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http://dx.doi.org/10.4134/BKMS.b210281

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.59, no.3, 2022 , pp. 617-642 More about this Journal
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
Magic square; sparse anti-magic square; Kotzig array; Latin square;
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