GENERALIZED FORM OF A 4X4 STRONGLY MAGIC SQUARE

Transcription

1 IJMMS, Vol. 1, No. Geeralized 1, (Jauary-Jue Form 016):87-9 of A 4x4 Strogly Magic Square Serials Publicatios 87 ISSN: GENERALIZED FORM OF A 4X4 STRONGLY MAGIC SQUARE Neeradha. C. K, ad Dr. V. Madhukar Mallayya Abstract I this paper, differet types of magic squares is briefly described. A geeric defiitio for the strogly magic square is give. A geeralized form of a 4x4 Strogly magic square ad some basic properties of geeralized 4x4 strogly magic squares (R-N SMS) are also discussed. Keywords: Magic Square, Magic Costat, Strogly Magic Square, R-N SMS. 1. INTRODUCTION A magic square is a square array of umbers where the rows, colums, diagoals ad co-diagoals add up to the same umber. The costat sum is A= a is called a called magic costat or magic umber. A matrix m strogly magic square (SMS) of order 4 with magic costat 34 if the followig coditio is satisfied: For all m, such that 1 m 4, 1 4, we have a a a a m, m, 1 m1, m1, 1 34 where it is assumed that if a subscript exceeds 4 it is replaced by 1 (wrappig aroud rows ad colums). [8]. MATHEMATICAL DEFINITION OF MAGIC SQUARE A magic square of order is a th order matrix [a ij ] such that Assistat Professor, Dept. of Sciece & Humaities Mar Baselios College of Egieerig & Techology, Thiruvaathapuram ckeeradha@yahoo.co.i Professor & HOD, Dept. of Mathematics, Mohadas College of Egieerig & Techology, Thiruvaathapuram mmallayyav@gmail.com

2 88 Neeradha. C. K, ad Dr. V. Madhukar Mallayya a ij for i 1,,... (1) j1 a ji for i 1,,... () j 1 aii, ai, i 1 (3) i1 i1 Equatio (1) represets the row sum, equatio () represets the colum sum, equatio (3) represets the diagoal & co-diagoal sum ad represets the magic costat. 3. TYPES OF MAGIC SQUARES (a) Semi-magic square: Semi-magic square is a square that fails to be a magic square oly because oe or both of the mai diagoal sums do ot equal the magic costat.[4] (b) Pamagic Magic Square: If all the diagoals icludig those obtaied by "wrappig aroud" the edges of a magic square sum to the same magic costat, the square is said to be a pamagic square.[4] (c) Trimagic squares: If the etries of a magic square is replaced by its square or cube produces aother magic square, called trimagic square.[5] (d) Multiplicative Magic Square: Istead of addig the umbers i each row, colum ad diagoal, oe ca apply some other operatio. For example, a multiplicative magic square has a costat product of umbers. A multiplicative magic square ca be derived from a additive magic square by raisig (or ay other iteger) to the power of each elemet, because the logarithm of the product of umbers is the sum of logarithm of each.[6] (e) Templar Magic Square: A magic square-type arragemet of the words i the Lati setece "Sator Arepo teet opera rotas" ("the farmer Arepo keeps the world rollig") is called templar magic square.[6] (f) Alpha Magic Square: Alpha magic square is a magic square for which the umber of letters i the word for each umber geerates aother magic square.[7]

3 Geeralized Form of A 4x4 Strogly Magic Square 89 (g) Strogly Magic Square (SMS): Let M be a magic square of order 4 ad let it be represeted by matrix as follows: M a a a a a a a a a a a a a a a a The umbers i each row, ad i each colum, ad the umbers that ru diagoally i both directios, all add up to the umber 34. M is called a strogly magic square if the followig coditio is satisfied: For all m, such that 1 m 4, 1 4, we have a a a a m, m, 1 m1, m1, 1 34 where it is assumed that if a subscript exceeds 4 it is replaced by 1 (wrappig aroud rows ad colums). [8] Give below is a example of a strogly magic square called Sri Rama Chakra. A = Alog with the usual properties of a magic square, Sri Rama Chakra has the stroger property that all sixtee x sub squares (allowig wraparouds) have etries addig to STRONGLY MAGIC SQUARE (SMS): GENERIC DEFINITION Let A= a ij be a matrix of order aij for j 1 a ji for j 1, such that i i 1,,... (4) 1,,... (5) i1 a ii 1 1 t0 k0 a ik, jl a (6) i, i p i1 for i, j 1,,... (7)

4 90 Neeradha. C. K, ad Dr. V. Madhukar Mallayya where the subscripts are cogruet modulo Equatio (4) represets the row sum, equatio (5) represets the colum sum, equatio (6) represets the diagoal & co-diagoal sum, equatio (7) represets the sub-square sum with o gaps i betwee the elemets of () () rows or colums ad is deoted as M 0C or M 0R ad is the magic costat. Note: The th order sub-square sum with colum gaps or k row gaps is geerally deoted as M () kc or M () kr respectively. 5. EXAMPLES OF STRONGLY MAGIC SQUARE A = is of order I A all sixtee sub-squares (allowig wrap-arouds) have etries addig to B is of order I B, all 3 3, sub-square with o gaps i betwee the elemets of rows or colums add to the magic costat GENERALISED FORM OF A 4X4 STRONGLY MAGIC SQUARE FOR REAL NUMBERS (R-N SMS) A strogly magic square of order 4x4 ca be costructed from real umbers usig the geeralized form give below. The geeral form of a 4x4 Strogly Magic Square (R-N SMS) is give by a b c d cd ac bc R-N SMS 'M s ' = c d a b 3 3 ca bc cd where is the magic costat ad a, b, c, d R.

5 Geeralized Form of A 4x4 Strogly Magic Square 91 If we look ito the sum of row elemets, colum elemets, diagoal elemets ad co-diagoal elemets it ca be see that its value is the magic costat.the strogly magic square has a peculiar feature that the sub-square sum with o gaps i betwee the elemets of rows or colums which is deoted as M 0C () or M 0R () is also. 7. PROPERTIES OF R-N SMS 1. The sum of the four corers of this square is.. The sum of the four corers of sub square M (3) 0C or M (3) 0R is ad the sum of each diagoally opposite corer pair is ρ/ 3. Form ay triagle T s each side of which is made of three umbers of M S '. A few examples of the etries of such triagles are show below: a Ts cd d a or abc Ts a b 3 3 bc cd 4. I the sum of the six umbers alog the sides of the triagle is the same for all such triagles ad is equal to There exist several trasformatios which, whe applied to a strogly magic square yields aother strogly magic square like (a) Cyclig of rows or colums (b) Iterchage of colums 1 ad 3 or rows 1 ad 3 (c) Iterchage of colums ad 4 or rows ad 4 (d) Diagoal reflectios (e) Replacemet of every elemet of R-N SMS by m 8. ADVANTAGES OF R-N SMS 1. Ay 4 real umbers are oly required i the makig of R-N SMS.. It is a layma s Magic square 3. A structure which eables to study the mathematical properties very easily. 4. It is easy to formulate 5. This magic square is valid o all real values.

6 9 Neeradha. C. K, ad Dr. V. Madhukar Mallayya 9. CONCLUSION While magic squares are recreatioal i grade school, they may be treated somewhat more seriously i algebra courses. The study of strogly magic squares is a emergig iovative area i which mathematical aalysis ca be doe. Here the geeralized form of a 4x4 strogly magic square for real umbers called R-N SMS is itroduced which is of great use i the study of Strogly Magic Square. 10. ACKNOWLEDGEMENT We express our sicere gratitude for the valuable suggestios give by Dr. Ramaswamy Iyer, Former Professor i Chemistry, Mar Ivaios College, Trivadrum i preparig this paper. Refereces [1] Adrews, W. S. Magic Squares ad Cubes, d rev. ed. New York: Dover, 1960 [] William Symes Adrews Magic Squares ad Cubes i [3] Aderso, D. L. (001). Magic Squares [olie]. Available from ctm.org/lessodetail.aspx?id=l63 [4] Kraitchik, M. "Magic Squares." Ch. 7 i Mathematical Recreatios. New York: Norto, pp , 194. [5] Tarry G. "Le carré trimagique de 18." Compte Redu de la 34ème Sessio Cherbourg Paris: AFAS-Masso, pp , [6] e.wikipedia.org/wiki/magic square. [7] Sallows, L. C. F. "Alphamagic Squares." Abacus 4, 8-45, [8] T.V. Padmakumar Strogly Magic Square APPLICATIONS OF FIBONANCCI NUMBER, VOLUME 6 New Publicatio, Proceedigs of The Sixth Iteratioal Research Coferece o Fiboacci Numbers ad Their Applicatios, Washigto State Uiversity, Pullma, Washigto, USA, July 18-, April 1995April 1995.