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How do you arrange 6 pencils so that each one touches the other five? And what about 7 or 8?

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You need 2 layers. First arrange 2 pencils that spread outwards touching by the erasers, like a V, then slide one with the pencil tip in between. Then you can repeat the process with 3 more, sideways, laying on the original, this achieves 6:

6 pencils

You can also achieve an arrangement of 7 by making 3 V's of 2, an arranging them like a ninja star (not really sure how else to explain it) you may need to tie them with the V's because 1 pencil has to be on the other, it could be tough to balance. This is 6, you can add one more, upright to achieve a 7th:

enter image description here

I believe 8 may be impossible.

Note: The matchstick version, and images, can be found here;

http://brainden.com/forum/index.php/topic/148-touch/

I did not use it for the first solution, I found that myself (really proud) but it's where the second answer came from.

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  • $\begingroup$ That's correct. $\endgroup$ – Numberknot Oct 25 '14 at 15:11
  • $\begingroup$ But there's one more solution of this. $\endgroup$ – Numberknot Oct 25 '14 at 15:12
  • $\begingroup$ @user6029 Don't worry. Post more questions like this and you have enough rep in no-time :). Are there solutions for 7 or more pencils? $\endgroup$ – Mathias711 Oct 25 '14 at 15:20
  • $\begingroup$ @number can you please re-accept this answer? $\endgroup$ – warspyking Oct 26 '14 at 0:20
  • $\begingroup$ @number I also added some images in. $\endgroup$ – warspyking Oct 26 '14 at 0:37
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The solution for 7 pencils - without using the ends - was only recently discovered:

http://arxiv.org/abs/1308.5164

Seven pencils

Seven mutually touching infinite cylinders - Sándor Bozóki, Tsung-Lin Lee, Lajos Rónyai. It was presented at "Gathering 4 Gardner" in May 2014, because the original 7-cigarette version, allowing the ends to be used, was popularised by a Martin Gardner column in Scientific American 50 years ago.

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    $\begingroup$ Wow. I didn't expect it to be possible, but it is. I edited your answer by adding the picture from the quoted paper, which clarifies the post even more. $\endgroup$ – Mathias711 Oct 25 '14 at 20:38
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    $\begingroup$ I cannot remember the pattern while @warspyking's one instantly stuck into my head. Also it is symmetrical: hence it is the elegant solution. $\endgroup$ – bunyaCloven Oct 26 '14 at 8:55
  • $\begingroup$ @warspyking: your solution does need the ends. If, for example, your yellow pencil doesn't end where it does (but goes on infinitely), it has to pass under or over the red pencil. If it passes over the red pencil yellow doesn't touch blue anymore. If it passes under the red one, red doesn't touch blue anymore. $\endgroup$ – oerkelens Oct 27 '14 at 8:01
  • $\begingroup$ And yes, this answer is an elegant solution, but the question was about finite "cylinders", not infinite ones. And for finite cylinders, I think warspyking's solution is even more elegant because it can actually be demonstrated easily in reality :) $\endgroup$ – oerkelens Oct 27 '14 at 8:03
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    $\begingroup$ @oerk I didn't realize what was meant by ends... $\endgroup$ – warspyking Oct 27 '14 at 17:50
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My 7 year old daughter came up with this. I can't find anything wrong with it... 6 pencils all touching each-other

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    $\begingroup$ They don't touch exactly. There would be a bit of space between the two pencils going diagonally like this [/] and the far right pencil that's almost vertical. $\endgroup$ – Deusovi Aug 13 '15 at 7:32
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    $\begingroup$ @ChaslyfromUK: Actually, yes. I can prove my objection. For two 'bundles' of two pencils to fully touch, they must be on parallel planes. But if all three planes are parallel, the two not in the middle must not be touching because they're too far apart. $\endgroup$ – Deusovi Aug 13 '15 at 15:17
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    $\begingroup$ @chaslyfromUK By that I just mean that all four pencils in two bundles are touching all the others. $\endgroup$ – Deusovi Aug 13 '15 at 15:33
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    $\begingroup$ @chaslyfromUK Sorry for being unclear - a bundle is just two pencils next to each other. Two parallel lines define a plane, so each "bundle" has its own plane defined by the centers (where the lead is in the pencils). Because the pencils are touching, the lead between two touching bundles has a consistent vertical distance, meaning the planes defined by touching bundles are parallel. The three planes are parallel and a fixed distance apart (because the pencil widths are constant), so the outer bundles cannot touch. $\endgroup$ – Deusovi Aug 13 '15 at 18:44
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    $\begingroup$ @chaslyfromUK: If the pairs aren't parallel, then they cannot touch each other. I'm not assuming that they're placed on a surface at all. If one of the pencils up there is not parallel to the one next to it, then it can't be touching the other ones because it'd be above or below them! And if two of the planes defined by the pairs of pencils are not parallel, then we have the same problem that I originally mentioned. $\endgroup$ – Deusovi Aug 15 '15 at 4:20
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If the pencils are cylinders, and do not have to have equal radii, nine mutually touching cylingers is possible. See page 15 of this paper.

enter image description here

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  • $\begingroup$ It's very late, but could you provide an image or something so that this isn't just a link-only answer? $\endgroup$ – boboquack Oct 21 '17 at 21:44

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