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Beginner puzzle

This puzzle is intended to be suitable for people who are new to puzzle solving.

Clarification: Both experienced solvers and new solvers are welcome to post solutions to this puzzle.


Which figure or figures shown below can be drawn with one continuous line without drawing a segment twice?

enter image description here


This puzzle comes from the Senior Kangaroo 2021 contest.

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  • $\begingroup$ I don’t understand the question sorry. Could you show an example. How doesn’t my answer work? $\endgroup$
    – PDT
    Commented May 12 at 9:00
  • $\begingroup$ @PDT For each of the four figures, you are to determine which can be drawn with the given conditions. If you were trying to draw the figures on paper with a pen, you place the pen tip on the paper and keep it on the paper while drawing the figure without retracing any part of the figure. $\endgroup$ Commented May 12 at 9:08
  • $\begingroup$ More than one is correct. $\endgroup$ Commented May 12 at 9:30

3 Answers 3

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Same answer as @Beastly Gerbil, but I find the below path much more clearer. Also it's easier to see that this will work for every number of circles:

enter image description here

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  • 1
    $\begingroup$ +1 This is the same solution I came up with except I went clockwise around the circles. $\endgroup$ Commented May 12 at 18:17
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    $\begingroup$ @WillOctagonGibson we can dismiss that discrepancy as the effect of the Coriolis force on different hemispheres (north vs south) of the planet. Refer to the appropriate Simpsons episode down under for more information. $\endgroup$ Commented May 13 at 13:32
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The rule is to count the number of odd vertices. An odd vertex is any point where an odd number of line or curve segments end at that point, including "dead ends" where just one segment is there. Then any path drawn without retracing can have no more than two odd vertices: possibly one at the beginning and possibly one at the end.

So with the first figure, you have one odd vertex on the left and one on the right, these being the "dead ends". That's just two, so you're good. But with the second figure you also have the points where the circle meets the "stubs" (three line or curve segments ending at these points), so a total of four odd vertices. That is a no-go, you have to draw the figure in two parts or retrace a segment to remove sone odd vertices.

Can you count the odd vertices in the third and fourth figures and see if they can be drawn without retracing or using multiple parts?

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    $\begingroup$ You should really use spoiler tags. $\endgroup$
    – bob
    Commented May 14 at 20:43
  • $\begingroup$ Maybe, but in this case I wanted people to see the key topological rule directly. I still require the reader to evaluate the last two cases. $\endgroup$ Commented May 14 at 21:13
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I would say

All of them except the second:

enter image description here

The 3 can all start at the end of the horizontal line and can loop around semi circles back and forth until they reach the other side. This can be done for infinitely many circles on a line.

However, the circle can't be looped round to get to the other side without crossing over itself

(Sorry for the bad drawings :P)

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  • $\begingroup$ Ah I see thought there could be only one solution mb $\endgroup$
    – PDT
    Commented May 12 at 9:14
  • $\begingroup$ @PDT I can see why you thought that, the wording is a bit ambiguous $\endgroup$ Commented May 12 at 9:17
  • $\begingroup$ @PDT I apologize for the ambiguity. I will try to fix the question. $\endgroup$ Commented May 12 at 9:22

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