A four by four grid of unit squares contains squares of various sizes (1 by 1 through 4 by 4), each of which are formed entirely from squares in the grid. In each of the 16 unit squares, write the number of squares that contain it. For instance, the middle numbers in the top row are 6s because they are each contained in one $1 \times 1$ square, two $2 \times 2$, two $3 \times 3$, and the one $4 \times 4$. What is the sum of all sixteen numbers written in this grid