Squares in a square

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In mathematics, a common mathematical puzzle involves finding the number of squares in a large n by n square grid. This number can be derived as follows:

The number of 1x1 boxes found in the grid is $n 2$ .

The number of 2x2 boxes found in the grid is $(n - 1) 2$ . These can be counted by counting all of the possible upper-left corners of 2x2 boxes.

The number of kxk boxes (1 =< k =< n) found in the grid is $(n - k + 1) 2$ ). These can be counted by counting all of the possible upper-left corners of kxk boxes.

It follows that the number of squares in a n by n square grid is:

$x = n^2 + (n-1)^2 + (n-2)^2 + (n-3)^2 + \ldots + 1^2$