How many (possibly overlapping) squares are in an mxn grid?

Given an n*m grid with n > m.

Orient the grid so n is its width.  Divide the grid into two portions, 
an m*m square on the left and an (n-m)*m rectangle on the right.
Count the squares that have their upper right-hand corners in the
m*m square.  There are m^2 of size 1*1, (m-1)^2 of size 2*2, ...
up to 1^2 of size m*m.  Now look at the n-m columns of lattice points
in the rectangle on the right, in which we find upper right-hand
corners of squares not yet counted.  For each column we count m new
1*1 squares, m-1 new 2*2 squares, ... up to 1 new m*m square.

Combining all these counts in summations:

         m                   m
total = sum i^2 + (n - m) sum i
        i=1                  i=1

        (2m + 1)(m + 1)m   (n - m)(m + 1)m
      = ---------------- + ---------------
                6                  2

      = (3n - m + 1)(m + 1)m/6

-- David Karr