Problem: There are $2017$ lines in the plane such that
no three of them go through the same point.
Turbo the snail sits on a point on exactly one of the lines
and starts sliding along the lines in the following fashion:
she moves on a given line until she reaches an intersection of two lines.
At the intersection, she follows her journey
on the other line turning left or right,
alternating her choice at each intersection point she reaches.
She can only change direction at an intersection point.
Can there exist a line segment through which she passes
in both directions during her journey?

Solution: \textbf{Color the regions} of the plane black and white in alternating colors.
Then:
\begin{claim*}
Turbo will always move in one orientation around black regions
and the other orientation around white regions.
\end{claim*}
This completes the proof, even if Turbo may pick
which of left/right she follows each time!

\begin{remark*}
  An example of a cyclic path:
  take a pentagon and extend its sides to form a star.
  Then Turbo can trace around the exterior of the star.
\end{remark*}