2019
DOI: 10.1002/jcd.21667
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Nearly magic rectangles

Abstract: Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares is magic rectangles which are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to even by odd nearly magic rectangles. We also prove necessary and sufficient conditions for the existence of a near… Show more

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Cited by 10 publications
(5 citation statements)
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“…). Define g : E(G) → [1, 4mn + 2m + n(2n − 3) − 1] such that g(e) = h(e) if e ∈ E(C 2m−1 ), g(v i u j ) = x i,j + (2m − 1) for 1 ≤ i ≤ 2n, 1 ≤ j ≤ 2m − 1 where x i,j is the (i, j)-entry of a (2n, 2m−1)-nearly magic rectangle that has (2i−1)-st row sum (2m−1)(1+4mn−2n)−1 2 = n(2m−1) 2 +m−1, (2i)-th row sum (2m−1)(1+4mn−2n)+1 2 = n(2m−1) 2 +m and column sum n(1+4mn−2n) (see [6]). Finally, g(e) = f (e) + (2n + 1)(2m − 1) if e ∈ K 2n .…”
Section: Resultsmentioning
confidence: 99%
“…). Define g : E(G) → [1, 4mn + 2m + n(2n − 3) − 1] such that g(e) = h(e) if e ∈ E(C 2m−1 ), g(v i u j ) = x i,j + (2m − 1) for 1 ≤ i ≤ 2n, 1 ≤ j ≤ 2m − 1 where x i,j is the (i, j)-entry of a (2n, 2m−1)-nearly magic rectangle that has (2i−1)-st row sum (2m−1)(1+4mn−2n)−1 2 = n(2m−1) 2 +m−1, (2i)-th row sum (2m−1)(1+4mn−2n)+1 2 = n(2m−1) 2 +m and column sum n(1+4mn−2n) (see [6]). Finally, g(e) = f (e) + (2n + 1)(2m − 1) if e ∈ K 2n .…”
Section: Resultsmentioning
confidence: 99%
“…In [2], the authors generalized the concept of magic rectangle to nearly magic rectangle as follows.…”
mentioning
confidence: 99%
“…In this article, we prove the existence of QMR a b d ( , : ) for all possible values of a and b, when d ab = 2 + 1 ∕ . In addition, if gcd a b ( , ) = 1, we prove that the condition A magic square is an n n × array whose entries are the consecutive numbers n 1, 2, …, 2 , each appearing once, such that the sum of each row, column, and the main and main backward diagonal is equal to n n ( +1) 2…”
mentioning
confidence: 87%
“…When a and b are of different parity, a magic rectangle MR a b ( , ) does not exist. Recently, Chai et al [2] introduced the concept of nearly magic rectangles. A nearly magic rectangle NMR a b ( , ) is defined as an a b × array with a odd and b even that contains each of the integers from the set ab {1, 2, …, } exactly once and the row sums are constant, while the column sums differ by at most one.…”
mentioning
confidence: 99%