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Centrally symmetric and magic rectangles. (English) Zbl 0898.05007

Let \(c\), \(n\in\mathbb{Z}\) such that \(c\geq 0\) and \(n>1\). A \(4\times n\) rectangular array of the integers in the set \(\{k\mid c+1 \leq| k|\leq c+2n\}\) is called a centrally symmetric rectangle with threshold \(c\), \(R(c;n)\), if all row and column sums are 0. A \(p\times q\) rectangular array of the integers between 1 and \(pq\) is a magic \((p\times q)\)-rectangle if each row sum is a constant and each column sum is another (if \(p\neq q)\) constant. The authors construct a centrally symmetric rectangle for each \(c\geq 0\) if \(n\) is even. If \(n\) is odd, they show that an \(R(c;n)\) exists precisely when \(c\leq {1\over 2} ((n-1)^2-4)\). Although the magic rectangle problem had already been solved, the authors use centrally symmetric rectangles to show that a magic \((p\times q)\)-rectangle exists for \(p,q>1\) if and only if \(p\equiv q \bmod 2\) and \((p,q) \neq(2,2)\).

MSC:

05B15 Orthogonal arrays, Latin squares, Room squares
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[1] Abraham, J., Perfect systems of difference sets — a survey, Ars Combin., 17A, 5-36 (1986)
[2] Andersen, L. D.; Hilton, A. J.W., Generalized Latin rectangles I: constructions and decompositions, Discrete Math., 31, 125-152 (1980) · Zbl 0443.05019
[3] Andersen, L. D.; Hilton, A. J.H., Generalized Latin rectangles II: embedding, Discrete Math., 31, 235-260 (1980) · Zbl 0476.05018
[4] Ando, K.; Gervacio, S.; Kano, M., Disjoint integer subsets having a constant sum, Discrete Math., 82, 7-11 (1990) · Zbl 0732.05002
[5] Andrews, W. S., Magic Squares and Cubes, ((1960), Dover: Dover New York), 257-266 · Zbl 1003.05500
[6] Bier, T.; Rogers, D. G., Balanced magic rectangles, European J. Combin., 14, 285-299 (1993) · Zbl 0782.05014
[7] Enomoto, H.; Kanoa, M., Disjoint odd integer subsets having a constant sum, Discrete Math., 137, 189-193 (1995) · Zbl 0819.05006
[8] Faudree, R. J.; Jacobson, M. S.; Lehel, J.; Schelp, R. H., Irregular networks, regular graphs, and integer matrices with distinct row and column sums, Discrete Math., 76, 223-240 (1989) · Zbl 0685.05029
[9] Freeman, G. H., Magic square designs, (Encyclopedia of Statistical Sciences, Vol. 5 (1985), Wiley: Wiley New York), 173-174
[10] Fu, H. L.; Hu, W. H., Disjoint odd integer subsets having a constant odd sum, Discrete Math., 128, 143-150 (1994) · Zbl 0795.05010
[11] Gyarfas, A.; Schelp, R. H., A matrix labelling problem, (Congressus Numer., 65 (1988)), 237-244 · Zbl 0725.05057
[12] Harmuth, T., Über magische Quadrate und ähnliche Zahlenfiguren, Arch. Math. Phys., 66, 286-313 (1881) · JFM 13.0145.02
[13] Harmuth, T., Über magische Rechtecke mit ungeraden Seitenzahlen, Arch. Math. Phys., 66, 413-447 (1881) · JFM 13.0146.01
[14] Jacroux, M. A., A note on constructing magic rectangles of even order, Ars Combin., 36, 335-340 (1993) · Zbl 0793.05026
[15] Phillips, J. P.N., Methods of constructing one way and factorial designs balanced for trend, Appl. Statist., 17, 162-170 (1968)
[16] Phillips, J. P.N., A simple method of constructing certain magic rectangles of even order, Math. Gazette, 9-12 (1968) · Zbl 0155.02902
[17] Rogers, D. G., On the general Erdős conjecture for perfect systems of difference sets and embedding partial complete permutations, European J. Combin., 12, 549-560 (1991) · Zbl 0755.05011
[18] Rogers, D. G., Critical perfect systems of difference sets, (Proc. 4th Czechoslovak Symp. Combinatorics. Proc. 4th Czechoslovak Symp. Combinatorics, Prachatice, 1990. Proc. 4th Czechoslovak Symp. Combinatorics. Proc. 4th Czechoslovak Symp. Combinatorics, Prachatice, 1990, Annals of Discrete Math., Vol. 51 (1992), North-Holland: North-Holland Amsterdam), 275-279 · Zbl 0770.05018
[19] Shen, X., Generalized Latin squares II, Discrete Math., 143, 221 (1995) · Zbl 0826.05013
[20] Shen, X.; Cai, Y. Z.; Liu, C. L.; Kruskal, C. P., Generalized Latin squares I, Discrete Appl. Math., 25, 155-178 (1989) · Zbl 0687.05008
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