A Technique for Constructing Odd-order Magic Squares Using Basic Latin Squares

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1 Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0- A Techque for Costructg Odd-order Mgc Squres Usg Bsc Lt Squres Tomb I. Deprtmet of Mthemtcs, Mpur Uversty, Imphl, Mpur (INDIA) tombrom@gml.com Abstrct- I ths pper, techque for costructg mgc squres (whe s odd) usg bsc Lt squre s developed. Mgc squres re prctclly mportt of the propertes of equlty the sum of ts rows, colums, dgols. The costructo s mde by fxg the pvot elemet d rrgg other elemets orderly mer. The costructo s llustrted wth umercl exmples.. Idex Terms- Lt squre (bsc), mgc squre (orml), pvot elemet, rotto, reflecto I. INTRODUCTION The Lt squres d Greco-Lt squres re used sttstcl reserch prtculrly grculturl sceces d desg of expermets wheres mgc squres re used puzzle gmes of cubes, ptter recogto d mgc crpet costructos, mgc squre cpher Cryptology etc. Bsc Lt Squres A bsc ( x ) Lt squre c be represeted wth Lt letters A, B d C s A B C [] B C A C A B B A C C A B B C A C A B C B A A B C A B C etc. A B C A C B B C A C A B B C A (Iter-chgg rows d colums) re other forms of ( x ) Lt Squres. I ll cses Lt letters re see oce ech row d colum. I Lt squre, the sums of rows d colums re equl but ot the sums of dgols. The bsc Lt Squre s represeted s Where, but d d [] Norml Mgc Squres 9 7 [] 8 6 Where, the sums of the rows, colums d dgols re equl. The bove ( x ) mgc squre (orml) c be expressed s A ;,,, Stsfyg d d ;,,, [] Sce the elemets re cosecutve d ot repeted d therefore orml mgc squre. Mgc squres (orml) my be clssfed rrgemet of o repeted tegers ( 0) rry of equl rows d colums such tht the sums of ts rows, colums d dgols re equl. For orml mgc squre, the followg propertes c be estblshed () Elemets or umbers ( 0) re cosecutve (b) Elemets re ot repeted (c) Sums of the rows, colums d dgols re equl d d for ll,,,..., (d) Equlty property of the rows, colums d dgols rem ultered for rottos d reflectos. There exsts dfferet ( x ) mgc squre ot stsfyg these propertes. Exmples of such mgc squres, ot stsfyg the bove propertes re: mgc squres (specl or rdom, prme umbers etc.) Exmples: () mgc squre (specl) O the other hd, ( x ) mgc squre (orml) wth umbers,,,.,9 s represeted s;

2 Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0-7 () Mgc squre (prme umbers) It stsfes; d d However, these mgc squres re ot orml becuse () the elemets re repeted d o-cosecutve d () the umbers (prme) re ot repeted but o-cosecutve. Symmetrc propertes of Bsc Lt Squres Lemm-: A ( x ) bsc Lt squre ( s odd) s symmetrc d o-duplcted. Let ( x ) bsc Lt squre be A B C B C A C A B Here, A d so o. A C for ll d C, B But [] The dgol elemets re ot equl or repeted oduplcted Lemm-: A ( x ) bsc Lt squre ( s eve) s symmetrc but duplcted Ag, let ( x ) bsc Lt squre be A B C D B C D A C D A B D A B C Clerly, for ll d Bsc Lt squres (of ll orders) re symmetrc. But A C A A d C C [6] The dgol elemets re equl or repeted duplcted Lemm-: Coversely, ( x ) squre ( s odd), stsfyg the symmetrc d o duplcto propertes s bsc Lt squre. Prof: If for ll d, the t follows tht s Bsc Lt squre. Lemm-: I bsc Lt squre ( s odd), oe of the sum of dgol s equl to the sum of rows or colums. Prof: It follows mmedtely tht bsc Lt squre (whe s odd), d or d holds. [7]. For costructg. METHODOLOGY ( s odd) mgc squre The techque of costructg mgc squre usg bsc Lt squre prcple c be expressed s follows: Let the ( x ) mtrx ;,,,... wth the cosecutve elemets/umbers of (... ), ( ), ( ), rrged Bsc... Lt squre formt be; gvg ) S for ll where S Ths codto wll be true for ll (odd or eve) due to bsc Lt squre property The pvot elemet (umber) the mddle cell, whe s odd c be defed s [9], Sce the pvot elemet s fxed, we select the row, ssocted wth t d ssg s the dgol of the ( x ) rry, fxg the pvot elemet the mddle d rrgg the other elemets orderly mer to get ew mtrx b ;,,,..., stsfyg symmetrc property of Lt Squre Hece, b d d S for ll d [0] Ag, sce sum of the colums of re ow the rows of b. Therefore, b S [] Hece, d s fulflled d b ;,,,... s mgc squre. Hece the theorem s estblshed s: The ( x ) squre, developed by usg bsc Lt squre formt whe the pvot elemet s fxed d rerrgg orderly mer represets mgc squre []. Steps for costructo of mgc squre ( s odd) The costructo of mgc squre by usg bsc Lt squre c be expressed the followg steps: Step-: Frst rrge the cosecutve umbers ( [8]... ),... ), (... ) bsc ( Lt squre form

3 Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0- Step-: Determe the pvot elemet to be ssged the mddle cell,, d select the row ssocted wth ths pvot elemet. Step-: Assg ths row s dgol elemets, fxg the pvot elemet the mddle d rrge other elemets orderly mer to gve the desred mgc squre. Note: () Check whether t stsfes the property or ot, b b d d For the cosecutve umbers ( ), the pvot elemet (P) d sum (S) re ) ) P d S [] () If the cosecutve umber ( 0), the t gves the lowest mgc squre. () If the cosecutve umber strts from s+ where s, the correspodg (v) d S ( P s ) ( ) s [] Mxmum d mmum elemets c be determed usg +, d -, []. Alterte Structures of x) mgc squres Let be mgc squre stsfyg the propertes () to (d). Equlty the sums of rows, colums d dgols wll rem uchged for rottos d reflectos The lterte structures of mgc squre c be expressed (clockwse or tclockwse rotto) ( k ); k,,... m ) s (k) Where (k) for ll 0,, 8,... [6]. More propertes () Ifte umber of mgc squres c be geerted by multplyg or ddg by umber p to ech elemet of the gve mgc squre. Or, f d s mgc squre, the p pre mgc squres (b) If the mmum elemet/umber s 0, the gves the lowest mgc squre (c) Sum of two mgc squres the sme rotto/reflecto gves mgc squre (d) (e) Sum of two mgc squres dfferet rotto re ot mgc squres. Product of two mgc squres s ot mgc squre Mgc squres the sme rotto/reflecto re ddtve. NUMERICAL EXAMPLES. To costruct ( x ) mgc squre Let the umbers be (,, ), (,, 6) d (7, 8, 9) Step-: Lt squre formt gves. [8] It gves the colum totls equl, for ll ) Here, P ) d S for Step-: Select the row ssocted wth the pvot elemet (sy, 6, ) d ssg t s dgol elemets, fxg the pvot elemet the mddle (sy,, 6) Step-: Rerrge the other elemets orderly mer to get ew ( x ) rry b,,, s whch represets the 7 mgc squre O lterte structures of mgc squre By rotto or reflex o, lterte structures of mgc squre A,,, c be expressed dfferet structures ( multples of 90 0 ) Let A Reflecto: d Rotto (+90 o 8 ): () A (m) (m) wth the rotto of m 90 0 clockwse or t-clockwse, where m s rel d postve or egtve. A () 8 () A (-) A () 9 () A (-)

4 Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0- A () I ll cses, () To costruct mgc squre d 8 A (-) d re fulflled. Let the cosecutve umbers be (,,,, ), (6, 7, 8, 9,0), (,,,,), (6,7,8,9, 0), (,,,, ) Step-: Arrgg bsc Lt squre formt, t gves ) P Stsfyg S for ll, ) d S 6 Step-: Select the row ssocted wth the pvot elemet () s (,,,, ) d ssg ths row s dgol elemets, fxg the pvot elemet () the mddle. Step-: Rerrge the other elemets orderly mer to get ew mtrx 7 8 b 7 6 where,,, Stsfyg d d. To costruct 7 mgc squre Let the cosecutve umbers be (,,,,, 6 7), (8, 9, 0,,,, ),.(,,, 6,, 8, 9). Followg the steps of rrgg 7 bsc Lt squre formt: Selectg the row ssocted wth the pvot elemet () d ssgg t s dgol elemets, fxg the pvot elemet the mddle d rerrgg the other elemets orderly mer to get ew (7 x 7) mtrx It stsfes d d where P d S 7. Costructo of 9 d mgc squres usg bsc Lt squres Selectg the row ssocted wth the pvot elemet d ssgg t s dgol elemets, fxg the pvot elemet the mddle d rerrgg the other elemets orderly mer, oe c costruct the mgc squres. : () Arrgg 9 bsc Lt squre formt ) P ) d S the requred 9 mgc squre s

5 Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0-8. Costructo of mgc squres of complex umbers (b) Arrgg bsc Lt squre formt; ) P ) 8 d S the requred mgc squre s Tkg two x mgc squres, mgc squres of complex umbers c be geerted. Let the two mgc squres, modfed wth the subtrcto of 7 d respectvely be d 7 gve d Hece mgc squre of complex umbers, c be geerted s 6 where P - + d S -6 + Smlrly, mgc squres of complex umbers of y order ( s odd) c be geerted.. CONCLUSION The techque c be used for fdg mgc squres from bsc Lt Squres of y order (, for s odd) esly wth shortest possble tme. I ths pper,costructo of odd order mgc squres usg bsc Lt squres s show. However, eve-order mgc squres c t be costructed drectly the sme process becuse of duplctos dgol elemets d therefore seprte techques re to be dopted REFERENCES []. Crl, B Boyer (Revsed by Ut, C. Merzbch): A Hstory of Mthemtcs, Revsed Edto, 998 []. Flery, S. d Flery, D.: I code: A Mthemtcl Jourey, Lodo s Profle Books, p6-, 000 []. Hez, H d Hedrcks J. R.: Mgc Squres Lexcor, Illustrted Self Publshed, 00 []. McCre, Judso: Mgc Squres of All Orders, Mthemtcs Techer, , 988