A3. A circle with the origin $(0,0)$ as its center and radius 13 passes through $n$ grid squares.

Determine $n$. A circle passes "through" a grid square if there are points of the square on both sides of the circle arc.
A3. It is sufficient to look at the circle only in the first quadrant, because by rotating it, you can create the rest of the circle on the grid paper. The points that lie exactly on the circle are $(0,13)$, $(5,12)$, $(12,5)$, and $(13,0)$. They are indicated with an open circle in the figure. If you walk along the arc of the circle from $(0,13)$ to $(13,0)$, you pass 12 horizontal grid lines and 12 vertical grid lines. When you pass such a line, you enter a new square. The number of squares you pass through is then $2 \times 12 + 1 - 2$, "+1" because of the square you start in and -2 because when you pass $(5,12)$ and $(12,5)$, you cross two grid lines but enter only one new square. For the quarter circle, this results in 23 squares and for the entire circle, 92 squares.

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