Puzzle ZDPX 

Workings

Hint
There are no squares of size 5x5.

Answer
There are 125 squares without a rook.

Reasoning
There are 61 squares of size 1x1.
There are 39 squares of size 2x2.
There are 20 squares of size 3x3.
There are  5 squares of size 4x4.

Giving a total of 61 + 39 + 20 + 5 = 125 squares.