RESEARCH ARTICLE. Linear Magic Rectangles

Transcription

1 Linear and Multilinear Algebra Vol 00, No 00, January 01, 1 7 RESEARCH ARTICLE Linear Magic Rectangles John Lorch (Received 00 Month 00x; in final form 00 Month 00x) We introduce a method for producing a variety of p r p s magic rectangles using Z p-linear transformations This adds significantly to the collection of known magic rectangles with non-coprime dimensions Keywords: magic rectangle; magic square; combinatorial design AMS Subject Classification: 05B15; 15B33; 05B30 1 Introduction Our purpose is to introduce a linear-algebraic construction for magic rectangles of size p r p s, where p is prime and 1 r s For fixed p, r, and s, this method allows for the production of many different magic rectangles It follows that, for many non-coprime dimensions, this construction will significantly augment the sparse collection of known magic rectangles A magic rectangle of size m n is a rectangular array containing integers 0, 1,, mn 1 such that the row sums are constant and the column sums are constant (The constants are different if m n) When m = n and both diagonal sums are equal to the row/column sums, one obtains a magic square The construction of magic squares is a venerable pastime, dating back to the Lo-Shu square of ancient China (ca 650 bce) A variety of magic square constructions may be found in [1]; connections among magic squares, magic rectangles, orthogonal latin squares, and statistical design are described in [6], [9], and [10] Efforts to construct magic rectangles are much more recent and far less numerous than those for their square counterparts Sun [11] showed that a magic rectangle of order m n exists if and only if m and n have the same parity, are both larger than 1, and are not both (We declare such sizes m n to be admissible) Similar (partial) results were achieved independently in [], and were further refined in [7] Other constructions of magic rectangles may be found in [3], [4], and [5] Invariably these constructions have a combinatorial flavor and produce a single example (and its obvious permutations) for each admissible size We take a different approach: instead of a combinatorial method producing one example for each of a large set of sizes, we introduce a linear-algebraic method that will produce a variety of examples for a rather more limited set of sizes, namely p r p s where p is prime This, together with the magic rectangle product theorem given in [], will increase the number of magic rectangle examples for many admissible sizes m n where gcd(m, n) > 1 The paper is organized as follows: The construction is given in Section, conditions we must place upon certain aspects of the construction are developed in Section 3, and in Section 4 we show that these conditions can be satisfied for each ISSN: print/issn online c 01 Taylor & Francis DOI: / xxxxxxxxx

2 John Lorch size of the form p r p s where r < s (The case r = s is addressed in Section 3) Along the way we will have occasion to perform arithmetic in both Z and Z p Arithmetic occurring in Z p with final result regarded as an integer will be indicated by angle brackets For example, if p = 5 then = Z Construction Let p be prime and 1 r s In this section we propose a method for producing magic rectangles of size p r p s using Z p -linear operators Locations in a p r p s magic rectangle can be described by elements of the vector space Zp r+s Rows are enumerated from the top, beginning with 0 and ending with p r 1; columns from left to right beginning with 0 and ending with p s 1 By expressing each row number in base p we can identify row locations with Z r p Similarly we can identify column locations with Z s p, and therefore any grid location (row,column) may be identified with an element of Z r p Z s p = Z r+s p By way of illustration, the symbol 5 in Figure 1 lies in location Z +3, where the first two entries indicate the row and the last three entries indicate the column Figure 1 A magic rectangle of size 3 Symbols in a p r p s magic rectangle can also be described by elements of Z r+s p These symbols consist of the numbers {0, 1,, p r+s 1} Each symbol Λ has a unique base-p expansion Λ = λ p r+s 1p r+s 1 + λ p r+s p r+s + + λ p p 1 + λ 1 p 0, where λ p j {0, 1,, p 1} for each j {0, 1,, r + s 1} Therefore we can make the identification Λ (λ p r+s 1,, λ p 0) Z r+s p We will find it convenient to write Λ = (Λ r, Λ s ), where Λ r = (λ p r+s 1,, λ p s) and Λ s = (λ p s 1,, λ p 0) For example, if p =, r =, and s = 3 as in Figure 1, then Λ = 5 corresponds to (Λ r, Λ s ) = (11, 001) We will routinely view these vectors, both for positions and symbols, as column vectors via transpose ( ) A B The magic rectangle construction is as follows Let M = be an (r + C D s) (r + s) matrix with entries in Z p, expressed in block form, where the sizes of A, B, C, D are r r, r s, s r, and s s, respectively Define a linear mapping T M : Z r+s p Z r+s p by T M (Λ) = MΛ The mapping T M uniquely determines a p r p s array with entries in {0, 1,, p r+s 1} by declaring T M (Λ) to be the

3 Linear Magic Rectangles 3 array location housing the number with base-p representation Λ (as described in the previous paragraphs) When p =, r =, and s = 3, the matrix M = determines the magic rectangle shown in Figure 1 To see that the number 5 is sent the to the correct location, recall that Λ = and so MΛ = 01101, which is indeed the location housing 5 as described above 3 Conditions In the construction of Section, clearly the matrix M must be nonsingular so that each location houses a unique number In this section we develop a list of conditions on the matrix M that guarantee a magic rectangle These conditions are summarized in Theorem 34 Throughout we let the matrix M, submatrices A, B, C, D, and the mapping T M be as in Section Lemma 31: If M and A are nonsingular and B has full rank then the rectangular array of numbers determined by T M has magic rows Proof : Let µ Z r p A number Λ = (Λ r, Λ s ) lies in the µ-th row of the array if and only if AΛ r = µ BΛ s (1) Since B is of full rank its kernel as a linear operator Z s p Z r p is of size p s r, so as Λ s ranges over Z s p the right-hand side of (1) achieves each element of Z r p exactly p s r times This implies, since A is nonsingular, that Λ r ranges over Z r p exactly p s r times Regarding Λ s and Λ r as integers following Section, that is, Λ r p s (λ p r+s 1p r λ p sp 0 ) and Λ s λ s 1 p s λ p 01, we have that the sum of all the integers in the µ-th row is p s r [p s ( p r 1 )] + [ (p s 1)] = ps (p s+r 1) where the left summand represents the sum of all the Λ r, each accounted for p s r times, and the right summand represents the sum of all the Λ s, each accounted for once This sum is independent of µ; the conclusion follows Next we seek conditions on M that ensure magic columns in a rectangle determined by T M Let ν Z s p A number Λ = (Λ r, Λ s ) lies in the ν-th column exactly when DΛ s = ν CΛ r () If we assume that C and D are of full rank, then as Λ r ranges over Z r p exactly once we have that Λ s ranges over Z s p a total of p r s times If r = s then, arguing

4 4 John Lorch just as in Lemma 31, we conclude that the array has magic columns Therefore, if r = s and M, A, B, C, D are all of full rank then the array determined by T M is a magic rectangle Matrices satisfying these conditions exist except when p = and r = s = 1 (eg, see Proposition 4 of [8]), and exist in great abundance as p and r = s become large On the other hand, problems arise in () when r < s In this case p r s < 1, thus stifling our attempt to mimic the proof of Lemma 31 However, we can say that the sum of the numbers in the ν-th column is p s ( p r 1 ) + S ν, (3) where the left summand is the integer sum of the Λ r s, while S ν denotes the integer sum (possibly depending on ν) of the corresponding Λ s s Since the magic sum along columns must be p r (p r+s 1)/ (obtained by adding all of the symbols together and dividing by the number of columns), we conclude from (3) that we require S ν = p r (p s 1)/ In the remainder of this section we introduce conditions on C and D that yield this value of S ν In arguments that follow we will be considering a Z p -matrix W of size s r with block form W = [ Ur U s r ], (4) where U r and U s r are of sizes r r and (s r) r, respectively Observe that if U[ r is ] nonsingular then the column span of W is identical to the column span of Ir, where V = U V s r Ur 1 and I r is the r r identity matrix We let v (1),, v (r) denote the columns of V and v (r) = v s r 1 Lemma 3: If a, b Z p with a 0 then the mapping x ax + b is a bijection of Z p onto itself Lemma 33: Assume r < s Let µ Z s p and let W be as in (4) above where U r is nonsingular and the last column v (r) of V consists of nonzero entries Define F : Z r p Z s p by F (x) = µ+w x Regarding elements of Z s p as base-p representations of integers 0, 1,, p s 1, we have x Z r p v 0 F (x) = pr (p s 1), where the sum is taken over the integers Proof : Since W has full rank our sum is exactly the sum over all vectors in [ the ] Ir image of F These vectors lie in the µ-translate of the Z p column span of, V

5 that is, they have the form Linear Magic Rectangles 5 µ s 1 µ 0 λ s 1 + λ s r+1, α w + α v (r) where µ s 1,, µ 0 are the components of µ, λ s 1,, λ s r+1, α Z p, and w = λ s 1 v (1) + + λ s r+1 v (r 1) Regarding these vectors as base-p representations of integers (Section ), the corresponding integers have the form s r 1 µ j + w j + αv j p p j + µ s r + α p p s r j=0 µ s r+j +λ s r+j p p s r+j, r 1 + (5) where w 0,, w s r 1 are the components of w We need to add these integers as λ s 1,, λ s r+1, α vary Begin by holding the λ s fixed and adding the expressions in (5) as α to varies over Z p Via Lemma 3 (note that v 0,, v s r 1 are nonzero) and elementary addition formulas we obtain α=0 j=0 p 1 s r 1 µ j + w j + αv j p p j + µ s r + α p p s r p 1 r 1 + µ s r+j + λ s r+j p p s r+j+1 s r r 1 = α p p j + µ s r+j + λ s r+j p p s r+j+1 α=0 j=0 = p(ps r+1 1) r 1 + µ s r+j + λ s r+j p p s r+j+1 (6) Next, adding in (6) by letting λ s r+1,, λ s 1 vary (with a total of p r 1 summands) we use Lemma 3 again to yield p r (p s r+1 1) This proves the lemma p 1 + p s r+ λ s r+1=0 = pr (p s r+1 1) = pr (p s 1) p 1 λ s 1=0 r 1 µ s r+j + λ s r+j p p j 1 + p s r+ ( p r 1 (p r 1 1) Theorem 34 : The matrix M determines a p r p s magic rectangle according to the construction given in Section provided that the matrices [ M, A, ] B, C, D are of full rank over Z p and that when r < s the matrix D 1 Ur C = satisfies U s r )

6 6 John Lorch (a) U r is an r r nonsingular matrix over Z p, and (b) the last column of V = U s r Ur 1 consists entirely of nonzero entries Proof : The case r = s was addressed earlier in the section, so we assume r < s The fact that M is nonsingular ensures that each location of the array is populated by a unique number Moreover, Lemma 31 guarantees that the array will have magic rows provided that A and B are of full rank Finally, in order for the array to have magic columns we require that S ν = p r (p s 1)/ for each ν Z s p where, by () and (3), S ν is the (integer) sum of all Λ s satisfying Λ s = D 1 ν D 1 CΛ r as Λ r ranges over Z r p According to Lemma 33, S ν has the required value when C, D are of full rank and D 1 C (or equivalently D 1 C) satisfies the conditions listed in items (a) and (b) above 4 Existence and Consequences Theorem 34 sets forth conditions on the matrix M (see Section ) that will guarantee a p r p s magic rectangle Observe that these conditions are open, so that as p, r, and s increase we should be able to find many matrices M that satisfy the conditions, perhaps by random search These will correspond to many magic rectangles In this section we establish the existence of such matrices M when r < s (The case r = s was addressed in Section 3) The following notation will be used: 0 m,n denotes the m n matrix consisting entirely of 0 s J m,n dentes the m n matrix with 1 s in the last (rightmost) column and 0 s elsewhere U m denotes the m m matrix with 1 s on the super-diagonal (immediately above the main diagonal) and 0 s elsewhere E m,n denotes the m n matrix with a 1 in the (n, 1)-position (n-th row; first column) and 0 s elsewhere Theorem 41 : Matrices M with entries in Z p satisfying the conditions of Theorem 34 exist for all primes p and all integers r, s with 1 r < s Proof : When p = the matrix I r U r E r,s r M = I r I r 0 r,s r J s r,r 0 s r,r I s r satisfies the conditions of Theorem 34, where the division into submatrices A, B, C, D is indicated by double lines Similarly, when p > the matrix I r I r 0 r,s r M = I r I r 0 r,s r J s r,r 0 s r,r I s r satisfies the conditions

7 Linear Magic Rectangles 7 For example, when p = 3, r =, and s = 3 the matrix M given in the proof of Theorem 41 gives rise to the following magic rectangle: The magic rectangle in Figure 1 is generated from a matrix not in the list presented in the proof of Theorem 41 In conclusion we indicate how these results augment the existing known magic rectangles in sizes other than p r p s It is known (eg, see []) that given magic rectangles of sizes m 1 n 1 and m n, one can construct a magic rectangle of size m 1 m n 1 n Now suppose that m n is an admissible size with m = ap r1 1 pr prk k and n = bp s1 1 ps psk k where p 1,, p k are prime, 1 r j s j for 1 j k, and a b is an admissible magic rectangle size Using the aforementioned product theorem, magic rectangles of size m n can be generated by a single a b magic rectangle together with p rj p sj rectangles generated by our construction References [1] W W R Ball and H S M Coxeter, Mathematical Recreations and Essays, thirteenth edition, Dover Publications, New York, 1987 [] T Bier and D G Rogers, Balanced magic rectangles, Europ J Combinatorics 14 (1993), [3] F S Chai, A Das, and C K Chandra, Construction of magic rectangles of odd order, preprint [4] J P De Los Reyes, A Das, and C K Chandra, A matrix approach to construct magic rectangles of even order, Australas J Combin 40 (008), [5] J P De Los Reyes, A Das, C K Chandra, and P Vellaisamy, On a method to construct magic rectangles of even order, Utilitas Math 80 (009), [6] J Dénes and A D Keedwell, Latin Squares and Their Applications, Academic Press, New York, 1974 [7] T R Hagedorn, Magic rectangles revisited, Discrete Math 07 (1999), 65 7 [8] J Lorch, Magic squares and sudoku, Amer Math Monthly 119 no 10 (01), (to appear) [9] J P N Phillips, The use of magic squares for balancing and assessing order effects in some analysis of variance designs, Appl Statist 13 (1964), [10] J P N Phillips, Methods of constructing one-way and factorial designs balanced for tread, Appl Statist 17 (1968), [11] R G Sun, Existence of magic rectangles, Nei Mongol Daxue Xuebao Ziran Kexue 1 (1990), 10 16