Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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CONSTRUCTIONS OF REGULAR SPARSE ANTI-MAGIC SQUARES

  • Chen, Guangzhou (Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control School of Mathematics and Information Science Henan Normal University) ;
  • Li, Wen (School of Science Xichang University) ;
  • Xin, Bangying (School of Science Xichang University) ;
  • Zhong, Ming (Central Primary School of Tingzi Town)
  • Received : 2021.04.06
  • Accepted : 2022.04.04
  • Published : 2022.05.31
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Abstract

For positive integers n and d with d < n, an n × n array A based on 𝒳 = {0, 1, …, nd} is called a sparse anti-magic square of order n with density d, denoted by SAMS(n, d), if each non-zero element of X occurs exactly once in A, and its row-sums, column-sums and two main diagonal-sums constitute a set of 2n + 2 consecutive integers. An SAMS(n, d) is called regular if there are exactly d non-zero elements in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order n ≡ 1, 5 (mod 6), and prove that there exists a regular SAMS(n, d) for any n ≥ 5, n ≡ 1, 5 (mod 6) and d with 2 ≤ d ≤ n - 1.

Keywords

Acknowledgement

The authors would like to thank Professor Zhu Lie of Soochow University for his encouragement and many helpful suggestions. The authors thank the anonymous reviewers for their careful check, constructive comments and suggestions that greatly improved the quality of this paper.

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