9.65 On a standard chessboard (where the edges of the small squares are considered black), draw a circle so that it is entirely within the black parts. Find the maximum radius of such a circle.

In a standard chessboard (where the edges of the small squares are considered black), draw a circle so that it is entirely within the black parts, find the maximum radius of such a circle.
[Solution] For the entire circle to be located within the black parts, the circle must pass through the two vertices of each black grid it intersects. Moreover, for two black grids sharing a common vertex, the two vertices cannot be the endpoints of a diagonal, such as points $A, B, D$ in the figure, because these three points are collinear and cannot form a circle. Therefore, the largest circle is the circle $O$ passing through points $A, B, C$.

Assuming the side length of the square grid is 1, the perpendicular bisectors of segments $A B$ and $B C$ intersect at point $O$ (see the figure). The radius of circle $O$ is
$$
r=\sqrt{\left(\frac{3}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}}=\frac{1}{2} \sqrt{10} .
$$