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There are curved lines drawn on a piece of paper. Lines can have two characteristics:

  1. One of its endpoints is on another characteristic 1 line
  2. One of its endpoints is on another characteristic 2 line, exactly halfway between:
    • Its two endpoints
    • A point where a characteristic 2 line touches it and an endpoint
    • Two points where a characteristic 2 line touches it

Is it possible for there to be a third characteristic such that, for all sets of lines

  • It is impossible for all possible characteristic 3 starting points connect to a unique characteristic 1 endpoint
  • It is impossible for all possible characteristic 2 starting points connect to a unique characteristic 3 starting point
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    $\begingroup$ I'm having a hard time understanding this question. You say that lines can have characteristic 1 or characteristic 2. What is a "characteristic 2 starting point"? Is "characteristic 3" something that lines can have? Can a line have both characteristics 1 and 2? What about 1 and 3, or 2 and 3? $\endgroup$ – f'' Sep 25 '15 at 0:53
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    $\begingroup$ It may be helpful to replace the "characteristics" with colors (assuming each line gets one characteristic) and drawing an example picture. $\endgroup$ – Deusovi Sep 25 '15 at 0:58
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Yes, if we define characteristic 3 curves like so:

  1. A characteristic 3 curve cannot touch a curve of another characteristic at any point.

This trivially satisfies both criteria.

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