A.K. Tolpygo

A broken line consisting of 37 segments is given, which is not closed and does not intersect itself. A straight line has been drawn through each segment.

What is the smallest number of different lines that could result?

#
Suppose there are eight lines. Then they have no more than 28 intersection points, but the broken line cannot have vertices not at the intersection points of the lines, except for the initial and final points, meaning it cannot have more than 29 segments. An example for nine lines is shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_9a42dc41a6897b3a1545g-17.jpg?height=660&width=672&top_left_y=1368&top_left_x=699)

(A circle is depicted, with the vertices of a regular nonagon marked on it. Each of them is connected to two others by a straight line. The broken line in question runs along these lines, marked in white in the figure.)

## Answer

9 lines.