Question 58
64 squares each with side length 1 are arranged in 8 rows and 8 columns to fill the inside of a square $ABCD$ with side length 8. We choose a number of small squares and color them following the 3 conditions specified below:
* The placement of the colored small squares within the square $ABCD$ is symmetric with respect to both of the diagonals $AC$ and $BD$.
* For each row the number of colored small squares is at most 1.
* For each column the number of colored small squares is at most 1.
How many possible arrangements of the location of the colored small squares satisfying the 3 conditions above are there? The arrangement where none of the small squares is colored is counted as one possibility.
* The placement of the colored small squares within the square $ABCD$ is symmetric with respect to both of the diagonals $AC$ and $BD$.
* For each row the number of colored small squares is at most 1.
* For each column the number of colored small squares is at most 1.
How many possible arrangements of the location of the colored small squares satisfying the 3 conditions above are there? The arrangement where none of the small squares is colored is counted as one possibility.