In the picture below (from this document), connect each pair of like-colored dots with a continuous path such that

  • No two paths intersect
  • No dot has three or more paths outside it enter image description here

Here is another image to clarify the second rule about paths outside of dots enter image description here

One thing I've realized is that each path will necessarily have to go outside of one or more of the dots. Because, e.g., if you connect the two green dots with a straight line, that leaves 5 pairs that will have to be connected by paths that each go outside of one of the green dots. So one of the green dots would have 3 (or more) paths outside it.

Beyond that, it has been trial and error (and more error). I am looking for hints about how to proceed. I think this is not the case, but maybe there is a good reason it is actually impossible?

  • $\begingroup$ Where is this puzzle from? Or did you come up with it yourself? $\endgroup$
    – bobble
    Commented Jan 21, 2023 at 3:51
  • $\begingroup$ I found it in this document which has a couple puzzles meant for middle or high school students. $\endgroup$ Commented Jan 21, 2023 at 3:59

2 Answers 2


As for a hint, consider trying a simpler version of the problem: Can you connect a 6 dot version of the same problem only allowing at most 1 line passing outside the circle?

Implicit hint: There is at least one solution to the puzzle as posted.

  • $\begingroup$ I found a solution for 6 dots right away, but something got lost in translation when I tried to scale up to the full puzzle. $\endgroup$ Commented Jan 21, 2023 at 13:54
  • 1
    $\begingroup$ The analogy I would use for the transformation would be to "unzip" the connection $\endgroup$ Commented Jan 21, 2023 at 16:15
  • $\begingroup$ <img src="cdn.discordapp.com/attachments/906528885561249804/…" width="100" height="100"> $\endgroup$ Commented Jan 21, 2023 at 16:34
  • $\begingroup$ Perfect. I actually solved it with a bit of luck before your comment about unzipping. That unzipping idea is the logical leap I couldn't seem to make on my own. $\endgroup$ Commented Jan 21, 2023 at 17:24
  • $\begingroup$ I find this really unsatisfying even if it is what the person asked for. Can it be done? How? I looked at the image linked in a comment and it seems to be to have 3 lines passing a dot, so i don't see how it helps. At the moment this answer is nothing more than "well, we smart ones know how to do it". Like maybe the answer should show how to do it for 6? Should explain why the linked image doesn't break the 3 lines rule? $\endgroup$ Commented Jan 18 at 19:28

There are several factors to the solution. Hints:

  • practice with fewer dots
  • think about balance and organization, not symmetry
  • be systematic

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