Example 6 (Problem from the 14th American Mathematical Invitational) A $7 \times 7$ chessboard has 2 squares colored yellow, and the rest of the squares are colored green. If one coloring can be obtained from another by a rotation in the plane of the board, then the two colorings are considered the same. How many different colorings are possible?
Select a pair of squares to be yellow, and the rest to be green, there are $C_{49}^{2}$ ways to choose.
If this pair of squares is symmetric about the center of the chessboard, then after rotating $90^{\circ}, 270^{\circ}$, it will coincide with another pair of squares, and after rotating $180^{\circ}$, it will coincide with itself. There are $\frac{48}{2}=24$ such pairs of squares. If this pair of squares is not symmetric about the center of the chessboard, then after rotating $90^{\circ}, 180^{\circ}, 270^{\circ}$, it will coincide with three other pairs of squares, respectively. Therefore, the number of different coloring methods is
$$
\frac{24}{2}+\frac{C_{19}^{29}-24}{4}=300 \text {. }
$$
(2) Mapping method and general correspondence method
When it is difficult to directly calculate the number of elements $|M|$ in set $M$, we try to establish a bijection $f$ from set $M$ to another set $N$. If the number of elements $|N|$ in set $N$ is easy to calculate, then the number of elements in $M$ is $|M|=|N|$. This counting method is called the mapping method. If the established mapping is only an injection or a surjection, then we get the inequality $|M| \leqslant|N|$ or $|M| \geqslant|N|$, thus obtaining the upper or lower bound of $|M|$.