Series Summation: Squares on a nxn Grid

How many squares are there in an $n \times n$ grid?

Answer

Consider the following method for an $n \times n$ chessboard:

Choose $1 \times 1$ squares: there are $$ n^2 $$ such squares.
Choose $2 \times 2$ squares: there are $$ (n-1)^2 $$ such squares, because there is one less degree of freedom on each axis.
Choose $3 \times 3$ squares: there are $$ (n-2)^2 $$ such squares, with two less degrees of freedom on each axis.
$\vdots$
Choose $n \times n$ squares: there is $$ 1^2 = 1 $$ such square.

Using the sum of squares formula, the total number of squares is:

1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n + 1)(2n + 1)}{6}

Notes and comments

Comment 1: For a chessboard (8x8) this number would be $$204$$ .