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Existence of weakly pandiagonal orthogonal Latin squares


A weakly pandiagonal Latin square of order n over the number set {0, 1, ..., n − 1} is a Latin square having the property that the sum of the n numbers in each of 2n diagonals is the same. In this paper, we shall prove that a pair of orthogonal weakly pandiagonal Latin squares of order n exists if and only if n ≡ 0, 1,3 (mod 4) and n ≠ 3.

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  1. Cao, H., Li, W.: Existence of strong symmetric self-orthogonal diagonal Latin squares. Discrete Math., 311, 841–843 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  2. Colbourn, C. J., Dinitz, J. H.: Handbook of Combinatorial Designs, 2nd Edition, Chapman & Hall/CRC, Boca Raton, FL, 2007

    MATH  Google Scholar 

  3. Denes, J., Keedwell, A. D.: Latin Squares and Their Applications, Academic Press Inc., New York, 1974

    MATH  Google Scholar 

  4. Atkin, A. O. L., Hay, L., Larson, R. G.: Enumeration and construction of pandiagonal Latin squares of prime order. Comput. Math. Appl., 9, 267–292 (1983)

    MathSciNet  MATH  Article  Google Scholar 

  5. Bell, J., Stevens, B.: A survey of known results and research areas for n-queens. Discrete Math., 309, 1–31 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  6. Bell, J., Stevens, B.: Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular n-queens solutions. J. Combin. Des., 15, 221–234 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  7. Hedayat, A.: A complete solution to the existence and nonexistence of Knut Vik designs and orthogonal Knut Vik designs. J. Combin. Theory Ser. A, 22, 331–337 (1977)

    MathSciNet  MATH  Article  Google Scholar 

  8. Xu, C., Lu, Z.: Pandiagonal magic squares. Lecture Notes in Computer Science, 959, 388–391 (1995)

    MathSciNet  Article  Google Scholar 

  9. Harmuth, T.: Über magische Quadrate und ähniche Zahlenfiguren. Arch. Math. Phys., 66, 286–313 (1881)

    MATH  Google Scholar 

  10. Harmuth, T.: Über magische Rechtecke mit ungeraden Seitenzahlen. Arch. Math. Phys., 66, 413–447 (1881)

    MATH  Google Scholar 

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Correspondence to Yong Zhang.

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Supported by National Natural Science Foundation of China (Grant Nos. 61071221, 10831002, 11071207 and 11201407), Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant No. 12KJD110007), and Natural Science Foundation of Jiangsu Province (Grant No. BK2012245)

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Zhang, Y., Li, W. & Lei, J.G. Existence of weakly pandiagonal orthogonal Latin squares. Acta. Math. Sin.-English Ser. 29, 1089–1094 (2013).

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  • Latin square
  • weakly pandiagonal
  • Knut Vik design

MR(2010) Subject Classification

  • 05B15