Given an 8 by 8 grid how many unique solutions would there be where a square is filled on each row and column such that it does not sit on the same row, column or diagonal as any other filled square?

Answer 1

What you ask is commonly referred to as the Eight Queens Puzzle. Wikipedia has a nice treatment of the problem. The answer is that there are 92 solutions if you allow solutions which are reflections or rotations of other solutions, and only 12 solutions if you don't.

Answer 2

Draw and label a diagram!

For the first row, 8 choices

For the second row, 7 choices

For the third row, 6 choices.

See the pattern?

The the possible ways are 8!