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In the below image, can you draw 6 straight lines that pass all the circles?
As soon as you start drawing lines you can't take your pen up until you draw all six lines.
hint: you don't have to keep the polyline inside the square.

Edit: Best answer is the one in which straight line passes center of the circles.

enter image description here

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    $\begingroup$ hint: you don't have to keep the polyline inside the square $\endgroup$
    – John Dvorak
    Commented Aug 12, 2014 at 8:40
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    $\begingroup$ Do you have to use all 6 lines? $\endgroup$
    – RobinvdA
    Commented Aug 12, 2014 at 14:09
  • $\begingroup$ it is possible with less than 6,but only with 6 lines you can pass the center of each circle.if you can do that with less than 6 do it in answer. $\endgroup$
    – Masoud Mohammadi
    Commented Aug 12, 2014 at 15:54

16 Answers 16

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I think this is one possible solution:

solution

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If you use just one massive line, you can make it pass through the center of all of the circles!

solution

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    $\begingroup$ that's not what i meant, and what kind of pen is it?it's just a big triangle not big line. $\endgroup$
    – Masoud Mohammadi
    Commented Aug 12, 2014 at 19:36
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    $\begingroup$ It's just a really big pen. It is not a triangle. $\endgroup$
    – waylon531
    Commented Aug 12, 2014 at 19:56
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    $\begingroup$ I signed up to +1 this. Hilarious! regarding it being a square and not a line, just extend the 'line' very far beyond the border and zoom out. problem solved, you have a line that looks like a line. $\endgroup$
    – n00b
    Commented Aug 13, 2014 at 20:00
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    $\begingroup$ This is a contender for the worst answer on the site, and it has pretty strong competition. The question is clearly a math question, not a troll request. Declaring that a square is a line does not make a line have two dimensions. $\endgroup$
    – Gilles 'SO- stop being evil'
    Commented Jan 14, 2015 at 19:28
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    $\begingroup$ I'm just waiting for enough reputation to downvote this solution :-P $\endgroup$
    – ABcDexter
    Commented Sep 8, 2015 at 10:22
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Here's an option that uses only 4 lines. You can extend the concept to place another 2 lines if you really want 6 ...

solution using 4 lines

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    $\begingroup$ This is what I immediately thought of when I read the question. +1. This works for arbitrarily small circles, since the lines can be arbitrarily long. And if you give me 0-size circles, I can still do it with infinitely long lines! $\endgroup$
    – Cruncher
    Commented Aug 12, 2014 at 14:35
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    $\begingroup$ it is possible with less than 6,but only with 6 lines you can pass the center of each circle, which i didn't mention in this puzzle.but still your answer is correct.can you pass center of each line with less than 6 lines? $\endgroup$
    – Masoud Mohammadi
    Commented Aug 12, 2014 at 15:56
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    $\begingroup$ @Riddle well that would be the same as passing through 0-size circles which works with infinitely long lines as Cruncher mentioned in his/her comment. $\endgroup$
    – Verena Praher
    Commented Aug 13, 2014 at 9:03
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    $\begingroup$ ... and whether you can draw two parallel lines without lifting your pen depends on whether they meet, which in turn depends on the geometry of the space in which the figure appears. Paste the figure to a torus (at an angle) and you can do it in one line! $\endgroup$
    – user1501
    Commented Aug 13, 2014 at 12:32
  • $\begingroup$ @ExpectoPatronum I guess the crux is that you can't draw infinitely long lines :) $\endgroup$
    – Cruncher
    Commented Aug 15, 2014 at 20:54
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By mapping the puzzle onto a cylindrical topography, I've solved the puzzle using only a single straight line.

solution pic 1solution pic 2

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  • $\begingroup$ it is possible with less than 6,but only with 6 lines you can pass the center of each circle, which i didn't mention in this puzzle.but still your answer is correct.can you pass center of each line with less than 6 lines? $\endgroup$
    – Masoud Mohammadi
    Commented Aug 12, 2014 at 15:57
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    $\begingroup$ Why have the sides match up evenly? Have the right side of the first row of circles match up with the left side of the second row of circles, and your line should go through the center of all the circles. $\endgroup$
    – Rob Watts
    Commented Aug 12, 2014 at 20:19
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    $\begingroup$ What definition of straight line are you using here? Presumably something other than "shortest path between two points"? $\endgroup$
    – jl6
    Commented Aug 14, 2014 at 12:40
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    $\begingroup$ Btw, @Simon can you create another image with the second row matching the first row like suggested by RobWatts? That way it will answer the question (with OP's additional clarification) perfectly with only one line. $\endgroup$
    – justhalf
    Commented Sep 4, 2014 at 5:17
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    $\begingroup$ @justhalf, that was created in Sketchup with the puzzle image mapped onto a cylinder as a texture (I have the joy of being Simon's boss) $\endgroup$
    – Caius Jard
    Commented Sep 18, 2014 at 14:53
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solution

Here's another way of thinking. This is using edges instead.

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    $\begingroup$ I like this one a lot. $\endgroup$
    – jliv902
    Commented Aug 13, 2014 at 19:27
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    $\begingroup$ This actually looks like art :) +1, I would hang this $\endgroup$
    – Cruncher
    Commented Aug 15, 2014 at 20:59
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  1. Print the picture.
  2. Fold the image such that there is a single stack of circles. This will take 6 folds.
  3. Draw one line across the middle of it.
  4. Profit.

solution

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    $\begingroup$ actually you just draw a line on one of circles, if you unfold your paper again just one of them has been crossed, it seems you just disappeared the other circles. $\endgroup$
    – Masoud Mohammadi
    Commented Aug 13, 2014 at 7:23
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    $\begingroup$ @Riddle: It's all about point of view. From my point of view the line has crossed the center point of all of the circles as they are stacked on top of each other. The problem with this exercise is that you haven't defined any real boundary conditions (things you are explicitly not allowed to do). This means there are many different ways of solving the puzzle that don't fit your preconception. $\endgroup$
    – NotMe
    Commented Aug 13, 2014 at 14:10
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    $\begingroup$ Just set the point of the pen on the center of the circle and push really hard with the pen, then it will have passed through all the circles. It is one line, going through the z-axis of the folded paper. $\endgroup$
    – Anssssss
    Commented Aug 15, 2014 at 21:18
  • $\begingroup$ @Anssssss: that's even better! $\endgroup$
    – NotMe
    Commented Aug 15, 2014 at 22:42
  • $\begingroup$ -1 for no ??? step $\endgroup$
    – Kaz Wolfe
    Commented Aug 17, 2014 at 9:02
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This one doesn't solve it the way you're supposed to, but also doesn't break any of the implied "rules", but kind of defeats the purpose of the puzzle (which is go outside of the square). Were you to make the circles smaller, it would eliminate this answer from plausibility. (Also the (3,3) circle line is a little dubious, if I had taken some more time to perfect this it would fit inside the circle better, but also shows how this solution could be easily invalidated by making the circles smaller)

solution

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    $\begingroup$ You better write that this way you manage to keep the polyline inside the square! +1 $\endgroup$
    – klm123
    Commented Aug 12, 2014 at 17:33
  • $\begingroup$ You can move the vertex at (1,2) upwards slightly to get a better fit for (3,3) up to the point where it equals the fit for (3,2). $\endgroup$
    – John Dvorak
    Commented Aug 13, 2014 at 18:15
  • $\begingroup$ Yeah, I didn't try super hard to get it to fit perfectly, as it kind of helped point out how this method is easily broken by making the circles smaller. $\endgroup$
    – DLeh
    Commented Aug 13, 2014 at 18:27
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Why not this?

This one also does what he wants.

solution

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    $\begingroup$ because you take your pen up for several times $\endgroup$
    – Masoud Mohammadi
    Commented Aug 13, 2014 at 7:19
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    $\begingroup$ @Riddle But if we r allowed to trace back our path then its doable this way without lifting the pen right. $\endgroup$
    – Hubble07
    Commented Aug 13, 2014 at 8:27
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    $\begingroup$ You don't need to pick your pen up. Just double back on the line you already drew. $\endgroup$
    – Roddy of the Frozen Peas
    Commented Aug 14, 2014 at 1:49
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    $\begingroup$ Wouldn't doubling back be drawing another line, though? So you could only have three lines. $\endgroup$
    – wchargin
    Commented Aug 14, 2014 at 14:42
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    $\begingroup$ No, there is no "depends" about it. There is no doubling back. The instant you try that is the instant you fail on this. Making a line is a forward motion (meaning not doubling back). $\endgroup$
    – Olive Stemforn
    Commented Jun 26, 2015 at 15:53
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solution

Similar but different answer. Great puzzle!

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my own answer to this question is
in this picture you can see 6 lines throw the center of all circles without taking your pen up.

solution

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For some reason I wanted the answer to have a line with an angle that was not a multiple of $45^\circ$.

solution

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solution

This could be the answer as well.

6 lines without lifting the pen.

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another possible answer!

here is my answer!

nice question!

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Hope this will also be considered as an answer.

solution

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Looking through the answers, there are several similar, but none the same as this one. So here's one more possible answer.

Possible answer/solution to the puzzle.

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  • $\begingroup$ your answer is exactly same as mine, if you rotate your picture once to right you can see that. $\endgroup$
    – Masoud Mohammadi
    Commented Nov 28, 2014 at 11:31
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below is my answer photo.
Just Simple and beautiful :D

My cool Answer


I guess my answer matched with at least this

Edit: Best answer is the one in which straight line passes center of the circles.

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    $\begingroup$ how would you draw those lines without lifting the pen? $\endgroup$
    – Kate Gregory
    Commented Dec 10, 2015 at 13:47
  • $\begingroup$ Cannot @KateGregory. That's why I add my last sentence there. $\endgroup$
    – Nai
    Commented Dec 11, 2015 at 8:02
  • $\begingroup$ Well, can you overlap lines? $\endgroup$
    – Mr Pie
    Commented Jul 10, 2018 at 5:07

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