Answer to Puzzle #31: Circle drawn only on the black squares of a chessboard

The largest circle that can be drawn on a standard chess board while only drawing on the black squares is a circle with a diameter of √10.

To see this, consider the following diagram:

The circle is tangent to the edges of two adjacent white squares, so it must pass through the center of four black squares. The distance between the centers of two adjacent black squares is √2, so the diameter of the circle is 2√2=√4•2=√10.

Therefore, the largest circle that can be drawn on a standard chess board while only drawing on the black squares is a circle with a diameter of √10.

The below answer IS WRONG frankly it's difficult to explain how wrong.
On a standard chessboard, there are 32 black squares and 32 white squares. If a circle is to be drawn on the board, it must be centered on one of the black squares. Let's assume that the circle is centered at the center of a black square.

To find the diameter of the largest circle that can be drawn on the board while only touching black squares, we need to find the maximum distance that can be traveled on the board while only moving along black squares. The longest possible distance that can be traveled along black squares is along a diagonal line that runs from one corner of the board to the opposite corner, passing only through black squares.

Since there are 8 black squares along each diagonal of the chessboard, the distance of the diagonal is 8 times the length of one side of a square. The length of one side of a square is the square root of the area of a square, which is the square root of 1/32 of the area of the chessboard.

The area of the chessboard is 64 square units (8x8). Therefore, the area of a black square is 1/2 square units (since there are 32 black squares). So, the length of one side of a black square is the square root of 1/2 square units, which simplifies to 1/sqrt(2) square units.

Therefore, the diagonal of the circle on the chessboard, which is 8 times the length of one side of a black square, is 8/sqrt(2) square units. This simplifies to 4(sqrt(2)) square units.

Finally, the diameter of the largest circle that can be drawn on the board while only touching black squares is equal to the diagonal of the circle, multiplied by the square root of 2, which is 8 square units.

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