Question 166
Suppose you want to distribute nine distinct integers greater than or equal to $1$ and less than or equal to $9$ into nine square boxes of a $3 \times 3$ grid in such a way that the sums of the numbers on each row and each column are all multiples of $3$. How many possible ways of distributing numbers are there to attain this goal? Regard two configurations, for which one can be obtained from rotating the other one or by flipping over the other one, to be distinct in counting the number of possible ways.