16. Every two adjacent sides of a flat (self-intersecting!) 14-sided polygon are mutually perpendicular; no two sides of it lie on the same line. What is the maximum possible number of self-intersection points of the sides of such a polygon?
16. Let's agree to consider that all sides of the 14-sided polygon are either "horizontal" or "vertical." Clearly, in this case, exactly 7 sides are horizontal (and 7 are vertical): after all, from each vertex, one horizontal (and one vertical) side extends. Summing over all 14 vertices, we count 14 horizontal sides; but in doing so, each side is counted twice (corresponding to its two vertices).

![](https://cdn.mathpix.com/cropped/2024_05_21_15011fcc7946713185f0g-083.jpg?height=466&width=788&top_left_y=254&top_left_x=249)

Fig. 52.

(further, the highest and the lowest of the horizontal sides of the 14-sided polygon cannot contain any intersection points with other sides (since there are no vertices of the polygon above, respectively below, these sides); the second from the top and the second from the bottom sides can contain only two intersection points (since there are only 2 vertices above, respectively below, these sides); the third from the top and the third from the bottom of the horizontal sides can contain up to 4 intersection points (since there are 4 vertices of the 14-sided polygon above and below these sides). Finally, the middle horizontal side by height can contain no more than 6 intersection points, since there are 6 vertices of the 14-sided polygon above and below it; however, the number 6 is actually impossible here, because otherwise each vertex above this middle side would be connected to one of the lower vertices and vice versa, and thus the vertices belonging to the "middle" side itself would not be connected to any other vertices.

Thus, the maximum possible number of intersection points of our 14-sided polygon is

$$
2+4+5+4+2=17
$$

the fact that such a number of intersection points is possible is shown by the example (Fig. 53).

![](https://cdn.mathpix.com/cropped/2024_05_21_15011fcc7946713185f0g-083.jpg?height=400&width=392&top_left_y=954&top_left_x=148)