Question 8
A magical square of dimensions $3 \times 3$ is a square with side $3$, consisting of $9$ unit squares, so that the real numbers written in the unit squares (one number in each unit square) satisfy the property: the sum of the numbers in the unit squares in any row is equal to the sum of the numbers in the unit squares in any column and is equal to the sum of the numbers in the unit squares in the two diagonals.
A rectangle of dimensions $m \times n$, $m \ge 3$, $n \ge 3$ is given, which consists of $mn$ unit squares. If in each unit square one number is written in such a way that each square of dimensions $3 \times 3$ is magical, then how many different numbers can be used at most to fill the rectangle?
A rectangle of dimensions $m \times n$, $m \ge 3$, $n \ge 3$ is given, which consists of $mn$ unit squares. If in each unit square one number is written in such a way that each square of dimensions $3 \times 3$ is magical, then how many different numbers can be used at most to fill the rectangle?