Exercise 12. A circle with diameter $2 n-1$ is drawn at the center of a chessboard $2 n \times 2 n$. How many squares are crossed by an arc of the circle?
## Solution to Exercise 12

![](https://cdn.mathpix.com/cropped/2024_06_04_f521a17d739d2f1da72bg-18.jpg?height=815&width=823&top_left_y=449&top_left_x=651)

Let \( \mathrm{r} = \mathrm{n} - \frac{1}{2} \) be the radius of the circle. The chessboard is crossed by \( 2 \mathrm{n} - 1 \) horizontal segments and \( 2 \mathrm{n} - 1 \) vertical segments, from edge to edge. The circle intersects each of these segments twice.

Since \( \mathrm{r}^2 \) is not an integer, the Pythagorean theorem allows us to assert that the circle cannot pass through the corner of a square.

Thus, for each square crossed by the circle, there are exactly two intersection points with the grid of the chessboard. Additionally, each intersection point of the circle with the grid is on the edge of exactly two squares.

Therefore, the number of cells crossed by the circle corresponds to the number of intersection points between the circle and the grid, which is \( 8 \mathrm{n} - 4 \).

We note that this is also the number of cells on the edge of the chessboard.

Comment from the graders: The exercise is very well done!