
A note on constructing and enumerating of magic squares
Author(s) -
Mohammad Reza Oboudi
Publication year - 2022
Publication title -
boletim da sociedade paranaense de matemática
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.347
H-Index - 15
eISSN - 2175-1188
pISSN - 0037-8712
DOI - 10.5269/bspm.46836
Subject(s) - magic square , diagonal , combinatorics , integer (computer science) , magic (telescope) , mathematics , order (exchange) , square (algebra) , column (typography) , discrete mathematics , arithmetic , computer science , physics , geometry , quantum mechanics , finance , connection (principal bundle) , economics , programming language
Let $n\geq1$ be an integer. A magic square of order $n$ is a square table $n\times n$, say $A$, filled with distinct positive numbers $1,2,\ldots,n^2$ such that all cells of $A$ are distinct and the sum of the numbers in each row, column and diagonal is equal.Let $M(n,s)$ be the set of all $n\times n$ $(0,1)$-matrices, say $T$, such that the number of $1$ in every row and every column of $T$ is $s$.In this paper for every positive integer $k$ we find a new way for constructing magic squares of order $4k$. We show that the number of magic squares of order $4k$ is at least $|M(2k,k)|$. In particular we show that the number of magic squares of order $4k$ is at least $\frac{{2k \choose k}^2}{2}$.