this is a very old and famous problem yet i saw no actual non-bullshit model for this in comments so:
the everytime the grid gets bigger it adds a square number of squares.
2^2, 3^2, 4^2 and so on
as 's comment says the pattern goes as:
1,5,14,30,55,91,140,204
RE: Math Brain Teasers 18:: ChessBoard - How Many Squares??