B Six Pencil Puzzle

[bob012345]

TL;DR Summary

Is it possible to arrange six pencils such that each one touches the other five?

Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.

 

[DaveE]

Unsharpened pencils (like cigarettes), I presume.

 

[Gavran]

This is a three dimensional problem. There is no solution if all six pencils occupy only one layer.

The source:
https://puzzleaday.wordpress.com/2022/11/23/six-pencils-all-touching/

 

[bob012345]

This is a three dimensional problem. There is no solution if all six pencils occupy only one layer.

https://puzzleaday.wordpress.com/2022/11/23/six-pencils-all-touching/

Did you figure this yourself before searching for the answer online? Also, please use spoiler bars so as not to give away the answer. Thanks.

 

[DaveC426913]

I failed, and had to cheat. So there was no point in me responding and ruining the puzzle.

 

[Gavran]

Did you figure this yourself before searching for the answer online?

No, I did not. Sorry for the spoiler bar.

By the way, there is another solution which can also be found by using Google.

In my opinion, this solution is not correct. What do others think?

 

[willyengland]

I think you're right.
The slope prevents certain pencils from being touched.

 

[DaveC426913]

In my opinion, this solution is not correct. What do others think?

I think you're right.
The slope prevents certain pencils from being touched.

I don't see why it would be incorrect.

Each pair of pencils is not 100% parallel in 3 dimensions, even though it may look like they are. They will be very slightly askew and only touch near their centres. This allows them to be in contact with two other pencils at each end.

Put another way, this structure might not work on a flat surface, where the four bottom pencils are fully in contact with the surface along their entire lengths. But it could be glued together so that all six touch each other; it just wouldn't sit flush with the four bottom pencils lying flat on the surface. It would rock a tiny bit.

 

[bob012345]

No, I did not. Sorry for the spoiler bar.

You can edit your original answer so that anyone can see the solution if they click on the “spoiler” button.

Spoiler

Hidden content goes here

Below I misspelled it so you can see the code.

[SPOILERR]
Hidden content goes here
[/SPOILERR]

 

[bob012345]

There are other ways to solve this. Can anyone find different solutions for six pencils? Also, can it be done with seven?

 

[DaveE]

By the way, there is another solution which can also be found by using Google.
View attachment 363688
In my opinion, this solution is not correct. What do others think?

Yes, my first guess. But then I thought:

The slope prevents certain pencils from being touched.

And then:

But it could be glued together so that all six touch each other; it just wouldn't sit flush with the four bottom pencils lying flat on the surface. It would rock a tiny bit.

Now I'm pretty confused. My intuition says it will work as a non-planar structure. But, every time you move a pencil to close the gap at a vertex, you change another vertex, creating another error to fix. Does this process converge? It seems like it ought to.

This sort of 3-D geometry hurts my brain, I'm not very good at these. Since I can't afford a Solidworks license, I may need to get 6 pencils...

BTW, in case anyone wants to use their brain and not glue. Let's make a naming convention. This is mine:
- Equilateral triangle, because symmetry is our friend.
- Each side of the triangle is named 0^o, 120^o, or -120^o. I wouldn't complain if you shortened this to 0, +, or -.
- Each leg of the triangle has inner and outer pencils. So, a pencil could be labeled "Inner 120^o", or "I+".

I'd vote to give you a fields medal if you could find all of the angles without computers or protractors.

 

[bob012345]

Yes, my first guess. But then I thought:


And then:


Now I'm pretty confused. My intuition says it will work as a non-planar structure. But, every time you move a pencil to close the gap at a vertex, you change another vertex, creating another error to fix. Does this process converge? It seems like it ought to.

This sort of 3-D geometry hurts my brain, I'm not very good at these. Since I can't afford a Solidworks license, I may need to get 6 pencils...

BTW, in case anyone wants to use their brain and not glue. Let's make a naming convention. This is mine:
- Equilateral triangle, because symmetry is our friend.
- Each side of the triangle is named 0^o, 120^o, or -120^o. I wouldn't complain if you shortened this to 0, +, or -.
- Each leg of the triangle has inner and outer pencils. So, a pencil could be labeled "Inner 120^o", or "I+".

I'd vote to give you a fields medal if you could find all of the angles without computers or protractors.

Glueing a gap in my mind is not a solution since these are mathematical pencils.

 

[DaveE]

Glueing a gap in my mind is not a solution since these are mathematical pencils.

Of course. But it means you're really close to a solution. Maybe pretend that your mathematical pencil has a diameter of pencil plus glue thickness. Caulking would definitely be cheating.

 

[bob012345]

Of course. But it means you're really close to a solution. Maybe pretend that your mathematical pencil has a diameter of pencil plus glue thickness. Caulking would definitely be cheating.

The original puzzle was for round cigarettes. I changed it to pencils but I realize some pencils are hexagons. One edge could be higher than another for two adjacent pencils but I think this is not in line with the spirit of the puzzle.

 

[DaveE]

Somehow I knew I would end up with this. Now I'm 99.9% sure it works.

Plus (bonus!) I used up some of the very many old broom sticks I am compelled to save. Also, I avoided painting my bathroom for at least an hour.

 

[bob012345]

Somehow I knew I would end up with this. Now I'm 99.9% sure it works.

View attachment 363705
View attachment 363706

Plus (bonus!) I used up some of the very many old broom sticks I am compelled to save. Also, I avoided painting my bathroom for at least an hour.

I’m not convinced they all touch.

 

[DaveE]

I’m not convinced they all touch.

It would be nice to have a more rigorous solution. Maybe some linear algebra since there are so many constraints?

 

[bob012345]

It would be nice to have a more rigorous solution. Maybe some linear algebra since there are so many constraints?

I can tell you that the solution in the Martin Gardner book matches the solution in post #3 above but hints there are others.

 

[willem2]

Here are 2 solutions with 7 infinite cylinders.
https://www.sciencedirect.com/science/article/pii/S0925772114000819
Major computation is needed to find solutions of 20 non-linear equations with 20 variables.
The only thing in that article that's easy, is that if you have 2 parallel infinite pencils, the maximum number is 4, so the solution from Gavran in post #6 with two pencils on the table, next to each other can't work.

 

[willyengland]

I tested "solution" #6 with real pens and can confirm that they don't all touch each other.
There's a small, visible gap. And I can slide a piece of paper into it.

 

[bob012345]

Here is Martin Gardner’s solution for 7 which also is a solution for 6 along side his original solution for 6.

Spoiler

Remove #7 and it is a different solution for 6 pencils.
/

 

[DaveE]

The only thing in that article that's easy, is that if you have 2 parallel infinite pencils, the maximum number is 4, so the solution from Gavran in post #6 with two pencils on the table, next to each other can't work.

"The case of parallel cylinders (lines) is excluded from our analysis. It is left to the reader to show that if two cylinders are parallel, then the maximum number of mutually touching cylinders is four."

Yes, but that doesn't address the non-planar case where the pencils that look parallel are not, they only touch in the middle. As described in post #8.

 

[DaveE]

And... Version 2. Now I'm 99.9% sure it doesn't work. Although it's hard to show in a photo. The cable ties are a nice easy solution which allows you to slide and rotate the sticks. There's always 1 gap you can't get rid of where an inner stick misses an outer stick (you get to choose which one, of course).

Also, no surprise, the asymmetric configuration with one leg below the other four crossing sticks doesn't work. But it's a lot closer, which surprised me.

Still not painting my bathroom, but I'm rapidly running out of excuses.

 

[Frabjous]

Still not painting my bathroom, but I'm rapidly running out of excuses.

Unless you the foresight to cut the sticks from your paintbrush handles.

 

[bob012345]

And... Version 2. Now I'm 99.9% sure it doesn't work. Although it's hard to show in a photo. The cable ties are a nice easy solution which allows you to slide and rotate the sticks. There's always 1 gap you can't get rid of where an inner stick misses an outer stick (you get to choose which one, of course).

I think there is no loss of generality by using square pegs which lay flat in the plane. Then the gap would be quite easy to see.

 

[DaveE]

I think there is no loss of generality by using square pegs which lay flat in the plane. Then the gap would be quite easy to see.

Which plane? No two sticks are coplanar, everything's twisted a bit. Anyway, the gap is easy to see in person, particularly because you can slide and twist things around without improvement. It's just hard to demonstrate here.

 

[bob012345]

Which plane? No two sticks are coplanar, everything's twisted a bit. Anyway, the gap is easy to see in person, particularly because you can slide and twist things around without improvement. It's just hard to demonstrate here.

Bending, twisting, forcing and distorting the pencils is not allowed. In the original puzzle with cigarettes it was a rule that they could not be bent or distorted to make it work.