Signed Magic Rectangles With Two Filled Cells in Each Column
Abdollah Khodkar, Brandi Ellis · Journal of Combinatorial Mathematics and Combinatorial Computing · 2024
A signed magic rectangle S M R ( m , n ; r , s ) is an m × n array with entries from X , where X = { 0 , ± 1 , ± 2 , … , ± ( m s – 1 ) / 2 } if m r is odd and X = { ± 1 , ± 2 , … , ± m r / 2 } if m r is even, such that precisely r cells in every row and s cells in every column are filled, every integer from set X appears exactly once in the array, and the sum of each row and of each column is zero. In this paper, we prove that a signed magic rectangle S M R ( m , n ; r , 2 ) exists if and only if m = 2 and n = r ≡ 0 , 3 ( mod 4 ) or m , r ≥ 3 and m r = 2 n .