Question 120
Let $n \ge 3$ be a positive integer. Consider an $n \times n$ square. In each cell of the square, one of the numbers from the set $M = \{1, 2, \dots, 2n-1\}$ is to be written. One such filling is called "good" if, for every index $i$, $1 \le i \le n$, row no. $i$ and column no. $i$, together, contain all the elements of $M$.
a) Prove that there exists $n \ge 3$ for which a *good* filling exists.
b) Prove that for $n = 2017$ there is no good filling of the $n \times n$ square.
a) Prove that there exists $n \ge 3$ for which a *good* filling exists.
b) Prove that for $n = 2017$ there is no good filling of the $n \times n$ square.