, 2019

Solution:

In what ways could we potentially fill a single row? The only possibilities are if it contains the numbers $(0,0,7)$ or $(0,1,6)$ or $(0,2,5)$ or $(0,3,4)$ or $(1,2,4)$. Notice that if we write these numbers in binary, in any choices for how to fill the row, there will be exactly one number with a $1$ in its rightmost digit, exactly one number with a $1$ in the second digit from the right, and exactly one number with a $1$ in the third digit from the right. Thus, consider the following operation: start with every unit square filled with the number $0$. Add $1$ to three unit squares, no two in the same row or column. Then add $2$ to three unit squares, no two in the same row or column. Finally, add $4$ to three unit squares, no two in the same row or column. There are clearly $6^{3}=216$ ways to perform this operation and every such operation results in a unique, suitably filled-in $3$ by $3$ square. Hence the answer is $216$.