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 Post subject: A Short Mathematical Problem
PostPosted: Tue Oct 08, 2013 9:48 pm 
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So, I have buckyballs as desk toy (as I'm sure many of us do), and I've noticed something when messing around with 6 hexagons. Basically here's the statement that I'm wondering if it is possible to prove (I haven't been able to find a case where it's not true):

    You have six congruent regular hexagons. You arrange these hexagons in a plane in such a way that every hexagon shares at least one side with another hexagon. By moving only one hexagon to a new location (where it still shares at least one side with another hexagon), it is possible to create an arrangement of hexagons with at least one line of symmetry.

Any ideas? I have absolutely no experience in this line of mathematics, so I'm interested to see what you guys can come up with.


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 Post subject: Re: A Short Mathematical Problem
PostPosted: Tue Oct 08, 2013 10:06 pm 
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ejisfun wrote:
You arrange these hexagons in a plane in such a way that every hexagon shares at least one side with another hexagon. By moving only one hexagon to a new location (where it still shares at least one side with another hexagon), it is possible to create an arrangement of hexagons with at least one line of symmetry.


You didn't implicitly state that the plane needs to originate in any amount of symmetry so couldn't you just build it in such a way that it is one step away from symmetry? Or am I missing something about the preface

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Tue Oct 08, 2013 10:24 pm 
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Can't you just make a line of 6 hexagons, then move one of the end ones to the other end?

000000 --> 00000 0 ----> 0 00000 --> 000000

[Pretend they are hexagons with two parallel sides vertical]

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Tue Oct 08, 2013 11:36 pm 
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No, the problem is to prove that for *any* starting configuration where they are mutually adjoining, a single move exists that will create symmetry.

Interesting.


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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 12:14 am 
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Ah okay. I'll give this a little think then

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 12:41 am 
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So... it appears to me that the answer is no. There are 82 hexahexes:

http://en.m.wikipedia.org/wiki/Polyhex_(mathematics)

so it suffices to check them all individually. I can find exactly one that doesn't seem to have a single move to make it symmetric! This is such a nice property, this problem must be known. Of course I could be mistaken.

Can anyone else find the one hexahex that can't be made symmetric?


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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 1:42 am 
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Here's a simple algorithm:

1. Find all 82 hexahexes (combinatorial problem)
2. Link all pairs of hexahexes that can interconverted
3. Find all hexahexes which are symmetric
4. Find any hexahexes which have no symmetric partner.

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 2:41 am 
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I'm interested in the symmetric properties of the 30,490 decahexes. Can anyone check these for me :lol:

If you plot the number of n-hexes against n on a log-linear scale, you get this:
Attachment:
File comment: Graph
Graph.png
Graph.png [ 19.46 KiB | Viewed 2569 times ]
I've no idea what this means ...

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 4:06 am 
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bhearn wrote:
Can anyone else find the one hexahex that can't be made symmetric?


It is 2 am so I might be totaally wrong but, assuming rotational symmetry counts, I've figured out all but the very last one and the 5th to last, second row down. Are either of these the one you couldn't find a solution to?

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 5:42 am 
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TheCubingKyle wrote:
I've figured out all but the very last one and the 5th to last, second row down. Are either of these the one you couldn't find a solution to?

No, those both have mirror image solutions.

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 9:45 am 
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By the problem statement, no, rotational symmetry does not count, though I don't think that matters, given the requirement that a hex be moved to a new location.

I posted this to Facebook, where I have a lot of mathy puzzly friends, and one friend found the same solution I did, and nobody said they knew this problem.

I posed it as

"Find a hexahex that cannot be turned into a hexahex with reflective symmetry by moving a single hexagon. It appears to me that there is exactly one hexahex with this property."

Martin Gardner would have liked this.


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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 11:40 am 
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So what's the answer? I bet I can find a way to make it symmetric! 8-)

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 11:54 am 
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bhearn wrote:
"Find a hexahex that cannot be turned into a hexahex with reflective symmetry by moving a single hexagon. It appears to me that there is exactly one hexahex with this property."
Question... must the hexahex after the move of a single hexagon be a different hexahex then the one you started with?
bhearn wrote:
Martin Gardner would have liked this.
He's not the only one. Please don't post the answer just yet.

Thanks,
Carl

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 11:59 am 
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SPOILER BELOW

My guess is the third down in the first column. You can't move either of the middle two hexagons without leaving one floating. You only have to check one of the remaining 4 in all spots, and I didn't see anything symmetrical when I did it.

SPOILER ABOVE

Nevermind, I found the symmetrical one here.

I'm pretty sure it's the one at this link http://i.imgur.com/20Gy6TJ.png. I won't actually post the image, but I'm pretty sure this is it!

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 12:29 pm 
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I will have to give this another think now that it's not 2 am then :lol:

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 12:31 pm 
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Gus wrote:
I'm interested in the symmetric properties of the 30,490 decahexes. Can anyone check these for me :lol:

If you plot the number of n-hexes against n on a log-linear scale, you get this:
Attachment:
Graph.png
I've no idea what this means ...

Looks to me like it means the number of n-hexes grows O(n^c) as n grows. Lots of things grow exponentially.

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 1:03 pm 
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wwwmwww wrote:
Question... must the hexahex after the move of a single hexagon be a different hexahex then the one you started with?

By the problem statement, no. And in fact, if it had to be different, then the 6-in-a-row configuration (and maybe some others) also would not be solvable.

theVDude wrote:
I'm pretty sure it's the one at this link http://i.imgur.com/20Gy6TJ.png. I won't actually post the image, but I'm pretty sure this is it!

Hmm. That's not the one I found. But now I don't see a solution for this one either. Bummer! I thought I had checked them all, though I did go through them rather quickly. Maybe I skipped this one because it is already symmetric? If you are allowed to moved a hexagon to the same place, then this one is also OK.


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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 1:25 pm 
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bhearn wrote:
Hmm. That's not the one I found. But now I don't see a solution for this one either. Bummer! I thought I had checked them all, though I did go through them rather quickly. Maybe I skipped this one because it is already symmetric? If you are allowed to moved a hexagon to the same place, then this one is also OK.
I've got the list down to 6. 4 if I'm allowed to move the hexagon back to its starting position.

Carl

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 1:29 pm 
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wwwmwww wrote:
I've got the list down to 6. 4 if I'm allowed to move the hexagon back to its starting position.

Well maybe there are more that I missed. But there were a few I was convinced had no solution -- until I saw it.


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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 1:48 pm 
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bhearn wrote:
Well maybe there are more that I missed. But there were a few I was convinced had no solution -- until I saw it.
I can't upload an image to my personal website from work and I didn't want to post a spoiler in the thread so I've emailed you a picture on the 4 I've pinned things down to plus the 2 that I think require the hexagon be placed back where it started. If I'm missing something let me know.

This comment has me puzzled:
bhearn wrote:
By the problem statement, no, rotational symmetry does not count, though I don't think that matters, given the requirement that a hex be moved to a new location.
Some of these I can get to or start with rotational symmetry but I fail to see how to give them mirror symmetry with a single move.

Carl

UPDATE: Ok... I'm now down to 3 plus 2 that I think require the hexagon be placed back where it started. 2 of these 3 start with rotational symmetry and I can give the 3rd rotational symmetry by moving a single hexagon.

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Last edited by wwwmwww on Wed Oct 09, 2013 5:49 pm, edited 1 time in total.

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 2:16 pm 
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All of these hexahexagons have mirror image symmetry - in the plane of the paper/screen. :wink:

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 7:09 pm 
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wwwmwww wrote:
This comment has me puzzled:
bhearn wrote:
By the problem statement, no, rotational symmetry does not count, though I don't think that matters, given the requirement that a hex be moved to a new location.

What I meant was that for what I thought was the unique solution, it has rotational symmetry, but can't be made mirror symmetric by moving one hexagon. But...

Quote:
UPDATE: Ok... I'm now down to 3 plus 2 that I think require the hexagon be placed back where it started. 2 of these 3 start with rotational symmetry and I can give the 3rd rotational symmetry by moving a single hexagon.

I think I agree with you. Of the three that have no solution even if you move a hexagon to the same place, two of them can be made 180°-rotationally symmetric. I was mis-seeing that as mirror symmetry. And the two where you have to move a hexagon to the same place, theVDude's and one other, I'm not sure how I missed those. Again, maybe because they're already mirror symmetric.

So, this problem doesn't seem to have quite as nice an answer as I thought, no matter how you formulate it. I don't see any tidy statement (e.g. allowing null moves, rotational symmetry) that yields a unique solution. Am I missing something (again)? This is the meta-puzzle! Find a simple problem statement about moving pieces of hexahexes that has a unique solution. :)

I guess "find a hexahex that lacks mirror symmetry and 180° rotational symmetry, that cannot be made into such a hexahex by moving a single hexagon" seems to work, but it's not very natural.


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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 8:10 pm 
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You could think of the set of hexhexes as nodes in a graph and the edges connecting hexhexes that can be turned into each other by moving a hexagon.

Then: "Find a hehex that is more than one hexagon move way from having any rotional or mirror symmetry".

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Wed Oct 09, 2013 8:14 pm 
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But I think with that formulation there are none. The original solution I found has rotational symmetry, but not 180° rotational symmetry.


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 Post subject: Re: A Short Mathematical Problem
PostPosted: Thu Oct 10, 2013 9:52 am 
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Wish I could see the images being posted.

Anyways, some thoughts that came to mind reading through this topic:

Which hexahex has the highest degree of reflectional symmetry? Which has the highest degree of rotational symmetry. I am thinking the Ring Hexahex is the answer to both, but I am not sure.
How many hexahexes have n-fold reflectional symmetry for each value of n? How about for n-fold rotational symmetry.

Defining a move as removing one Hexagon from a hexahex to make a pentahex and than replacing the hexagon to form a different hexahex, am I correct that any hexahex can be transformed into any other hexahex in a finite number of moves? If so, what is the maximum number of moves needed to transform between a given starting and ending hexahex.

Defining a move such that a hexagon rotates about a vertex shared with a neighboring hexagon and moves to an adjacent empty cell of the hexagonal grid, is it possible to transform any hexahex into any other hexagex in a finite number of moves?
If such a restrictive move definition divides the hexahexes into multiple orbits, how many orbits exist, and how many hexahexes exist in each orbit?
What is the maximum number of moves to transform a hexahex into a different hexahex with at least one line of mirror symmetry or at least 180 degree rotational symmetry?
How does the problem change if you allow any vertex shared by exactly two hexagons to serve as a hinge provide there is enough empty grid space to allow both parts of the hexahex to complete a move about the hinge.(i.e. starting with a stick hexahex, if only one hexagon is allowed to move, only the end hexes can move, but with this more flexible set of movement rules, it can be bent in half to form a v-shped or check-mark shaped hexahex).
Any ideas on how to make such a Transforming Hexahex as a physical model?
How many permutations would such a transforming hexahex have if the hexagons where made to show position and orientation.
Has such a puzzle involving hinged tiles stuck in a plane ever been built before?

How about extending these lines of thought to other n-hexes and even other planar polyforms?

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Thu Oct 10, 2013 10:30 am 
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Must all the hexagons remain touching after the one move?

I've also narrowed the field, and found that some can only be given reflexive symmetry so long as the hexagons are no longer a regular hexahex.

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Thu Oct 10, 2013 12:56 pm 
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bhearn wrote:
This is the meta-puzzle! Find a simple problem statement about moving pieces of hexahexes that has a unique solution. :)

I guess "find a hexahex that lacks mirror symmetry and 180° rotational symmetry, that cannot be made into such a hexahex by moving a single hexagon" seems to work, but it's not very natural.
I don't think this one is any more natural but I think it does have a unique answer... and its different than the answer to yours.

Of the Hexahexes that lack any symmetry (mirror or rotational) what is the only one which can't be given mirror symmetry by moving a single hexagon.

The answer to my question can be given 180° rotational symmetry by moving a hexagon. The answer to Bob's question is rotationally symmetric... its just not 180° rotationally symmetric.
EMarx wrote:
I've also narrowed the field, and found that some can only be given reflexive symmetry so long as the hexagons are no longer a regular hexahex.
I don't think I agree with this statement. I'll post a link to a picture tonight that will show the results I was able to get.

Carl

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Thu Oct 10, 2013 2:22 pm 
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EMarx wrote:
Must all the hexagons remain touching after the one move?

I've also narrowed the field, and found that some can only be given reflexive symmetry so long as the hexagons are no longer a regular hexahex.


It's not a hexahex if they aren't all touching!

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Thu Oct 10, 2013 10:14 pm 
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Here is the image I set Bob. Don't open unless you want to be spoiled.

http://wwwmwww.com/Puzzle/hexahex2.png

The hexahexes with a red hexagon can be turned into hexahexes with mirror symmetry by moving the red hexagon. This may not be the only one that could be moved and I also don't say where to move it to so I don't spoil everything. The ones with a yellow hexagon already have mirror symmetry and the only way I see for them to keep it is to put the yellow hexagon right back where it started. The ones that are all green I don't see a way to give them mirror symmetry just by moving a single hexagon.... even if that moved hexagon doesn't remain touching. If EMarx's statement above is correct I've missed something... which is entirely possible.

If you spot any errors here please let me know.

Thanks,
Carl

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Fri Oct 11, 2013 12:35 am 
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The second yellow hexahex can be given mirror symmetry if you move a hexagon and they don't remain touching. The first one seems to be the unique case.

There were several others that I "solved" using my method, but your detailing has made me realize your more eloquent solutions.

Edit: "Unique" along with the three green outliers.

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Sat Oct 12, 2013 3:27 pm 
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EMarx wrote:
The second yellow hexahex can be given mirror symmetry if you move a hexagon and they don't remain touching.
I'm still not seeing it. Is this the shape you are turning it into?
Attachment:
hexahex3.png
hexahex3.png [ 3.68 KiB | Viewed 2067 times ]

If so this has 180 degree rotational symmetry and not mirror symmetry. What am I missing? As I see it the shape has mirror symmetry to start with and regardless of which hexagon you pick to move the only way I see to give it mirror symmetry is to place it back where it started.

Carl

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Sat Oct 12, 2013 3:53 pm 
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...............0
...............0
.............0..0
..................
...............0
...............0

Line of symmetry right down the middle. :lol:

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Sat Oct 12, 2013 4:32 pm 
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EMarx wrote:
Line of symmetry right down the middle. :lol:
Thanks!!! I stand corrected. Not sure how I missed that.

Carl

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Sat Oct 12, 2013 4:42 pm 
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Does it even matter though? What's the question we're trying to answer at this point? It's been lost in the shuffle.

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 Post subject: Re: A Short Mathematical Problem
PostPosted: Sat Oct 12, 2013 5:50 pm 
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EMarx wrote:
Does it even matter though? What's the question we're trying to answer at this point? It's been lost in the shuffle.
Oh the question I was hoping to answer just then was are there really any hexahexes which can only be given reflexive symmetry by moving a single hexagon so long as the hexagons are no longer touching (i.e. a regular hexahex). And it does appear that if one considers a move to not be the null move then the answer is yes... there is one. I had been saying "no" until you pointed out what I wasn't seeing.

This could be turned into another answer to Bob's question "This is the meta-puzzle! Find a simple problem statement about moving pieces of hexahexes that has a unique solution."

Carl

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