Journal of lgebra Number Theory: dvances and lications Volume 7 Number Pages -6 SQURING THE MGIC SQURES OF ORDER STEFNO BRBERO UMBERTO CERRUTI and NDIR MURRU Deartment of Mathematics University of Turin via Carlo lberto Torino Italy email: Nadir.murru@unito.it bstract In this aer we resent the roblem of counting magic squares and we focus on the case of multilicative magic squares of order. We give the exact number of normal multilicative magic squares of order with an original and comlete roof ointing out the role of the action of the symmetric grou. Moreover we rovide a new reresentation for magic squares of order. Such reresentation allows the construction of magic squares in a very simle way using essentially only five articular matrices.. Introduction magic square is defined as an n n matrix of integers where the sum of the numbers in each line (i.e. in each row in each column and in each diagonal) is the same. Magic squares have a very rich history (see Mathematics Subject Classification: 55 5B5. Keywords and hrases: additive magic squares multilicative magic squares normal magic squares symmetric grou. Received ril Scientific dvances Publishers
STEFNO BRBERO et al. e.g. the beautiful book of Descombes [6]) and they have many generalizations. There are a lot of contributes to their theory from several different fields (see e.g. the classical book of ndrews []). very difficult job is counting how many magic squares there are for a given order and a given line sum but if we relax the requirements and we ask for the so-called semi-magic squares (i.e. magic squares without the condition on the diagonals) the counting is easier. For order Bona [] has found that the number of semi-magic squares is r r r where r is the line sum (let us observe that in [] semi-magic squares are called magic squares). For a general theory of counting semi-magic squares you can see ([] Chater 9). The classic magic squares which are called normal are n n matrices whose entries consist of the numbers n and in each line the sum of the numbers is the same. Their number is known for the orders and 5. The case of order is esecially interesting. There are exactly normal magic squares of order (u to symmetries of the square i.e. in total they are = 7 ). They were enumerated for the first time in 69 by Frenicle de Bessy. The Frenicle method was analytically exanded and comleted by Bondi and Ollerenshaw []. Of course one can think to different kinds of magic squares only changing the oeration. Then a multilicative magic square is an n n matrix of integers where the roduct of the numbers in each line is the same. This is not a new idea for examle multilicative semi-magic squares of order are studied in [7]. There are several different methods to construct multilicative magic squares. The most obvious method is to transform an a additive magic square = ( a ij ) into M = ( b ij ) for any base b. More intriguing ways are exlained in [6] (multilicative magic squares are usually constructed by using orthogonal latin squares or geometric rogressions).
SQURING THE MGIC SQURES OF ORDER In [9] the roblem of finding the exact number of multilicative magic squares of order has been osed. In order to solve such question in [] a corresondence between additive and multilicative magic squares is roosed. The authors claimed that such function is an isomorhism even though it is only an injective morhism as ointed out in []. Since such revision is only a note the above roblem is not comletely clarified and the correct number of multilicative magic squares is given without a rigorous roof. In the next section we clarify the above questions roosing an original and comlete roof. Moreover in the last section a novel reresentation for some magic squares is roosed allowing an easy way to construct them.. Counting Multilicative Magic Squares of Order normal multilicative magic square n n is created using n rimes n. ll the divisors of k = n must aear once time in the matrix and in each line the roduct is the same. The magic constant is k and it is well-known that in a normal multilicative magic square in each line any rime comares with ower one and exactly two times. For examle a normal multilicative magic square of order is. In the following we use M k n to indicate the set of normal multilicative magic squares n n with magic constant k. hard roblem to solve is to find the order of M as osed in [9]. In order to answer to this question M k is made in corresondence with the set of normal additive magic squares []. k
STEFNO BRBERO et al. normal additive magic square is a matrix n n whose entries consist of the numbers n and in each line the sum of the n( n ) numbers is the same. The magic constant is. From now on we name the set of normal additive magic squares n n. n It is well-known that the order of is 7 or u to symmetries of the square []. Can we find a similar result for M? In [] the following corresondence is defined: k f : M. k n n If f is bijective M k is determined. The function f mas M = ( mij ) M k n into = ( a ) ij. The element m ij is made in n corresondence with a ij where a ij is the base number between and n whose base exression is the n-string associated to m. This string is constructed as follows: arrange the n rime numbers in ascending order; if k is a factor of m ij lace a in the k-th osition of the string; if not lace a. Such a function f is surely injective but unfortunately it is not surjective and the order of M k is not = 7. Indeed we can consider the grou S which ermutes the four rimes involved in M k. Theorem. The order of S divides the order of M. k Proof. Surely when a ermutation ρ S acts on M M k then ρ ( M ) is still an element in M. Furthermore if ρ is not the identity it k does not fix any element of M i.e. any orbit has elements. k Therefore by Burnside lemma S must be a divisor of M. k ij
SQURING THE MGIC SQURES OF ORDER 5 s immediate consequence of the revious theorem 7 can not be the order of M since does not divide 7. The correct answer is k given in the following theorem: Theorem. The order of M k is. Proof. If we consider and we write its entries in base then in each line the sum of the digits in the first osition multilied by and added to the sum of the digits in the second osition must yield. The only ossible combinations are = 7 = 6 6 = 5. So if has a line in which the digits in the second osition have sum we are in the only two situations where indicates the first osition of the number in base. Considering the reresentation in base of these strings we have one of the following situations:. Now if we try to create M M k using the inverse of f if has a line which resents one of these situations we can not have a normal multilicative magic square. In fact if we are in the first situation for examle then the rime will not aear in this line. Similarly if a line of has digits in the second osition with sum the ossible cases are
6 STEFNO BRBERO et al. which corresond in base to the strings. Once more if we try we fail to obtain M M k. Finally if has a line whose digits in the second osition have sum 6 we have the ossibilities. With similar arguments as the ones used before it is easy to see that the last two situations do not allow to obtain a normal multilicative magic square. Therefore the squares from which we obtain a normal multilicative magic square are only those with lines whose entries in base have digits in the first and second osition comosed by strings. In [] all the squares in are classified. The squares in generated using only these strings i.e. which corresond to squares in M k are only those in category one ([]. 5). They are exactly 5 unless of symmetries of the square. Therefore our arguments and the injectivity of f allow us to conclude that the order of M k is 5 =.
SQURING THE MGIC SQURES OF ORDER 7. New Reresentation for Magic Squares In this section we see a reresentation for normal additive magic squares which corresond to normal multilicative ones. In [5] a lower bound for the distance between the maximal and minimal element in a multilicative magic square is given. In order to do that the relation between additive and multilicative magic squares is highlighted recalling that a multilicative magic square can be factorized as i where i s are additive magic squares. Moreover focusing i i on magic squares of order in [5] the Hilbert basis (comosed by magic squares) for such magic squares is exlicited. However such reresentation and the relation between additive and multilicative magic squares are not really manageable in order to construct additive and multilicative magic squares of order. Here the roosed reresentation allows to determine all (and not only) the normal additive magic squares of order which corresonds to normal multilicative magic squares. In this way they can all be easily constructed essentially using only 5 basic matrices. We consider M = ( ) M by means of f we have the m ij k corresondent = ( aij ) and we consider its entries in base. Now we decomose into four matrices whose entries are only or so that the entries in osition ij of the matrices form the string a. ij Examle. From the normal multilicative magic square
STEFNO BRBERO et al. using f we obtain the normal additive magic square 5 7 9 6 5 = and it can be decomosed into. Since is derived from a matrix in k M by means of f the matrices are all and only those having in each line exactly two ones. We call forms these matrices which allow us to construct any magic square in corresonding to magic squares in. k M These forms are exactly 6. Theorem. There are 6 different matrices with entries or such that in each line there are exactly two ones. Proof. The strings that we can use to generate these forms are only six.
SQURING THE MGIC SQURES OF ORDER 9 If we choose as a diagonal a string with different extremes then we have only two ossible different forms. For examle.. On the other hand if we choose as a diagonal a string with same extremes then we have four ossible forms. Since we have four strings with different extremes and two strings with same extremes our forms are. 6 = From these 6 forms we can individuate five fundamental forms: = = B = = D C
STEFNO BRBERO et al.. = E ll the remaining can be found acting on these five forms with the grou of symmetries of the square. D Combining any four forms as we have done in the revious examle the resulting matrix is always an additive magic square not necessarily normal and it corresonds to a multilicative one. Theorem. If are fundamental forms or matrices obtained from fundamental forms through the action of D then is always an additive magic square which corresonds to a multilicative magic square. Given a magic square constructed using our forms we can consider all the ermutations of the forms. fter a ermutation we have still a magic square. Thus we can use the grou S over our forms. Examle. We consider
SQURING THE MGIC SQURES OF ORDER which corresonds to. 6 5 9 7 5 Permuting the ositions of the forms we obtain which is equal to
STEFNO BRBERO et al. 6 9 5 7. 5 We have seen that using our forms we obtain always an additive magic square but it is not necessarily normal. Finally let us see how to utilize the forms in order to generate all and only the normal additive magic squares corresonding to normal multilicative magic squares. The orbit of the fundamental forms with resect to the action of D are = { } B = { B B B B } C = { C C C C } D = { D D D D } E = We easily observe that { E E }. = = U E E =. Furthermore for any form in the orbits B C D there is another form in the same orbit such that their sum is U. We call class the set ( B C D) whose elements are all the magic square obtained combining the forms belonging to the orbits B C D
SQURING THE MGIC SQURES OF ORDER (e.g. B C D or B C D are elements of ( B C D) ). In the next theorem we show all the classes which rovide normal additive magic squares. Remark. We obtain normal additive magic squares which corresonds to all the normal multilicative magic squares only from the classes ( C D E) ( B B C C) ( B B C D) ( B B D D) ( B C C D) ( B C D D) ( C C D D). We have to made clear some details () When we choose a form in the orbit C the forms available in the orbit D are only two and vice versa excet for the class ( C C D D). () When we have a class with two forms from the same orbits their sum must not be U. () We can not take two times the same form in the same class. Considering this remarks we can count the magic squares obtainable from these classes and we check that they are exactly. For the class ( C D E) we can choose forms from the orbit from the orbit C only from the orbit D and from E. Thus from this class we can obtain normal additive magic squares unless of ermutations. Thus we have = 76 normal additive magic squares from ( C D E) and we write
STEFNO BRBERO et al. ( C D E) = 76. Similarly we find ( B B C C) = ( ) 6 = ( B B C D) = ( ) = 76 ( B B D D) = ( ) 6 = ( B C C D) = ( ) = 76 ( B C D D) = ( ) = 76 ( C C D D) = ( ) 6 = and 76 76 76 76 =. Remark. ll the magic squares that can be reresented through our notation can be classified and identified by the membershi class. Remark. Such reresentation allows to construct in a simle way all the normal multilicative magic squares of order. We conclude this aer with a further examle. Examle. Let us consider the factorization 5 67 of. We take a magic square in M 5 5 67 5 6 567 5 = 67 567 5 67 67 5 67 5 5 67 567 567 its image through f is
SQURING THE MGIC SQURES OF ORDER 5. 6 5 7 9 5 = This square belongs to the class ( ) D D C C in fact it can be decomosed as follows:. References [] W. S. ndrews Magic Squares and Cubes Dover Publications New York 96. [] M. Bona new roof of the formula for the number of magic squares Mathematics Magazine 7 (997) -. [] M. Bona Introduction to Enumerative Combinatorics McGraw Hill 7. [] Sir H. Bondi and Dame K. Ollerenshaw Magic squares of order four Phil. Trans. R. Soc. Lond. 6 (9) -5. [5] J. Cilleruelo and F. Luca On multilicative magic squares The Electronic Journal of Combinatorics 7 #N. [6] R. Descombes Les Carrés Magiques Vuibert. [7] D. Friedman Multilicative magic squares Mathematics Magazine 9(5) (976) 9-5.
6 STEFNO BRBERO et al. [] C. Libis J. D. Phillis and M. Sall How many magic squares are there? Mathematics Magazine 7 () 57-5. [9] Macalester College Problem of the Week Nov. 99. [] L. Sallows C. Libis J. D. Phillis and S. Golomb News and letters Mathematics Magazine 7() () -. g