POWERS OF 3RD ORDER MAGIC SQUARES

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1 Fuzzy Sets, Rough Sets ad Multivalued Operatios ad Applicatios, Vol. 4, No. 1, (Jauary-Jue 01): Iteratioal Sciece Press POWERS OF 3RD ORDER MAGIC SQUARES Sreerajii K.S. 1 ad V. Madhukar Mallayya Abstract: Powers of 3rd order magic squares are defied ad some properties of such powers are discussed here. AMS Classificatio No: MSC 000, 15A36 Keywords: Magic square, Semi Magic Square, Magic Costat, Powers, Odd, Eve etc. 1. INTRODUCTION A magic square is a square matrix i which if we add up the umbers i the square horizotally, vertically or diagoally, the sum remais the same. We have various refereces to magic squares i the aciet ad medieval literature. I Chia, such squares were kow eve before Christia era. There it was called Loh Shu. I Idia, Bhaskaracharya (1th Cetury A.D), Naraya Padit (14th Cetury A.D) have writte verses to frame such squares. I Idia, a procedure called Vedic Method is beig used to costruct magic squares eve before ceturies. A 4 th order magic square is foud i a work of Varahamihira. Oe amog the famous magic squares i Idia is Sree Rama Chakra which is of 4 th order. It takes a importat role i Hidu astrology. Several magic squares of various orders are available to us such as Euleria Magic Squares, Kubera Chakra, Mahadeva Soori s Magic Square, Mars Square, Nasik Magic Square, Claude - Gaspar Bachet Magic square, Milleium-000, Topsyturvy Magic Squares, Strogly Magic square, Sree Rama Chakra, Ramauja s Square, Khajuraho Squares, Wishig Caps, Albrochet Durer s Melacholia-I etc. I real life situatios, some problems relatig to divisio of objects equal i umbers ad value ca be easily solved by costructig a magic square i accordace with the give coditios. Apart from the recreatioal ad mythological aspects of magic squares, it is foud that they posses several advaced mathematical properties. Here we deal with some properties of powers of 3rd order magic squares.. NOTATIONS AND MATHEMATICAL PRELIMINARIES.1 Magic Square A magic square of order is a square array [a ij ] of umbers such that 1, Departmet of Mathematics, Mar Ivaios College, Thiruvaathapuram majusree1118@yahoo.com, crimstvm@yahoo.com

2 38 Sreerajii K.S. ad V. Madhukar Mallayya a k, for i 1,,3, , (1) j 1 ij j 1 a k, for i 1,,3, , () ji a k, ad a k (3) j 1 ij i 1 i, i 1 Where k is a costat ad the above metioed a ij s ad a ji s are the row ad colum elemets ad a ii s & a i, i + 1 s are the left ad right diagoal elemets of the magic square respectively.. Magic Costat The costat k i the above defiitio is kow as the magic costat or magic umber. Magic costat of the magic square A is deoted as (A). For example, the below table is a 3rd order magic square with = Semi Magic Square I defiitio.1, if oly coditios (1) ad () are satisfied, the that square array is kow as a semi magic square..4 Additio of Two th Order Magic Squares If A ad B are two th order magic squares, the their sum A + B is agai a magic square each elemet of which is obtaied by addig the correspodig elemets of A ad B. I geeral, if A = [a ij ad B = [b ij ' the A + B = C = [c ij, where c ij = a ij + b ij. [Here (A + B) = (A) + (B).].5 Multiplicatio of a Magic Square by a Scalar Multiplicatio of a th order magic square A = [a ij with a scalar 'c' is deoted by ca ad is obtaied by multiplyig every elemet of A by c. i.e., ca = [ca ij ]. [Here ca is a magic square ad (ca) = c (A)]..6 Liear Combiatio of Two Magic Squares If A = [a ij ad B = [b ij are two magic squares ad & are ay two real umbers, the the liear combiatio ' A + B' of A ad B is defied as A + B = C = [c ij ], where c ij = a ij + b ij. [Here A + B is a magic square ad ( A + B) = (A) + (B).]

3 Powers of 3rd Order Magic Squares 39.7 Powers of a Magic/semi Magic Square If A is a magic/semi magic square, the the kth power of A is defied as A k = A.A.A (k times, provided k is a positive iteger). Here the procedure of matrix multiplicatio is used. For example, if A = The, A PROPOSITIONS Propositio 3.1: If A is a 3 rd order magic square, the A is a semi magic square. Proof: Give that A is a 3 rd order magic square. The by [3], A ca be represeted as Table (i) a x a x y a y A a x y a a x y a y a x y a x Table (ii) b c c. A A A c b c c c b

4 40 Sreerajii K.S. ad V. Madhukar Mallayya where b = 3a + x y ad c = 3a x + y Clearly, A is a semi magic square ad (A ) = b + c = 9a. Note: Hereafter Table (i) ad Table (ii) are used to represet a 3 rd order magic square ad its square respectively. Corollary 3.1: If (A) = e, the (A ) = e, where A is a 3rd order magic square. Proof: From propositio 3.1, e = 3a ad (A ) = 9a = (3a) = e. Propositio 3.: If A is a 3 rd order magic square, the A 3 is also a magic square. Proof: d x( c b) d ( x y)( b c) d y( c b) 3 A A A d x y b c d d x y c b [where d = a(b + c)].. ( )( ) ( )( ) d y( b c) d ( x y)( c b) d x( b c) It ca be verified that A 3 is a 3 rd order magic square ad (A 3 ) = 3d = 3a(b + c). Corollary 3.: If (A) = e, the (A 3 ) = e 3, where A is a 3 rd order magic square. Proof: From propositio 3., (A 3 ) = 3a(b + c) = e.e = e 3 (from propositio 3.1 ad corollary 3.1). Propositio 3.3: If A is a 3 rd order magic square, the A k is a magic square wheever k is a positive odd iteger. Proof: Give that A is a 3 rd order magic square. We eed to show that A k is a magic square, wheever k is a positive odd iteger. If k is a positive odd iteger, k will be of the form k = 1; = 1,, 3,.... We ca prove this by mathematical iductio o. Here A is a magic square ad from propositio 3., A 3 is also a magic square. So the claim is correct whe = 1,. Assume that the argumet is true whe = m. i.e., A m 1 is a magic square. The it is eough to show that the argumet is true whe = m + 1. i.e., A (m+1) 1 = A m+1 is also a magic square. Now, A m+1 = A m 1. A. Sice A m 1 is a 3rd order magic square, it ca be represeted as The, it ca be show that a x a x y a y m 1 A a x y a a x y a y a x y a x d x ( c b) d ( x y )( b c) d y ( c b) m 1 A d x y b c d d x y c b ( ) ( ) ( )( ) d y ( b c) d ( x y )( c b) d x ( b c)

5 Powers of 3rd Order Magic Squares 41 where d' = a'(b + c). It ca be verified that A m + 1 is a magic square of order 3 ad (A m + 1 )) = 3d' = 3a' (b + c). Hece the claim is correct whe = m + 1. So it is correct for = 1,, 3,... i.e., if A is a 3rd order magic square, the A k is a magic square wheever k = 1, = 1,, 3,.... Propositio 3.4: If A is a 3 rd order magic square, the A k is a semi magic square wheever k is a positive eve iteger a A k will take the form k A Proof: Proceedig as i propositio 3.3, here k =, = 1,,3,.... We use mathematical iductio o. From propositio 3.1, A is a semi magic square. Now, b c c bc c bc 4 A A. A c bc b c c bc c bc c bc b c Clearly A 4 is a semi magic square ad takes the required form. So the claim is correct whe = 1,. Assume that the argumet is true whe = m. i. e., A m is a semi magic square ad takes the required form. Now whe = m + 1,. b c c ( m 1) m A A A c b c c c b b c c ( b c) c ( b c) c ( b c) b c c ( b c) c ( b c) c ( b c) b c Clearly, A (m+1) is a semi magic square of the required form ad (A (m+1) ) = b + c + b + 4 c = ( + ) (b + c). Propositio 3.5: If (A) = e, the (A k ) = e k, where A is a 3 rd order magic square. Proof: Give that A is a 3 rd order magic square. Let us seek the help of mathematical iductio for provig the proposed claim. Case (i) whe k is eve : The, by propositio 3.4 A k is a semi magic square. Sice k > 0 is eve k will be of the form k = ; = 1,, 3,.... By corollary 3.1, A is a semi magic square ad (A ) = e. From Table (ii), e = b + c (1)

6 4 Sreerajii K.S. ad V. Madhukar Mallayya So the claim is correct whe = 1. Assume that the argumet is true whe = m. i.e., A m is a semi magic square ad (A m ) = e m. Now, by propositio 3.4 e m = + () Now, for = m + 1, from propositio 3.4, (A (m+1) ) = ( + ) (b + c). Now, ( + ) (b + c) = e m. e = e (m+1) ) [from (1) ad ()]. i.e., A (m+1) ) is a 3 rd order semi magic square ad (A (m+1) ) = e (m+1) ), where e (A). Hece the claim is true for all the positive eve itegers. Case (ii) Whe k is odd : The, by propositio 3.3 A k is a magic square. Sice k > 0 is odd k will be of the form k = 1; = 1,, 3,... From corollary 3.1, (A) 3a. i.e., e = 3a (3) By propositio 3., A 3 is a magic square ad (A 3 ) = 3a(b + c) = e.e = e 3 [by (3) ad (1)]. The argumet is true whe = 1,. Assume that our claim is correct whe = m. i.e., A (m 1) ) is a magic square ad (A m 1 ) = e m 1. From propositio 3.3, e m 1 = 3a' (4) For = m + 1, by propositio 3.3, the (A (m+1) 1 ) = (A m+1 ) = 3a'(b + c). Now, 3a'(b + c) = e m 1. e = e m+1 [by (1) ad (4)]. Therefore, A m+1 is a magic square ad (A m+1 ) = e m+1, where e is (A) ad this completes the proof. Propositio 3.6: If A is a 3 rd order magic square, the A 3 ca be represeted as A 3 = 3( A + J) where, are itegers ad J (Here J is also a 3 rd order magic square with all etries equal to 1.) Proof: From propositio 3., d x( c b) d ( x y)( b c) d y( c b) 3 A A A d x y b c d d x y c b. ( )( ) ( )( ) d y( b c) d ( x y)( c b) d x( b c) [where d = a(b + c)]. Usig the priciple of additio ad multiplicatio by a scalar of magic squares, A 3 ca also be represeted as A 3 = [d a(b c)] J + (b c)a = 3( A + J), where = (x y ), ad = a[3a ] ad a x a x y a y A a x y a a x y a y a x y a x

7 Powers of 3rd Order Magic Squares 43 Hece we are doe. Corollary 3.3: If A is a 3 rd order magic square, the A 3 0(mod3). i.e., all the etries of A 3 is divisible by 3. Proof: It immediately follows from propositio 3.6. Note: The above propositios ca also be proved by usig the cocept of Dot Product of magic squares (Ref : Sreerajii K.S ad Dr. V. Madhukar Mallayya, Eige Values ad Dot Products of Third Order Magic Squares, Idia Joural of Mathematics ad Mathematical Scieces, Jue 01). 4. CONCLUSION It is foud that the square of a 3 rd order magic square is a semi magic square ad cube of such magic square is agai a magic square. Geerally, for a 3 rd order magic square A, A k is a semi magic square or magic square accordig as k is a positive eve or odd iteger. If (A) = e, the (A k ) = e k. Fially we represeted A 3 as a liear combiatio of A ad J ad proved that, the cube of a 3rd order magic square is cogruet to 0(mod3). Ackowledgemet The authors express deep sese of gratitude to former Prof. Dr. Rama Swamy Iyer, for his valuable suggestios i preparig this paper. REFERENCES [1] Bibhutibhusa Datta ad Awadhesh Naraya Sigh (Revised by Kripa Sakar Shukla), Magic Squares i Idia, Idia Joural of History of Sciece, 199 Editio, [] Dave Poutey, O Powers of Magic Square Matrices, Liverpool Mathematical Society, Presidetial Address, April 008. [3] Richard Lodholz, Mathematics of Magic Squares, Washigto Uiversity Middle School Mathematics Teacher s Circle, April 008.