Determine the largest positive integer N, such that given any N-gon (not necessarily convex), there exists a line (infinitely extended in both directions) that contains exactly 1 edge of the N-gon.
The figure in blue is an example of a 20-gon that doesn't satisfy the conditions of the challenge. The dotted lines indicate why there is no line which contains exactly 1 edge of the 20-gon This shows that N\neq 20. It does not imply that N > 20 or N < 20 .
To determine the largest positive integer N that satisfies the given condition, we need to carefully analyze the problem.
First, let's consider some simple cases.
For a triangle (3-gon), any line drawn will intersect with exactly 2 edges of the triangle. So, N cannot be 3.
For a square (4-gon), any line drawn will intersect with exactly 2 edges of the square. So, N cannot be 4 either.
Now, let's consider an N-gon with N greater than 4. We'll try to find a pattern.
When N is 5, we can draw a diagonal line that will intersect with exactly 1 edge of the pentagon.
When N is 6, we can draw a line passing through two opposite vertices of the hexagon, which will intersect with exactly 1 edge.
Now, let's consider an N-gon where N is greater than 6. The line that we draw can pass through at most 2 vertices of the N-gon. In order for the line to intersect with exactly 1 edge, the line must not pass through any other vertex.
Since any line passing through 2 vertices of an N-gon where N is greater than 6 will also pass through other vertices, we conclude that no N-gon where N is greater than 6 can satisfy the condition given in the question.
Therefore, the largest positive integer N that satisfies the condition is 6.