18. This square should be cut by a broken line into two parts of equal area so that each segment of the broken line is parallel to the side or diagonal of the square, and the sum of the lengths of the segments parallel to the sides equals the length of the side, while the sum of the lengths of the segments parallel to the diagonals equals the length of the diagonal. What is the minimum number of segments such a broken line can have?

$$
7-8 \text { grades }
$$
18. It is obvious that each segment of the broken line must be shorter than the side or diagonal of the square parallel to it, since otherwise, this segment alone would already divide the square into two parts. From this, it follows that the number of segments of the broken line cannot be less than four. We will now show,

![](https://cdn.mathpix.com/cropped/2024_05_21_a21e90374409a05a4ea7g-58.jpg?height=340&width=679&top_left_y=1863&top_left_x=340)

Fig. 45

![](https://cdn.mathpix.com/cropped/2024_05_21_a21e90374409a05a4ea7g-58.jpg?height=397&width=403&top_left_y=1852&top_left_x=1312)
that there exists a broken line with four segments that meets all the requirements of the problem. Divide the given square into 36 equal squares. Construct the broken line shown in Figure 45. It is easy to verify that this broken line is the one we are looking for.