Puzzle ZUKB 

Workings

Hint
There are no larger squares than size 4 x 4.

Answer
There are 110 squares without a rook.

Reasoning
There are 60 squares of size 1 x 1.
There are 35 squares of size 2 x 2.
There are 12 squares of size 3 x 3.
There are  3 squares of size 4 x 4.

Giving a total of 60 + 35 + 12 + 3 = 110 squares.