Problem 5. Small Island and Big Island have a rectangular shape and are divided into rectangular counties. In each county, a road is laid along one of the diagonals. On each island, these roads form a closed path that does not pass through any point more than once. Here is how Small Island is organized, with a total of 6 counties (see Fig. 1). Draw how Big Island might be organized if it has an odd number of counties. How many counties did you get?

[7 points] (A. V. Shapovalov)

![](https://cdn.mathpix.com/cropped/2024_05_06_611ab840a0f62c05d9afg-05.jpg?height=307&width=313&top_left_y=1453&top_left_x=300)

Fig. 1

![](https://cdn.mathpix.com/cropped/2024_05_06_611ab840a0f62c05d9afg-05.jpg?height=307&width=360&top_left_y=1453&top_left_x=791)

Fig. 2
Answer. Figure 2 provides an example for 9 counties.

Comment. We will show that examples do not exist for 7 counties (or fewer), while at the same time pointing out a property characteristic of all such examples.

All roads can be divided into two types: some roads connect the top-left corner of a county with the bottom-right, while others connect the bottom-left with the top-right. If the roads of different types always alternated when traversing a closed path, their total number would be even (just like the groups of boys and girls in the previous problem). On the Large Island, where the number of counties is odd, there must, therefore, be two roads of the same type in a row.

For example, suppose two roads $A B$ and $B C$ of the same type are in a row (see the figure). In addition to the counties in which they are constructed, two more counties, shaded in gray, adjoin vertex $B$. By the condition, the roads in these counties do not pass through point $B$. Let's draw them.

Now there are four roads on the figure. To have a total of seven, three segments are needed to connect them into a single chain. To do this, it would be necessary to connect some two ends of the roads in the "gray counties" with a segment, which is impossible.

![](https://cdn.mathpix.com/cropped/2024_05_06_611ab840a0f62c05d9afg-06.jpg?height=355&width=471&top_left_y=616&top_left_x=128)