Example 8: On an $8 \times 8$ chessboard, select two small squares such that these two squares share exactly one common point. How many different ways are there to do this?
For each selection method, there is a point corresponding to it. This point is the common point of the two selected small squares, which is the intersection of the horizontal and vertical lines on the chessboard, not on the boundary of the chessboard.

From Figure 19-1, it can be seen that each point corresponds to two different selection methods. That is, taking two black squares or two white squares, the corresponding point is the same (point $A$). Therefore, the mapping here is a multiple mapping (multiple $k=2$).

Since on an $8 \times 8$ chessboard, the 7 internal horizontal lines and 7 internal vertical lines have $7 \times 7=49$ intersections, there are a total of $49 \times 2=98$ different selection methods.