On the Techniques for Constructing Even-order Magic Squares using Basic Latin Squares

Transcription

1 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue 9, September 0 ISSN O the Techques for Costructg Eve-order Magc Squares usg Basc Lat Squares Tomba I. Departmet of Mathematcs, Mapur Uversty, Imphal, Mapur (INDIA) tombarom@gmal.com Abstract: Tomba (May, 0) troduced a techque for costructo ( x ) magc squares (whe s odd) usg basc Lat Squares by fxg the pvot elemet ad arragg other elemets a orderly maer [0]. However, eve-order magc squares ca t be costructed usg the same procedure because of duplcato dagoal elemets. Aga, Tomba (July, 0) developed a techque for costructg ( x ) magc squares (whe s doubly-eve) usg basc Lat square by fxg the colum assocated wth the elemets, adacet to the pvot elemet ad arragg a orderly maer, makg symmetrc trasformatos that geerates a magc parametrc costat (T) ad fally derved by mor adustmet o the par-umbers of satsfyg T []. The techque ca provde weak magc squares for sgly-eve cases. I ths paper, a techque for costructg sgly-eve magc squares s developed ad llustrated wth sutable examples. Key-words: Lat square (basc), sgly eve ad doubly eve magc square (ormal), weak magc squares, magc parametrc costat (Tomba s costat), AMS Classfcato No: A- 05 ad A-. INTRODUCTION I a Lat square, Lat letters are see oce each row ad colum whereas the sums of rows ad colums are equal but ot the sums of dagoals. The dea of Lat Squares ad basc Lat squares, ormal magc squares have studed [0] ad []. A ( x ) basc Lat square s symmetrc ad oduplcated (f s odd) ad symmetrc but duplcated (f s eve). (a) (b) (c) Normal magc square has the followg propertes; Elemets or umbers ( 0) are cosecutve ad ot repeated Sums of the rows, colums ad dagoals (usg dagoal otato as d ) are equal to the magc sum, S b b d d,,,..., Equalty property of the sum of rows, colums ad dagoals rema ualtered for rotatos ad reflectos. b (wth dagoal otato d ) satsfyg the followg propertes are weak magc squares; A ( x ) array (a) (b) (c) Elemets or umbers ( 0) are cosecutve ad ot repeated Sums of dagoals are equal to the magc sum, S d d S,,,..., Sums of the rows, colums are equal to the magc sum (S), except for some ad b b S,,,..., (d) Equalty property of the rows, colums ad dagoals rema ualtered for rotatos ad reflectos. The the matrx b s a weak magc square. Let a be a magc square satsfyg the propertes (a) to (c). The alterate structures of a magc square ca be expressed a (k) for (clockwse or atclockwse rotato) as ( k ); k,,... m ) where, a = (k) for all 0, 4, 8,.... TECHNIQUE (For costructg doubly-eve magc squares) The techque of costructg doubly-eve magc square usg basc Lat square prcple ca be expressed as follows: Let the matrx a [] a ;,,,... wth the cosecutve umbers be arraged basc Lat square format as; a a... a a a a3... a a where, a S for all,... a a... a a S = ( ) [] Ths codto s true for all (odd or eve) due to basc Lat square 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, assocated wth the umbers adacet to the pvot elemet, assg t as the dagoal elemets ad arrage the other elemets a

2 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue 9, September 0 ISSN orderly maer to get a ew matrx satsfyg the property of d d S for all ad [3] Aga, make symmetrc trasformatos of other elemets, retag the dagoal elemets u-chaged to fd out the extreme corer blocks ad cetral block of ( x ) each. Revertg ( 4) rows ad colums systematcally, a magc parametrc costat (T) ad a set of sub magc parametrc costats are geerated. For doubly-eve cases, by adustg the values the parametrc costats ad sub parametrc costats, the requred magc squares (doubly-eve) ca be costructed as b b b b b b b b b b b b Satsfyg b b d d [4] b ;,,,... s a (doubly-eve) magc square. 3. THEOREMS 3.: Odd order Magc Squares derved usg basc Lat Squares For ay ( s odd), a magc square costructed usg basc Lat square, the followg theorem holds; Theorem : A ( x ) matrx b, developed by usg basc Lat square format whe the pvot elemet s fxed, assged the row assocated wth the pvot elemet (keepg the pvot elemet the mddle cell) as dagoal elemet ad arragg a orderly maer represets a magc square for ay odd. [0] 3.: Doubly-eve Magc Squares derved usg basc Lat Squares For ay ( s doubly-eve), a magc square costructed usg basc Lat square, the followg theorem holds; Theorem : A ( x ) matrx b, developed by usg basc Lat square format whe the colum, assocated wth the umbers adacet to the pvot elemet s selected, assg t as the dagoal elemets ad arrage the other elemets a orderly maer, makg symmetrc trasformatos gves a ew matrx satsfyg the equalty property of dagoal sums. Selectg a sutable cetral block (retag the dagoal elemets uchaged) ad reverg the ( 4) rows ad colums systematcally, the system geerates a magc parametrc costat (T) ad sub-magc parametrc costats. Wth some adustmet o the par-umbers satsfyg T, t geerates magc squares of ay doubly-eve. [] 3.3: Sgly-eve Weak Magc Squares derved usg basc Lat Squares For ay ( s sgly-eve), the followg theorem holds Theorem 3: A ( x ) matrx b, developed by usg basc Lat square format whe the colum, assocated wth the umbers adacet to the pvot elemet s selected, assg t as the dagoal elemets ad arrage the other elemets a orderly maer, makg symmetrc trasformatos of other elemets gves a ew matrx satsfyg the equalty property of dagoal sums. Selectg a sutable cetral block (retag the dagoal elemets uchaged) ad revertg the rows ad colums systematcally, the system geerates a magc parametrc costat (T) ad sub-magc parametrc costats. Wth some adustmets o the par-umbers satsfyg T, t geerates weak magc squares oly for ay sgly-eve. [] 4. METHODOLOGY (For costructg sgly-eve magc squares) The techque of costructg sgly-eve magc square usg basc Lat square prcple ca be expressed as follows: Let the matrx a ;,,,... wth the cosecutve umbers be arraged basc Lat square format as; a a... a a a a3... a a where, a S for all... a a... a a S = ( ) [5] Ths codto s true for all (odd or eve) due to basc Lat square property. Fd T=. Select the colum, assocated wth the mmum ad maxmum elemets, assg t as the dagoal elemets ad arrage the other elemets a orderly maer to get a ew matrx satsfyg d d S for all ad Aga, retag the dagoal elemets u-chaged, make symmetrc trasformatos, to fd the extreme corer blocks ad cetral block of ( x ) each. Revertg ( 4) rows ad colums systematcally, a magc parametrc costat (T) ad a set of sub magc parametrc costats are geerated. Revertg the dagoal elemets (keepg the cetral block uchaged) ad selectg the parumbers satsfyg magc parametrc costat T, assgg the par-umbers properly ad makg mor adustmets the values the parametrc costat ad other elemets, the requred magc squares (sgly-eve) ca be costructed as; b b... b b b b, b, b b b... b b Satsfyg b b d d [6] b ;,,,... s a (sgly-eve) magc square.

3 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue 9, September 0 3 ISSN Hece, for ay sgly-eve, the followg theorem s developed; Theorem 4: A ( x ) matrx b, developed by usg basc Lat square format whe the colum assocated wth the maxmum ad mmum umbers s selected, assg t as dagoal elemets ad arrage the other elemets a orderly maer get a ew matrx satsfyg the equalty property of dagoal sums. Aga, make symmetrc trasformatos, retag the dagoal elemets u-chaged to fd the extreme corer blocks ad cetral block of ( x ) each. Reverg ( 4) rows ad colums systematcally (keepg the cetral block uchaged), the system geerates a magc parametrc costat (T) ad sub-magc parametrc costats. Revertg oe of the dagoal elemet (keepg the cetral block uchaged), assgg the par-umbers satsfyg the propertes of T selectve postos, followed wth some adustmets gves magc squares of ay sgly-eve. 5. STEPS FOR CONSTRUCTION 5.: Doubly-eve magc squares Costructo of magc square usg basc Lat square s expressed the followg steps: Step-: Frst arrage the cosecutve umbers to or ( a to a ) basc Lat square format. Fd T=. Step-: Fd the rage of a a ad select the colum assocated wth ths rage assg t as dagoal elemets ad arrage other elemets a orderly maer, gvg dagoals sums equal Step-3: Retag the dagoal elemets uchaged, make trasformatos of other elemets to costruct the extreme corer ad cetral blocks of ( x ) each. Step-4: Revertg ( 4) rows ad colums a systematc maer, a magc parametrc costat (T) ad a set of sub-magc parametrc costats are geerated... The reverso process wll satsfy the followg codto. () k for k = 0,,.... Step-5: Ma adustmets should be made o the elemets correspodg to the magc parametrc costat (T), whereas mor adustmets should be made o other elemets of sub-magc parametrc costats to get the magc square satsfyg the propertes of b b d d Note: For sgly-eve, t geerate weak magc squares oly. 5.: Sgly-eve magc squares Costructo of magc square usg basc Lat square s expressed the followg steps: Step-: Frst arrage the cosecutve umbers to or ( a to a ) basc Lat square format. Fd T=. Step-: Select the colum assocated wth maxmum ad mmum umbers, assg t as dagoal elemets ad arrage other elemets a orderly maer, gvg dagoals sums equal Step-3: Retag the dagoal elemets uchaged, make trasformatos of other elemets to costruct the extreme corer ad cetral blocks of ( x ) each. Step-4: Revertg ( 4) rows ad colums a systematc maer, a magc parametrc costat (T) ad a set of sub-magc parametrc costats ( T T... ) ca be determed. The reverso process wll satsfy the followg codto. () k for k = 0,,... Step-5: Revert oe of the dagoal elemets keepg the cetral block uchaged. Select the par-umbers satsfyg T ad arraged o selectve bass whch s ormally complcated. Step-6: Adustmets should be made o the elemets correspodg to the magc parametrc costat (T), whereas mor adustmets should be made o other elemets to get the magc square satsfyg the propertes of b b = d d 6. NUMERICAL EXAMPLES (Doubly-eve ad sgly-eve cases) 6.: To costruct a (4 x 4) magc square (doubly-eve) Step-: The umbers ( to 6) 4 rows ad 4 colums, arraged basc Lat square format be; The arragemet gves the colum totals equal, a 34 for all. Here, S = 34 ad T = Step-: Select the colum assocated wth the umbers 8 ad 9 (say 3, 8, 9, 4) ad assg t as dagoal elemets.

4 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue 9, September 0 4 ISSN Rearragg the other elemets a orderly maer gves a ew matrx satsfyg d d = Step-3: Retag the dagoal elemets u-chaged, make trasformatos of other elemets to geerate the extreme corer ad cetral blocks of ( x ) each Extreme corer blocks: , 8, 4 7, Step-4: Here, ( 4) = 0 for = 4. No reverso process eeded. Step-5: Hece, o adustmet eeded ad therefore the costructo of (4 x 4) magc squares s completed Step To costruct (8 x 8) magc square (doubly-eve) Step-: Let the cosecutve umbers be (,, 3,... 64) 8 rows ad 8 colums be arraged basc Lat Square format as: Step-: Select the row assocated wth the elemet (3 ad 33) ad assg ths as dagoal elemets. Rearragg the other elemets a orderly maer gves a ew matrx gvg the dagoal sums equal satsfyg the dagoal sums equal Step-3: Retag the dagoal elemets u-chaged, make trasformatos of other elemets to costruct the extreme corer ad cetral block of ( x ) each Extreme corer blocks: Let the Cetral block be assumed as: Note: Takg the cetral block dfferet forms, dfferet magc squares ca be costructed. Step-4: Here, ( 4) = for = 8. Revertg 3 th colum, 5 th colum ad 3 th row, 5 th row (keepg the dagoal elemets, extreme corer blocks ad cetral block uchaged), a magc parametrc costats (T) ca be determed Here, T = 65 ad sub-magc parametrc costats: T = 49, T = 8, T 3 = 59 T 4 = 7 Par-umbers satsfyg T : [8, 37], [7, 38], [6, 39], [56, 9], [49, 6], [4, 4], [48, 7], [8, 57], [, 64], [3, 34], [30, 35] ad [9, 36] Other par-umbers satsfyg T: () I extreme corer blocks: [6, 4], [3, 6], [, 53], [5, 60] [6, 59], [3, 5] () I cetral block: [40, 5] ad [3, 33]

5 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue 9, September 0 5 ISSN Step-5: Adustmets o the values of magc parametrc costats, T are made to get the magc square as follows; Fally the ecessary magc square of 8x8 s costructed as; As stated earler, for sgly-eve cases, the techque geerates weak magc squares oly 6.3 To costruct a (6 x 6) weak magc square (Sgly-eve) Step-: Let the cosecutve umbers be (,,... 36) 6 rows ad 6 colums be arraged basc Lat square format; Satsfyg a = S for all, where, S =. The pvot elemet les betwee 8 ad 9. Fd T = Step-: Select the colum assocated wth the elemet (8, 9) as (4,, 8, 9, 6, 33) ad assg ths colum as dagoal elemets, Rearrage the other elemets a orderly maer to get a ew matrx havg the dagoal sums equal Step-3: Retag the dagoal elemets u-chaged, make symmetrc trasformatos of other elemets to get the extreme corer ad cetral blocks of ( x ) each. Extreme corer blocks: Cetral block assumed: Step-4: Here, ( 4) = for = Revertg the 3 rd row ad 3 rd colum (retag the dagoal elemets ad cetral block uchaged), a magc parametrc costat T s geerated Par-umbers satsfyg T: [6, ], [7, 0], [, 36], [7, 30] [5, ], [3, 6], [4, 3] ad [5, ] Other par-umbers satsfyg T: () I extreme corer blocks: [34, 3], [9, 8], [, 35], [4, 33] [7, 0], [3, 5] () I cetral block: [4, 3] ad [8, 9]

6 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue 9, September 0 6 ISSN Step-5: After smple adustmets the par-umbers, satsfyg T, we get weak magc square, geerated from basc Lat Square Note: Adustmet to make row ad colum sums equal wll affect the sum of the dagoals ad therefore geerates weak magc squares oly. 6.4: To costruct a (6 x 6) magc square (Sgly-eve) Step-: Let the cosecutve umbers be (,,...36) 6 rows ad 6 colums be arraged Lat square format as; It satsfes a = S for all, where, S =. Here, T = 37 Step-: Select the colum assocated wth the mmum ad maxmum umbers as (, 8, 5,, 9, 36) ad assg ths colum as dagoal elemets, Rearrage the other elemets a orderly maer to get a ew matrx gvg dagoal sums equal, Step-3: Retag the dagoal elemets u-chaged, make symmetrc trasformatos of other elemets to get the extreme corer ad cetral blocks of ( x ) each. Extreme corer blocks: Cosder the cetral block as Step-4: Here, ( 4) = for = 6. Revertg 3 rd row ad 3 rd colum (retag the dagoal elemets ad cetral block uchaged), the magc parametrc costat T s geerated. Par-umbers satsfyg T: [3, 4] [4, 3] [4, 33] [0, 7] [8, 9] [34, 3] [7, 0] [8, 9] Step-5: Revert oe of the dagoal elemets (keepg the cetral block uchaged). Select the par-umbers satsfyg T a very selectve maer (whch s ormally very complcated, ot satsfed for may cases) as; Step-6: Make smple adustmets the value of the magc parametrc costats ad other elemets to get the magc square, geerated from basc Lat Squares

7 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue 9, September 0 7 ISSN Step-4: Here, ( 4) = 3 for = : To costruct a (0 x0) magc square Step-: Let the cosecutve umbers be (,,...00) 0 rows ad 0 colums be arraged Lat square format as; It satsfes a = S for all, where, S = 505. Here, T = 0 Step-: Select the colum assocated wth the mmum ad maxmum umbers ad assg ths colum as dagoal elemets, Rearrage the other elemets a orderly maer to get a ew matrx gvg dagoal sums equal, Step-3: Retag the dagoal elemets u-chaged, make symmetrc trasformatos of other elemets to get the extreme corer ad cetral blocks of ( x ) each Revertg 3 rd 5 th 7 th rows ad 3 rd 5 th 7 th colums (retag the dagoal elemets ad cetral block uchaged), the magc parametrc costat T s geerated The par-umbers satsfyg T: [4, 60] [4, 59] [58, 43] [44, 57] [6, 95] [6, 85], [6, 75], [36, 65], [50, 5] [49, 5] [48, 53], [54, 47], [5, 96] [44, 57] [96,5] [86, 5], [76, 5], [66, 35] Other par-umbers satsfyg T: () I extreme corer blocks: [34, 3], [9, 8], [, 35], [4, 33], [7, 0], [3, 5] () I cetral block: [4, 3] ad [8, 9] () I dagoals: [00,], [89,], [67,34], [9,0], [83, 8], [73, 8], [64, 37] Step-5: Cosder the cetral block as Revert oe of the dagoal elemets (keepg the cetral block uchaged). Select the par-umbers satsfyg T a very selectve maer (shftg ts postos 90 0 )

8 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue 9, September 0 8 ISSN Step-6: Make smple adustmets the value of the magc parametrc costats ad other elemets to get the magc square, geerated from basc Lat Squares Precautos I the costructo of sgly-eve magc squares usg basc Lat squares, selecto of proper cetral block, assgg the par-umbers satsfyg the magc parametrc costat T, s ormally complcated. I may cases, t wll geerate weak magc squares. Example:6.6: Usg step-4 of example 6.4. Let 5 be the cetral block. 6 If we proceed wthout assgg the par-umbers satsfyg T properly, after makg adustmets, t geerates Weak magc square as CONCLUSION The techque ca be used for fdg magc squares from basc Lat Squares of ay order (, for s sgly-eve). The costructo s doe by fxg the colum, assocated wth the maxmum ad mmum elemets, assgg t as dagoal elemet ad arragg other elemets a orderly maer, makg symmetrc trasformatos, revertg oe the dagoal elemets (keepg the cetral block u-chaged), selectg ad arragg the par-umbers satsfyg the magc parametrc costat, T a very selectve maer ad fally adustg the elemets o the magc parametrc costat ad other elemets. Weak magc squares of dfferet structures ca be geerated depedg upo the choce of cetral block, assgg the parumbers satsfyg T. ad therefore expected to provde more applcatos cryptology tha that of the actual magc squares for ay (odd or eve). Ackowledgemet I would lke to express my deep sese of grattude to Prof K. S. Bhamra ad Prof M. Rat Sgh, Departmet of Mathematcs, Mapur Uversty for ther valuable suggestos gve whle developg the formulae/ techques. REFERENCES []. Abe, G.: Usolved Problems o Magc Squares; Dsc. Math. 7, 3-3, 994 [] Barard, F. A. P: Theory of Magc Squares ad Cubes; Memors Natl. Acad. Sc. 4, 09-70, 888. [3] Carl, B Boyer (Revsed by Uta, C. Merzbach): A Hstory of Mathematcs, Revsed Edto, 998 [4] Gaapathy G ad Ma K: Add-o Securty Model of Publc-key Cryptosystem Based o Magc Square Implemetato, Proceedgs of the World Cogress o Egeerg ad Computer Scece, (Vol-), Oct -7, Sa Fracsco, USA, 009 [5] Flaery, S. ad Flaery, D.: I code: A Mathematcal Jourey, Lodo s Profle Books, p6-4, 000 [6] Hez, H ad Hedrcks J. R.: Magc Squares Lexcor, Illustrated Self Publshed, 00 [7] Hrayama, A. ad Abe, G: Researches Magc Squares; Osaka, Japa: Osaka Kyokutosho, 983. [8]. McCrae, Judso: Magc Squares of All Orders, Mathematcs Teacher, , 988 [9] Pckover, C. A.: The Ze of Magc Squares, Crcles, ad Stars: A Exhbto of Surprsg Structures Across Dmesos. NJ: Prceto Uversty Press, 00 [0] Tomba I. : A Techque for Costructg Odd-order Magc Squares usg Basc Lat Squares, Iteratoal Joural of scetfc ad research Publcatos, Volume-, Issue-5, May 0 pp [] Tomba I. : A Techque for Costructg Eve-order Magc Squares usg Basc Lat Squares, Iteratoal Joural of scetfc ad research Publcatos, Volume-, Issue-7, July 0 pp