When we cut the sheet along the grid lines into two pieces and rearrange those pieces without overlap, we can make new rectangles with different dimensions.
For a pair w and h, let F(w,h) be the number of distinct rectangles that can be made from a sheet with dimensions w × h . For example, F(2,1) = 0, F(2,2) = 1, F(9,4) = 3 and F(9,8) = 2. Note that rectangles congruent to the initial one are not counted in F(w,h). Note also that rectangles with dimensions w × h and dimensions h × w are not considered distinct.
For an integer N, let G(N) be the sum of F(w,h) for all pairs w and h which satisfy 0 < h ≤ w ≤ N. We can verify that G(10) = 55, G(103) = 971745 and G(105) = 9992617687.