Counting all squares on a chessboard grid

On a standard $8\times8$ chessboard, how many squares of every size (from $1\times1$ up to $8\times8$) can be traced along the grid lines?

$204$. A $k\times k$ square can start in $9-k$ positions along each axis, so there are $(9-k)^2$ of them. Summing gives $1^2+2^2+\dots+8^2=204$.