Acta Mathematica Sinica, English Series

, Volume 29, Issue 6, pp 1089–1094 | Cite as

Existence of weakly pandiagonal orthogonal Latin squares

Article

Abstract

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.

Keywords

Latin square weakly pandiagonal Knut Vik design 

MR(2010) Subject Classification

05B15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

10114_2013_2274_MOESM1_ESM.tex (19 kb)
Supplementary material, approximately 19.4 KB.

References

  1. [1]
    Cao, H., Li, W.: Existence of strong symmetric self-orthogonal diagonal Latin squares. Discrete Math., 311, 841–843 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Colbourn, C. J., Dinitz, J. H.: Handbook of Combinatorial Designs, 2nd Edition, Chapman & Hall/CRC, Boca Raton, FL, 2007MATHGoogle Scholar
  3. [3]
    Denes, J., Keedwell, A. D.: Latin Squares and Their Applications, Academic Press Inc., New York, 1974MATHGoogle Scholar
  4. [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)MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Bell, J., Stevens, B.: A survey of known results and research areas for n-queens. Discrete Math., 309, 1–31 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. [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)MathSciNetMATHCrossRefGoogle Scholar
  7. [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)MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Xu, C., Lu, Z.: Pandiagonal magic squares. Lecture Notes in Computer Science, 959, 388–391 (1995)MathSciNetCrossRefGoogle Scholar
  9. [9]
    Harmuth, T.: Über magische Quadrate und ähniche Zahlenfiguren. Arch. Math. Phys., 66, 286–313 (1881)MATHGoogle Scholar
  10. [10]
    Harmuth, T.: Über magische Rechtecke mit ungeraden Seitenzahlen. Arch. Math. Phys., 66, 413–447 (1881)MATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHebei Normal UniversityShijiazhuangP. R. China

Personalised recommendations