Magic rectangles and modular magic rectangles

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Journal of Statistical Planning and Inference 51 (1996) 171-180

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Magic rectangles and modular magic rectangles A n t h o n y B. E v a n s * Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, USA

Abstract Magic rectangles are m × n matrices with entries 1..... ran, all row sums being equal and all column sums being equal. Sun established necessary and sufficient conditions for the existence of magic (m, n) rectangles. We introduce modular magic rectangles, variants of magic rectangles, and study two classes of modular magic rectangles: Pseudomagic and complete magic rectangles. We construct classes of pseudomagic, modular magic rectangles that are not magic rectangles, and classes of complete, modular magic rectangles. This suggests the problem of determining the spectra of pseudomagic, modular magic rectangles that are not magic rectangles; complete, modular magic rectangles; and complete, magic rectangles.

1. Introduction A magic (m, n) rectangle is an m x n matrix with entries 1 . . . . . mn, for which all the row sums are the same and all the column sums are the same. The entries of each row of a magic (m, n) rectangle must sum to n(mn + 1)/2, and the entries of each column of a magic (m, n) rectangle must sum to m(mn + 1)/2. A magic square of order n is a square matrix of order n with entries 1 . . . . . n 2, for which all the row sums, all the column sums, and b o t h m a i n diagonal sums are the same. Thus the entries of any row, column or m a i n diagonal must sum to n(n 2 + 1)/2. A modular magic (m, n) rectangle is an m x n matrix with entries 1 .... , mn, for which all the row sums are congruent m o d u l o mn and all the column sums are congruent m o d u l o mn. The a r r a y (1) is trivially a magic square, a magic (1, 1) rectangle, and a m o d u l a r magic (1, 1) rectangle. A magic square of order n is a magic (n, n) rectangle, and a magic (m, n) rectangle is a m o d u l a r magic (m, n) rectangle. Nontrivial magic squares exist for all orders greater than 2. Simple constructions of these can be found in Ball (1963). See D6nes and Keedwell (1974, C h a p t e r 6) for information on constructions of different types of magic squares using Latin squares.

* E-mail: [email protected]. 0378-3758/96/$15.00 © 199f--Elsevier Science B.V. All fights reserved SSDI 0 3 7 8 - 3 7 5 8 ( 9 5 ) 0 0 0 8 1 - X

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A.B. Evans~Journal of Statistical Planning and Inference 51 (1996) 171 180

Sun (1990) showed that a necessary and sufficient condition for the existence of a nontrivial magic (m, n) rectangle is m, n > 1, and m and n either both odd or both even, but not both 2. Bier and Rogers (1993), apparently unaware of the previous publication, achieved partial results. In this paper we will prove that the existence criteria for modular magic rectangles are virtually the same as for magic rectangles. We will introduce the concepts of pseudomagic and complete, modular magic rectangles, establish the existence of many pseudomagic, modular magic rectangles that are not magic rectangles, and construct classes of complete, modular magic rectangles. This suggests three open problems: Determining the spectra of pseudomagic, modular magic rectangles that are not magic; of complete, modular magic rectangles; and of complete, magic rectangles.

2. Modular magic rectangles For what values of m and n does there exist a modular magic (m, n) rectangle? We will establish necessary and sufficient conditions for their existence that are almost identical to those for magic (m, n) rectangles. We will next turn our attention to pseudomagic and complete, modular magic rectangles. Pseudomagic, modular magic (m, n) rectangles viewed modulo mn, have the row and column sums that we would expect of a magic (m, n) rectangle. This condition is not strong enough to force a modular magic rectangle to be magic: We will establish the existence of many pseudomagic, modular magic rectangles that are not magic rectangles. When viewed modulo m, the columns of a complete, modular magic (m, n) rectangle are permutations, and when viewed modulo n, the rows are permutations. We will give construction of classes of complete, modular magic rectangles. Theorem 1. T h e r e e x i s t s a nontrivial m o d u l a r m a g i c (m, n) r e c t a n g l e if and only if m, n > 1 and m - n modulo 2. Proof. It is clear that for a nontrivial modular magic (m, n) rectangle to exist m and n must both be greater than 1. Suppose m is even and n is odd and suppose there exists a modular magic (re, n) rectangle M = (mlj). Now, for some r and all i = 1,...,m,Z~=l mij = r m o d u l o m . Thus I + 2 + ... + m n = mr = O m o d u l o m . But 1 + 2 + ... + m n = m n ( m n + 1)/2 = m / 2 m o d u l o m , a contradiction. Hence there exists a nontrivial modular magic (m, n) rectangle only if m, n > 1 and m --- n modulo 2. The existence of magic (m, n) rectangles, and hence modular magic (m, n) rectangles for m , n > 1 and m = nmodulo 2, m and n not both 2, was proved by Sun (1990). For the case m and n both 2, the array

is a modular magic (2, 2) rectangle.

[]

A.B. Evans~Journal of Statistical Planning and Inference 51 (1996) 171-180

173

If M = (mig) and N = (hij) are matrices with integer entries then we will say that M is congruent to N modulo p if m o =- n~j m o d u l o p for all i,j, and Mmod p will denote the uniquely determined matrix, with entries from the set {0 . . . . . . p - 1}, that is congruent to M m o d u l o p. Let m and n be odd, greater than one, and relatively prime. If M is a magic (m, n) rectangle with row sum r and column sum c then r is congruent to n/2 m o d u l o mn and c is congruent to m/2 m o d u l o mn. This need not hold for m o d u l a r magic rectangles. If M is a m o d u l a r magic (m, n) rectangle, each column sum congruent to c m o d u l o ran, and each row sum congruent to r m o d u l o ran, then nc = mr =- 0 m o d u l o mn, and so c = 0 m o d u l o m and r = 0 m o d u l o n. If c =- m/2 m o d u l o n then we shall say that M is column pseudomagic, and if r = n / 2 m o d u l o m then we shall say that M is row pseudomagic. We shall say that M is pseudomagic if M is both row and column pseudomagic. If M is a pseudomagic, m o d u l a r magic (m, n) rectangle then the row and column sums of M m o d u l o any divisor of mn will be the correct sums for a magic (m, n) rectangle. We will construct some classes of pseudomagic, m o d u l a r magic rectangles that are not magic rectangles. We shall say that M is row complete modulo p (just row complete ifp = n) if each row of M contains n/p representatives of each congruence class m o d u l o p. Similarly we shall say that M is column complete modulo p (just column complete if p = m) if each column of M contains m/p representatives of each congruence class m o d u l o p. M is complete if M is both row complete and column complete. Thus if M is row complete then each row of Mmodn is a permutation of {0 . . . . . n - 1}, and if M is column complete then each column of Mmod m is a permutation of {0 . . . . . m -- 1 }. In general a magic (m, n) rectangle need not be either row or column complete. L e m m a 1. I f m and n are odd, greater than one, gcd(m, n ) = g c d ( m - 1, n ) = 1, and there exist positive integers a, b, relatively prime to m, for which a + b = n, then there exist pseudomagic, complete modular magic (m, n) rectangles.

Proof. Let Lm = (mij) be an m x n matrix, mij = i i f j ~< a, and nb*(m + 1)/2 - ab*i if j > a , where b b * - l m o d u l o m . Let Ln=(ni~) be an m x n matrix, n o = j if i ~ (m + 1)/2, and m(n + 1)/2 - (m - 1)j if i = (m + 1)/2. Let M be the uniquely determined matrix, with entries from {1 . . . . . ran}, satisfying M = Lm m o d u l o m and M = L n m o d u l o n . Then M is a pseudomagic, complete, m o d u l a r magic (re, n) rectangle. []

Corollary 1. I f m and n are distinct odd primes then there exist pseudomagic, complete, modular magic (m, n) rectangles. Proof. As the transpose of a m o d u l a r magic (m, n) rectangle is again a m o d u l a r magic (m, n) rectangle, we m a y assume that m < n. Thus gcd(m - 1, n) = gcd(m, n) = 1. There exist positive integers a and b relatively prime to m, satisfying a + b = n as

174

one of n L e m m a 1.

A.B. Evans~Journal of Statistical Planning and Inference 51 (1996) 171-180

1, n []

2 must be relatively prime to n. The result then follows from

As a variant of L e m m a 1 we present the following less powerful but symmetric result.

Corollary 2. I f m and n are odd, m,n > 1, and g c d ( m , n ) = g c d ( m , n - 1 ) = g c d ( m - 1 , n ) = 1 then there exist pseudomagic, complete modular, magic (re, n) rectangles. Proof. Let L , , = ( m i j ) be an m x n matrix, m o = i if j ~ ( n + 1)/2, and n(m + 1 ) / 2 - ( n 1)i if j = (n + 1)/2. Let Ln = (nij) be an m x n matrix, nii = j if i ¢ (m + 1)/2, and m(n + 1)/2 - (m - 1)j if i = (m + 1)/2. Let M be the uniquely determined matrix, with entries from { 1 . . . . . mn}, satisfying M = Lm modulo m and M = L n m o d u l o n . Then M is a pseudomagic, complete, modular magic (m,n) rectangle. [] Pseudomagic, modular magic (m, n) rectangles have the 'correct' row and column sums m o d u l o mn, but still need not be magic rectangles. In the next theorem we list some parameters m, n for which we can construct pseudomagic, modular magic (m, n) rectangles that are not magic rectangles.

Theorem 2. There exist pseudomagic, modular magic (m,n) rectangles that are not magic rectangles in the following cases. (i) m = 3, n odd, and n --- 2 m o d u l o 3. (ii) ( m , n ) = (5,7), (5,9), (5,11), (5,13), (5, 17), (5, 19), (5,23), (7,11), (7, 13), (7, 17), (7, 19), (11, 13), (11, 17), (11, 19), (13, 17), (13, 19), (13,23), (17, 19), and (29,31). Proof. (i) Using the construction of L e m m a 1, the entries in the first column will be 1, (n - 1)/2, and n + 1, which sum to (3n + 3)/2, not (9n + 3)/2, the correct column sum for a magic (3, n) rectangle. (ii) These parameters are established by constructing pseudomagic, modular magic rectangles, using L e m m a 1 or Corollary 2, and checking row and column sums. [] It should be noted that T h e o r e m 5 provides a simple way to show that a complete, modular magic rectangle, constructed as in Corollary 2, is not a magic rectangle.

Lemma 2. Let p, q and r be odd and pairwise relatively prime. If there exist complete, modular magic (p, q) and (p, r) rectangles then there exist complete, modular magic (p, qr) rectangles. Proof. Let M be a complete, modular magic (p,q) rectangle and N a complete, modular magic (p,r) rectangle. F r o m a p x q r matrix L as follows. Let

A.B. Evans~Journal of Statistical Planning and Inference 51 (1996) 171-180 L1 =(Mmodp ... Mmodp), L2-----(Mmodq ... Mmodq), a n d

175

L 3 = ( N 1 ... N r ) , w h e r e

Ni is the p x q matrix in which each column is the ith column of Nmod,. If L is the uniquely determined matrix, with entries from {1 . . . . . pqr}, for which L = L x m o d u l o p , L -= L2 m o d u l o q, and L = L a m o d u l o r , then L is a complete, m o d u l a r magic (p, qr) rectangle. [] Lemma 3. Let p and q be distinct primes and let r be a positive integer. I f there exists a complete, modular magic (p, q) rectangle then there exists a complete, modular magic (p, qr) rectangle. Proof. Let M be a complete, m o d u l a r magic (p, q) rectangle. F r o m a p x qr matrix L as follows. Let L1 = (Mmodp ... Mmodp), L2 = ((Mmoaq + Ko) ... (Mmodq + Kt)), where t = q ' - I - 1 and

Ki =

iq

...

iq ]

iq

...

iq

f(iq) f(iq) -

'

... f(iq) - (p - l)iq m o d u l o q' if gcd(q,p - lq) = 1,

[ iq iq]

or

Ki =

iq f(iq) Lf(iq)

... iq , .'. f(iq) ... f(iq)

f(iq) -

- ((p/2) - 1)iqmoduloq', if gcd(q,p - 1) # 1.

If L is the uniquely determined matrix, with entries from {1 . . . . . pq'), for which L - L1 m o d u l o p and L - L2 m o d u l o q', then L is a complete, m o d u l a r magic (p,q') recangle. [] Lemma 4. Let A be a magic ( m l , n l ) rectangle and let B be a modular magic ( m 2 , n 2 )

rectangle. (i) I f A and B are row complete and g c d ( m l n l , n 2 ) = 1 then there exists a row complete, modular magic (ml m2, nl n2) rectangle. (ii) I f A and B are column complete and g c d ( m l n l , m 2 ) = 1 then there exists a column complete, modular magic ( m i r a 2 , n 1 n2) rectangle. (iii) I f A and B are complete and gcd(ml n l, m2n2) = 1 then there exists a complete, modular magic (ml mE, n l n2) rectangle.

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A.B. Evans / Journal of Statistical Planning and Inference 51 (1996) 171-180

(iv) I f A and B are complete magic rectangles and gcd(ml nx,m2n2) = 1 then there exists a complete, magic (ml m2, nl n2) rectangle. Proof. Following the construction given by Bier and Rogers (1993) for magic rectangles, if A = (aij) and B = (bkt) then define C = (c,s), 1 <~ r <~ mlm2, 1 <~ s <~ nln2, by C,s=ai~+(bkl-- l ) m l n l , where r = i + ( k - 1 ) m l and s = j + ( l 1)nl. C is a modular magic (m~ m2, n~ n2) rectangle. We need only prove (i) as the proof of (ii) is similar, (iii) follows from (i) and (ii), and (iv) is a consequence of (iii) and the proof in Bier and Rogers (1993). Let us assume that A and B are row complete and that g c d ( m l n l , n 2 ) = 1. When we fix r we also fix both i and k, and j and l can vary independently over 1..... n~ and 1. . . . . n 2 , respectively. Let us pick a ~ {1 ..... n~} and b e { 1 .... ,n2}. There is a unique j for which a~j = a m o d u l o n ~ , and a unique l for which b = a + (bk, -- 1)mini modulo n2. Thus there is a unique s for which crs = a modulo n~ and crs = bmodulo n2, and so C is row complete. [] The existence of many complete, modular magic rectangles can readily be inferred from the preceding lemmas. The following theorem lists some easily recognized parameters for which the existence question for complete, modular magic rectangles has thus been settled.

Theorem 3. I f m and n are odd, m, n > 1, then there exists a nontrivial complete, modular magic (re, n) rectangle in the following cases. (i) At least one o f m, n prime and gcd(m,n) (ii) m = 3p, n = 3q, one o f p , q prime, p,q > (iii) m = 5p, n = 7q, one ofp, q prime, p,q > (iv) m = 7p, n = 5q, one ofp, q prime, p,q >

= 1. 1, gcd(p,q) = gcd(3,pq) = 1. 1, gcd(p,q) = gcd(35,pq) = 1. 1, gcd(p,q) = gcd(35,pq) = 1.

Proof. As the transpose of a complete modular magic rectangle is again a complete, modular magic rectangle, we are free to assume that m is prime. Let n = ql "'" qt be the factorization of n into pairwise relatively prime, prime powers, q~ a power of a prime Pi. By Corollary 1 there exists a complete, modular magic (m, p~) rectangle for each i, and so, by Lemma 3, there exists a complete modular magic (m, ql) rectangle for each i, and so, by Lemma 2, there exists a complete, modular magic (m, n) rectangle. (ii)--(iv) follow from (i), the fact that the magic square of order 3 is complete, the existence of a complete magic (5, 7) rectangle, constructed in the next section, and Lemma 4(iii). []

3. Complete, magic rectangles Consider the following construction of a complete, modular magic (5, 7) rectangle, using a construction similar to that of Lemma 1.

A.B. Evans~Journal of Statistical Planning and Inference 51 (1996) 171-180

M 5

177

Ii°°°22i] E11234511 1

1

1

4

4

2 3 4

2 3 4

2 3 4

1 1 3 3 0 0

and

My--

0

1

2

3

4

5

0

1 2 1

2 5 2

3 4 1 4 3 4

5 0 5

.

The complete, modular magic (5, 7) rectangle M, congruent to M5 modulo 5 and 7, is as follows.

M 7 modulo

M=

35 21 7 13 14

15 1 22 23 29

30 16 2 33 9

10' 31 17' 8 24

32' 4 11' 18 25

12 19 26 28 5

21" l 34 ' 20 J

Each column sum of M is 90, and each row sum of M, except for the first and third, is 126, the first row sum being 161 and the third 91. Thus each column sum is congruent to 20 modulo 35, and each row sum is congruent to 21 modulo 35. How can we correct the row sums and so obtain a complete, magic (5, 7) rectangle? That is the focus of this section - - correcting the 'incorrect' row and column sums of a complete, modular magic (m, n) rectangle to obtain a complete, magic (m, n) rectangle. In the example just presented there is a simple correction that can be applied. In some columns two of the entries primed. Swapping the primed pairs in each column corrects the row sums while leaving the column sums unchanged. The corrected magic (5, 7) rectangle is as follows.

2 M'= 13 14

1 22 23 29

16 2 33 9

31 10 8 24

4 32 18 25

19 26 28 5

641 27[

3/ 201

As our swapping did not have any effect on either row or column completeness, the new magic rectangle is complete. For the remainder of this section we will restrict ourselves to the construction given in Corollary 2. For a complete, modular magic (m, n) rectangle M = (mo) we will use ri to denote the ith row sum and cj to denote thejth column sum, and we define the ith row error to be r e ( i ) = n ( m n + 1 ) / 2 - r i and the jth column error to be ce(j) = m(mn + 1)/2 - c~. Obviously a complete, modular magic rectangle is a complete, magic rectangle if and only if all its row and column errors are zero. We will use (a, b) to denote the unique integer in {1 . . . . . ran} that is congruent to amodulo m and bmodulon, ~/ to denote (n(m + 1)/2 - (n - 1)i, (n + 1)/2) - (i,(n + 1)/2) and fl~ to denote ((m + 1)/2, m(n + 1)/2 - (m - 1)j) - ((m + 1)/2,j). The following theorem

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A.B. Evans~Journal of Statistical Planning and Inference 51 (1996) 171-180

gives information about row and column errors that can occur in the constrution used in Corollary 2. Theorem 4. Suppose m and n are odd, greater than one, and that g c d ( m , n ) - - g c d ( m - 1, n ) = g c d ( m , n - 1 ) = 1. Then the following hold for the complete, modular magic (m, n) rectangle M constructed in Corollary 2. (a) re((m + 1)/2) = ce( (n + 1)/2) = O. (b) r e ( i ) = O o r + mn and c e ( j ) = O or +_ran. (c) I f 1 <~ i < (m + 1)/2 then re(i) >~ O, and if (m + 1)/2 < i <<.m then re(i) ~< O. I f 1 ~/O, and if(n + 1)/2 < j <~ n then ce(j) ~< O. (d) l f i ~ (m + 1)/2 then re(i) = 0 if and only if (m + 1)/2 - i and cti have the same sign. I f j ~ (n + 1)/2 then ce(j) = 0 if and only if (n + 1)/2 - j and flj have the same sign. n

Proof. (a) If i = (m + 1)/2 then eel = 0 and rl = ~ = 1 ((m + 1)/2,j) = n--1 ~ k = O (km + (m + 1)/2) = n(mn + 1)/2. Hence re((m + 1)/2) = 0. Similarly ce((n + 1)/2) = 0. n n-I (b) If i 4: (m + 1)/2 then r~ = E j = I (i,j) + ~ = ~ , k = 0 (kin + i) + ~ = mn(n - 1)/2 + ni + oq. Thus re(i) = n(m + 1 - 2i)/2 - ~i. But I~/[ < m n and so n/2 - 3ran~2 < re(i) < n/2 + 3ran~2. As re(i) is a multiple of ran it follows that re(i) = 0 or ___ran. Similarly c e ( j ) = 0 or _ mn. (c) F r o m the equation re(i) = n(m + 1 - 2i)/2 - ~i we see that i f / < (m + 1)/2 then re(i) > - 0ti > - mn and so re(i) = 0 or ran. If i > (m + 1)/2 then re(i) < - cti < m n and so re(i) = 0 or - mn. Similarly for ce(j). (d) This follows from (b), the equation r e ( i ) = n(m + 1 - 2 i ) / 2 - 0q, and the equation c e ( j ) = m(n + 1 - 2j)/2 - fit. [] Thus for a complete, m o d u l a r magic (m, n) rectangle, constructed as in Corollary 2, the 'incorrect' rows have now error +_ mn and there are the same n u m b e r of rows with row error mn as there are rows with row error - ran. Similarly for columns. This suggests a strategy for constructing magic rectangles that m a y be complete. Start with a complete, m o d u l a r magic (m, n) recrangle M = (rag), constructed as in Corollary 2. Pair each row il, of M, with row error mn with a r o w i2, of M , that has row error - mn. We then look for pairs (milj~,mi2jl) . . . . . (mi~jr,mi2jr) for which Zk= 1 mi2jk -- m~ljk = mn. Swapping entries mi~jk with m~2~k corrects the row sums of rows il and i 2 without affecting column sums. Similarly for columns. We shall call this procedure the simple swapping procedure. Theorem 5. Suppose m and n are primes, 2 < m < n, and that god(m, n - 1) = 1. Then a magic (m, n) rectangle can be constructed using the simple swapping procedure if and only if ((mn + 1)/2 - J1 + m6)mod r a n > ((mn + 1)/2 -- •l)mod m n for all t~, 1 ~< 6 ~< (n - 1)/2, where 1 = (0, 1) and amod,,~ denotes the unique integer in {1 . . . . . mn} that is congruent to a m o d u l o mn.

A.B. Evans~Journal of Statistical Planning and Inference 51 (1996) 171-180

179

Proof. Suppose that M is the complete, modular magic (m, n) rectangle, constructed as in Corollary 2. When using the simple swapping procedure the sum of the entries swapped out of any column of M must be congruent to zero modulo n. With our construction, as m < n, no proper subset of the entries in any given column of M can have sum congruent to zero modulo n. Hence we can only construct a magic (m, n) rectangle from M, using the simple swapping procedure, if all the column sums of M are correct. Next suppose that re(i) = mn and re(k) = - mn. Let S be the sequence { m k j - - mij: j = 1 . . . . . n , j v~ (n + 1)/2}. Each element of S is congruent to zero modulo n and k - i modulo m. Thus the sum of any m elements of S is congruent to zero modulo mn. Suppose that no sequence of m elements of S sums to exactly mn. As r k = r i + 2 mn some sequence of m elements of S must have a positive sum. Let S1 be a sequence of m elements of S with the least possible positive sum. Thus sum S 1 >~ 2mn and, if as e $1 and a t e S - S 1 then a ~ < a t , and so sum S ~ 2 m n . But sum S + C t k - - ~ i = rk -- rl = 2mn and so, as ~k is positive and ~ is negative, sum S < 2mn, a contradiction. Hence the simple swapping procedure can correct any incorrect row sums. Thus we can construct a magic (m, n) rectangle from M, using the simple swapping procedure, if and only if all the column sums of M are correct. The column sums of M are all correct if and only ifce(j) = 0 forj < (n + 1)/2, if and only ifflj is positive for j < (n + 1)/2, if and only if ((m + 1)/2, m(n + 1)/2 - (m - 1)j) > ((m + 1)/2,j) for j < (n + 1)/2. Setting 5 = (n + 1)/2 - j yields the desired result. [] Using the simple swapping procedure we can easily construct another example of a magic (5, 7) rectangle. However, attempts to construct other new magic rectangles using the simple swapping procedure have so far failed. This suggests that the construction of Corollary 2 is a poor starting point for the construction of magic rectangles. Thus we should search for other constructions of modular magic rectangles, that could then be analyzed and modified, using techniques similar to those of this section, to provide new examples of magic rectangles. Any mention of completeness is missing from this discussion. To implement the simple swapping procedure we have started with a complete, modular magic rectangle, constructed by the method or Corollary 2. If M is such a complete, modular magic (m, n) rectangle, and M is a magic (m, n) rectangle, constructed from M using the simple swapping procedure, then the column sums of M must be correct. The procedure may rearrange, but cannot change, the entries of any column of M, and so column completeness is preserved. Is row completeness also preserved? As the (m + 1)/2th row sum of M is always correct, any pair of swapped entries must be congruent modulo n, unless they are in the (n + 1)/2th column. Thus we have proved the following.

Corollary 3.

Suppose m and n are primes 2 < m < n, and that gcd(m,n - 1) : 1, M is the complete modular magic (m, n) rectangle, constructed as in Corollary 2, and M is a magic (m, n) rectangle, constructed f r o m M using the simple swapping procedure.

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A.B. Evans / Journal of Statistical Planning and Inference 51 (1996) 171 - 180

M_ is a complete, maoic (re, n) rectanole if the (n + 1)/2th columns of M and M are the same.

While the attempt, in this section, to construct complete magic rectangles has mostly failed, it should be noted that the construction of Corollary 2 yields modular magic rectangles that are not just complete, but also pseudomagic. Thus, when using the methods of this section, each failed construction of a complete, magic rectangle yields a construction of a pseudomagic, modular magic rectangle that is not a magic rectangle. We list three open problems.

Problem 1. Determine the spectrum for pseudomagic, modular magic rectangles that are not magic rectangles. Problem 2. Determine the spectrum for complete, modular magic rectangles. Problem 3. Determine the spectrum for complete magic rectangles.

Acknowledgements I would like to thank the referee and editor for calling my attention to the paper by Sun (1990).

References Ball, W.W.R. (1963). Mathematical Recreations and Essays. Macmillan, New York. Bier, T. and D. Rogers (1993). Balanced magic rectangles. European J. Combin. 14(4), 285-299. D6nes, J. and A.D. Keedwell (1974). Latin Squares and Their Applications. Academic Press, New York. Sun, R.G. (1990). Existence of magic rectangles. Nei Mongol Daxue Xuebao Ziran Kexue 21, 10-16.