6. A chessboard is formed from an $8 \times 8$ grid of alternating black and white squares, as shown. The side of each small square is $1 \mathrm{~cm}$.
What is the largest possible radius of a circle that can be drawn on the board in such a way that the circumference is entirely on white squares or corners?
Solution
To go from one white square to another, it is necessary to pass through a corner.

But it is not possible for a circle to pass through the point where two white squares meet (such as $Y$ in the diagram alongside) and both the diagonally opposite corners of this point ( $X$ and $Z$ ), since these points form a straight line and not an arc of a circle.

Is it possible for a circle to pass through exactly one of $X$ and $Z$, for every choice of $Y$ on the circle? In other words, the circle is alternately "close" to a diagonal and an edge of white squares.
The diagram alongside shows one such circle, whose centre is as shown. The circle has diameter $\sqrt{1^{2}+3^{2}}$, from Pythagoras' theorem.
A circle passing through $Y$ and neither $X$ nor $Z$, for every point $Y$ on the circle, such as the dashed circle in the diagram alongside, has a smaller radius.

Therefore the largest possible radius of the circle is $\frac{1}{2} \sqrt{10}$.