Hi everyone. So I have a question here. Find the number of squares contained in an n x n square array, where n is larger or equal to 2.

Ok so my solution: For 1x1 Square array, no of squares=1

Then keep on doing till a pattern is formed, which is

.

However, when I flipped to the solutions, they proposed an alternative method which I do not understand. They examined 2 x 2 squares and noted that the bottom-left hand corner of each square is unique. If you consider the (n x n) array as (n+1)(n+1) grid (why????) with both x and y coordinates such that x=0,1,2,.....,n-2 (????) repeat for y. The number of points equals (n-1)(n-1) (????)and this the number of 2x2 squares is (n-1)(n-1).

More generally, number of r x r squares is (n-r+1)(n-r+1).

Actually I dont really understand the method which the book wrote and had trouble visualizing it. Any helps?

Thanks.