Existence of Combinatorial Configurations

Abstract

A square table n × n filled with the positive integers 1, 2, …, n ² is called a magic square of order n if the sum of all numbers in each row, the sum of all numbers in each column, and the sum of all numbers in the two main diagonals are equal to each other. This constant sum is called a magic sum. The magic sum of a magic square of order n is

$$\displaystyle \frac 1n(1+2+\dots +n^2)=\frac 1n\cdot \frac 12n^2(n^2+1)=\frac {n(n^2+1)}{2}. $$