1.67 Given that a $2 \times 2$ square paper piece covers no fewer than 7 nodes on a grid paper with a side length of 1, how many nodes does it cover exactly?
［Solution］If a square piece of paper covers two nodes that are $2 \sqrt{2}$ apart, then the square exactly covers 4 grid squares, thus covering 9 nodes.

According to the given information, the square piece of paper covers at least 7 nodes. If it is not the aforementioned case, then the 7 nodes must be 3 nodes each in the 1st and 2nd rows, and 1 node in the 3rd row, as shown in the figure below. Consider $\triangle A B C$. Its area is 2, which is exactly half the area of the square piece of paper. It is easy to prove that for a triangle within (including the boundary) the square, the triangle's area equals half the area of the square if and only if one side of the triangle coincides with one side of the square and the third vertex of the triangle lies on the opposite side of the coinciding side of the square. Since the square piece of paper covers $\triangle A B C$, it follows that it must also exactly cover 4 grid squares, thus covering 9 nodes.

Clearly, a $2 \times 2$ square piece of paper can cover at most 9 nodes, so when the square piece of paper covers 7 nodes, it covers 9 nodes.