Find the total number of possible squares

Question

on the chessboard

Answer 1

A good way is to take two pieces of graph paper and cut both of them into 8by8 square pieces

It is easy to see that there is 1 large 8x8 square on a chessboard

Now use one of the 2 pieces and cut it down to a 7x7 square template.
Try placing this template on the 8x8 sheet (your chessboard)
you will see that you can do this in only 4 ways
so there are 4 squares of 7x7

now cut your 7x7 template down to a 6x6 template, and if you place it on top of your chessboard you will be able to do this in 9 ways

mmmhh, let's see

1 square of 8x8
4 squares of 7x7
9 squares of 6x6
16 squares of 5x5
25 squares of 4x4
36 squares of 3x3
49 squares of 2x2 , and
64 squares of 1x1, this one is easy to see

so total number of squares
= 1+4+9+16+25+36+49+64
=
204