Puzzle ZYLQ 

Workings

Hint
How many squares of size 1 x 1 are there?

Answer
There are 116 squares without a rook.

Reasoning
There are 61 squares of size 1 x 1.
There are 37 squares of size 2 x 2.
There are 15 squares of size 3 x 3.
There are  3 squares of size 4 x 4.

Giving a total of 61 + 37 + 15 + 3 = 116 squares.