$1 \cdot 79$ Draw a circle with a radius of 100 on a square grid paper where each side of the square is 1 unit long. The circle does not pass through any of the grid paper's nodes nor is it tangent to any grid line. How many squares can this circle possibly pass through?
［Solution］It is easy to see that a circle with a diameter of 200, which does not pass through any node and does not touch the grid lines, intersects exactly 200 horizontal lines and 200 vertical lines, and intersects each line at two points. Thus, there are 800 intersection points, which divide the circumference into 800 arcs, each of which lies within a square, so the circumference passes through at most 800 squares.

Let us consider whether it is possible for two arcs to belong to the same square. If such a case can occur, we call such a square a "singular square." Below, we will prove that there may be no singular squares, or there may be one but at most one.
Suppose the circumference intersects a side $AB$ of a square at two points, and let us examine the position of the center $O$. In this case, the distances from point $O$ to $A$ and $B$ are both greater than 100, while the distance from $O$ to the line $AB$ is less than 100. Therefore, the position of point $O$ must be outside the two circles with centers at $A$ and $B$ and radii of 100, and above the horizontal grid line parallel to and 100 units away from line $AB$. Specifically, point $O$ must lie within the curved triangle shown in the figure. Clearly, along each side of the square, a similar curved triangle can be drawn within the square, but the 4 curved triangles cannot completely cover the entire square, and when the center falls in the uncovered part, there is no singular square. In summary, the number of squares the circumference passes through is either 799 or 800.