Two squares of a $7 \times 7$ board are painted yellow and the rest is painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many non-equivalent color schemes can we obtain?

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Solution

Since the board has $7 \cdot 7=49$ squares, there are $\frac{49 \cdot 48}{2}=1176$ ways to choose two of them to be colored yellow. When they are not diametrically opposite, we can rotate them by $90^{\circ}$ three times to obtain configurations that generate equivalent schemes. To count the pairs of squares that are diametrically opposite, choose one square distinct from the center in $49-1=48$ ways. The other square is completely determined by this choice. However, since the order of choosing between them is irrelevant, the total number of pairs is $\frac{49-1}{2}=24$. Each of these 24 schemes can be rotated once by $180^{\circ}$ to generate another equivalent one. Therefore, the number of schemes is

$$
\frac{1176-24}{4}+\frac{24}{2}=300
$$