Loyal Lines

Given an NN-gon, we say that a line is loyal if it intersects the perimeter of the NN-gon along the entire length of exactly one edge. Determine the largest integer NN such that every non-degenerate NN-gon has a loyal line.

Details and assumptions

  • An NN-gon is non-degenerate if no three consecutive vertices are collinear, or equivalently, that no two consecutive edges are on the same line.
  • The above image is an example of a 20-gon that doesn't satisfy the conditions of the problem. The 10 dotted lines are all the lines that contain at least one edge of the 20-gon, however, because all of these lines contain 2 edges, none of them are loyal lines. This shows that the answer is not 20.
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