectg frst odd colums less th d lst eve colums, greter th 1, d reflectg g the sme frst odd rows d lst eve rows. The mgc squre geerted by ths process gves dfferet cetrl block s compred to tht of Nordgre [9]. The pr-umbers stsfyg the mgc prmetrc costt, T c be used for checkg the result very comprehesve mer. Ackowledgemet I would lke to express my deep sese of grttude to Prof. K. S. Bhmr (retred), Deprtmet of Mthemtcs, Mpur Uversty for hs vluble suggestos redered whle developg the techque. REFERENCES [1]. Abe, G.: Usolved Problems o Mgc Squres; Dsc.Mth. 17, pp 3-13, 1994 [] Brrd, F. A. P: Theory of Mgc Squres d Cubes; Memors Ntl. Acd. Sc. 4, pp09-70, [3] Crl, B Boyer (Revsed by Ut, C. Merzbch): A Hstory of Mthemtcs, Revsed Edto, 1998 [4] Gpthy G d M K: Add-o Securty Model of Publc-key Cryptosystem Bsed o Mgc Squre Implemetto, Proceedgs of the World Cogress o Egeerg d Computer Scece, (Vol-1), Oct -7, S Frcsco, USA, 009 [6] Hez, H d Hedrcks J. R.: Mgc Squres Lexcor, Illustrted Self Publshed, 001 [7] Hrym, A. d Abe, G: Reserches Mgc Squres; Osk, Jp: Osk Kyokutosho, [8]. McCre, Judso: Mgc Squres of All Orders, Mthemtcs Techer, pp , 1988 [9] Rold P. Nordgre: New Costructos for Specl Mgc Squres, Itertol Jourl of Pure d Appled Mthemtcs, Vol-, No., pp , 01 [10] Pckover, C. A.: The Ze of Mgc Squres, Crcles, d Strs: A Exhbto of Surprsg Structures Across Dmesos. NJ: Prceto Uversty Press, 00 [11] Tomb I. : A Techque for Costructg Odd-order Mgc Squres usg Bsc Lt Squres, Itertol Jourl of scetfc d reserch Publctos, Vol-, Issue-5, pp , My 01 [1] Tomb I. : A Techque for Costructg Eve-order Mgc Squres usg Bsc Lt Squres, Itertol Jourl of scetfc d reserch Publctos, Volume-, Issue-7, July 01 [13] Tomb I.: O the Techque for Costructg Eveorder Mgc Squres usg Bsc Lt Squres, Itertol Jourl of scetfc d reserch Publctos, Volume-, Issue-9, September, 01 [5] Flery, S. d Flery, D.: I code: A Mthemtcl Jourey, Lodo s Profle Books, pp16-4, 000