Tania,

What you need to do to count them all is count the number of squares of size1, of size 2, ... , of size 8. To count the 64 = 8x8 squares of size 1 are easy enough but what about those of size 2, ..., 8? Let's have a look at the number of squares of size 2. One way to do this is to think where the top right hand corner (TRHC) of the square can be. Think of the lines on the checkerboard and the points where they intersect as line segments and points in the first quadrant - you can label these points as (0,0), (0,1), ... (8,8). Look at your checkerboard and notice that the TRHC of a square of size 2 can be at
(2,2). (2,3),...(2,8), (3,2), ... (3,8),..., (8,2),..., (8,8), in total 49 = 7x7 possibilities for the TRHC and hence 49 = 7x7 squares of size 2 are possible. Now, how would you count the squares of size 3 (and so on)?

Penny