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 Post subject: hexagonal/pentagonal magics
PostPosted: Wed Aug 02, 2006 10:26 am 
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Juozos asked about hexagonal magics, because he'd like to see how they're made. I for one don't have one and have never seen one in person, although it seems like their stringing should be fairly analagous to that of regular magics, with flips being 120 degrees instead of 90. Any detailed info would be appreciated.

I also suggested having a magic with pentagonal faces. Intuitively, this also seems doable. It turns out that six pentagons in a loop can lie flat in the plane, but I'm not sure about more. Does anyone know anything more about this? Has a pentagonal magic ever been made?


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PostPosted: Wed Aug 02, 2006 11:19 am 
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I don't think that petagonal pieces would lay flat without gaps. But at least you could build a dodecahedron instead of just a cube. Thats assuming that peices with an odd number of sides would work. Do to the layout of the strings it may not be possible. But I have not completely thought it out, so I could be wrong. I think the best way to test it would be with triangular pieces, so they could lay flat without gaps. It may be possible to build a simple mock-up out of stiff cardboard and twine. It wouldn't be strong and it wouldn't last long, but it would be a proof of concept.


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 Post subject: Re: hexagonal/pentagonal magics
PostPosted: Wed Aug 02, 2006 11:51 am 
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Bram wrote:
Juozos asked about hexagonal magics, because he'd like to see how they're made. I for one don't have one and have never seen one in person, although it seems like their stringing should be fairly analagous to that of regular magics, with flips being 120 degrees instead of 90. Any detailed info would be appreciated.

I also suggested having a magic with pentagonal faces. Intuitively, this also seems doable. It turns out that six pentagons in a loop can lie flat in the plane, but I'm not sure about more. Does anyone know anything more about this? Has a pentagonal magic ever been made?


I am pretty sure that there are many more designs to be made. Not just pentagonal, but anything that is a regular polygon. Of course for smoother movement, it should be required to be at most hexagon.

I think rhombi, triangles (not only isosceles), rectangles, are also possible.
There is no limit for the imagination as long as it is accompanied by a little logic.

But this is just a mathematicians' honest opinion, proof ommitted! :)


Peter


PS In fact I am planning to send some trinagle ones for 3-d printing as I have a 3-d file format ready.... but I have no time! LOL
(the triangle lines looks exactly like the ones on a triangular side of the Trajber's Octahedron http://twistypuzzles.com/cgi-bin/puzzle.cgi?pid=401 )

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 Post subject:
PostPosted: Wed Aug 02, 2006 12:03 pm 
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Maybe also mix different shapes in one puzzle? Or solid pieces, not only flat tiles. Combine both ideas and make a twistable soma cube...


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 Post subject:
PostPosted: Wed Aug 02, 2006 12:29 pm 
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You can't mix different shapes in a puzzle, but you can have somewhat irregular shapes - the main criterion is that each piece must be the mirror image of the ones before and after it. You could use non-flat pieces too, but those might have problems with the strings coming off.

Pentagons lie flat with gaps. The obvious puzzle is 'invert the dodecahedron'. No idea how solvable that one is.

I'm pretty sure that odd-sided things can be made, because hexagons are just triangles with the corners chopped off, and those have been made already.


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 Post subject:
PostPosted: Wed Aug 02, 2006 12:29 pm 
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StefanPochmann wrote:
Maybe also mix different shapes in one puzzle? Or solid pieces, not only flat tiles. Combine both ideas and make a twistable soma cube...


That would be ideal regarding tetrahedrons. It is not good for cubes, as the "3-tile moves" are limited to half, making the puzzle too simple.
How do I know? I even tried and tested one of those LOL
(and making such a combination of cubes is quite easy!)


:)

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 Post subject:
PostPosted: Wed Aug 02, 2006 12:35 pm 
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I don't follow your reasoning about cubes - isn't there a circuit going over each face of the cube once? And couldn't you technically use the end-to-end flips if you really wanted to string it that way?

For that matter, couldn't you string a regular square tile with an end-to-end line-up included in one of the flips? Or even have a complicated stringing which involved two adjacent tiles having to flip through each face multiple times before hitting a repeat?


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 Post subject:
PostPosted: Wed Aug 02, 2006 12:40 pm 
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Bram wrote:
I don't follow your reasoning about cubes - isn't there a circuit going over each face of the cube once? And couldn't you technically use the end-to-end flips if you really wanted to string it that way?

For that matter, couldn't you string a regular square tile with an end-to-end line-up included in one of the flips? Or even have a complicated stringing which involved two adjacent tiles having to flip through each face multiple times before hitting a repeat?


Of course, there can be more ways. In my case, I used solid cubes!

If you see how a 3-tile magic behaves, a tile can go under the other tile. However, if we are talking about cubes here and not flat tiles, that is not possible (unless at least one of them iwas a ghost-cube and the tiles could overlap in a metaphysical way - just kidding LOL).

Hmm... I wish I had taken a video to demonstrate it... it is difficult to use words... :oops:

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 Post subject:
PostPosted: Wed Aug 02, 2006 12:56 pm 
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I only post this link. Other shapes of magic tiles were disscused here. Idea to use solids looked to me as a most promissing one. My thoughts on the topic will come later.

viewtopic.php?t=690&highlight=magic

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 Post subject:
PostPosted: Wed Aug 02, 2006 1:05 pm 
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Juozas Granskas wrote:
I only post this link. Other shapes of magic tiles were disscused here. Idea to use solids looked to me as a most promissing one. My thoughts on the topic will come later.

viewtopic.php?t=690&highlight=magic


Now that is exactly what I am talking about!!! :D


Regarding triangle tiles, imagine a normal square one, cut diagonally into two isoceles right triangles. Then think of an 8 tile magic having all squares cut into two. That will make it a 16-triangle-tile magic. The last step would be to position the strings of half of those triangles to their other side to preserve stringing consistency. I think that would work!!! ;)


Peter

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 Post subject:
PostPosted: Wed Aug 09, 2006 8:47 pm 
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kastellorizo,

Could you please post a picture of how you plan to string the triangular tiles? I can't seem to come up with a working idea.


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 Post subject:
PostPosted: Wed Aug 09, 2006 11:43 pm 
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kwm wrote:
kastellorizo,

Could you please post a picture of how you plan to string the triangular tiles? I can't seem to come up with a working idea.


Well... so far anything I say regarding triangular tiles is theoretical as none of them have been physically demonstrated yet.

I have mentioned two types of them, isosceles and equilateral. I don't know if they are possible, and it will not be nice to give a picture/drawing that may not actually work (I am trying to find an excuse here because I am extremely busy LOL). But I believe the description is quite succinct, and if you try it, please let us know if it actually works! :)

In general though, most constructions that I had thought are possible, had eventually worked!
e.g. the double loop magics, or the amazing single-loop odd-number-tile magics aka magic myths. Unfortunately, the cube-tile magics have not worked... at least not yet!!! ;)




Pantazis

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 Post subject:
PostPosted: Thu Aug 10, 2006 1:37 am 
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I have never even heard of a hexagonal magic tile. that'd be cool if they do exist though.

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 Post subject:
PostPosted: Thu Aug 10, 2006 3:17 am 
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ScottBedard wrote:
I have never even heard of a hexagonal magic tile. that'd be cool if they do exist though.


There is one here:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pid=491


:)

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 Post subject:
PostPosted: Thu Aug 10, 2006 5:23 am 
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I have one in my collection :lol:
Image


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 Post subject:
PostPosted: Thu Aug 10, 2006 6:08 am 
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excalibur wrote:
I have one in my collection


Me too. I'd put it on my site if it weren't so darned difficult to draw pictures of how to fold the thing.

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 Post subject:
PostPosted: Thu Aug 10, 2006 10:22 am 
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Since the hexagonal tiles already exist and we know that they work, could someone please post a stringing guide for them? Thanks.


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 Post subject:
PostPosted: Sat Aug 19, 2006 4:54 am 
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Bram wrote:
You can't mix different shapes in a puzzle


Ok, admittedly I thought about something like triangles and squares together, but ... it's possible to have different shapes by "bandaging" tiles, for example the leftmost two of a regular magic, which then has 6 squares and one 2x1 rectangle (and is still solvable). And if you take more tiles and bandage more, you could have a magic made up of many different polyominoes.


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 Post subject:
PostPosted: Sat Aug 19, 2006 11:01 am 
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StefanPochmann wrote:
Bram wrote:
You can't mix different shapes in a puzzle


Ok, admittedly I thought about something like triangles and squares together, but ... it's possible to have different shapes by "bandaging" tiles, for example the leftmost two of a regular magic, which then has 6 squares and one 2x1 rectangle (and is still solvable). And if you take more tiles and bandage more, you could have a magic made up of many different polyominoes.



Very very true.

If the isosceles (orthogonal) triangle case *does* work, then an 8-square tile magic would be nothing more than a bandaged 16-triangle tile magic. And like you correctly said, many more combinations could be possible.

The real challenge though, is testing such examples to see if they actually work. :)



Pantazis


PS And my tiles have still not arrived!!! (I am getting reeeeaaaally frustrated now)

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 Post subject:
PostPosted: Fri Sep 08, 2006 8:18 pm 
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Hey all,

I've been working on the idea of pentagonal magic pieces and came up with some ideas. I don't have any proof since its easier for me to just try it than figure out the mate. And I would mess around by myself, but have no way of producing the pieces, especially accurately. So I look forward to your ideas thoughts etc.

Heres the file list with a quick description. I'll attach the pics in numerical order over a couple entries.

Pentagon.jpg - basic piece it is 1/5 the length "point to point" from point to first cut. (i hope someone understands that line)

Pent2.jpg - Arrangement for 10 tiles.

Pent4.jpg - Arrangement for 6 tiles.

Pent3.jpg - Stringing patter for 10 tile

Pent5.jpg - First string for 6 tile

Pent6.jpg - Opposite string for 6 tile

Okay now for the pics...


Attachments:
pent6.jpg
pent6.jpg [ 31.11 KiB | Viewed 4795 times ]
pent5.jpg
pent5.jpg [ 31.09 KiB | Viewed 4796 times ]
pent4.jpg
pent4.jpg [ 22.25 KiB | Viewed 4795 times ]
pent3.jpg
pent3.jpg [ 43.95 KiB | Viewed 4793 times ]
pent2.jpg
pent2.jpg [ 26.61 KiB | Viewed 4791 times ]
Pentagon.jpg
Pentagon.jpg [ 7.46 KiB | Viewed 4789 times ]

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 Post subject:
PostPosted: Fri Sep 08, 2006 9:35 pm 
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I'm no expert, but I don't think that will work. From what I've seen and my own experiments, the strings need to be placed on in a semetrical pattern. I've been playing with triangular tiles using cardboard and thread. I'm starting to think that tiles with an odd number of sides create a parity problem so that the string will begin to interfere with itself after a few flips. Again, I could be wrong. Cardboard and thread are an easy way to test your designs.


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 Post subject:
PostPosted: Sat Sep 09, 2006 5:24 am 
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I believe that it *will* work. The reason is similar to the magic myths. If someone watches the string from the beginning and follows it, it will go through ALL paths that are used by the strings at one time, in a moebius way. Because of this, the magic will be solid and always connected.
(and unlike the magic myths case, this needs not to be 3-d).

To me, it looks like a step of the natural progression from a square tile to a hexagonal one. By making the angles bigger, the premade paths will have to move away from the centre, in order to enable standard string-exchanges between two pairs of strings.

The drawings look brilliant. Good luck if you construct a protoype. ;)


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 Post subject:
PostPosted: Sat Sep 09, 2006 2:45 pm 
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wronginthehead, why are you diagramming for the strings covering four pieces instead of two? It sure looks like stringing for two will work fine if stringing four works.


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PostPosted: Sat Sep 09, 2006 9:25 pm 
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Bram, the first reason I was going to do four instead of two was because i haphazardly built one out of wood. The angles weren't right, gouging the tracks was difficult and at the time had no idea how to transfer the lines to the other side accurately. Oh and it took forever. But after doing some quick measurements and some math found that by running a string from the rubiks magic, you would be able to cover four of the pentagons. It was also a few weeks ago, so I may have misrememberd.

My other thought was since the regular magic tile had 4 sides and the string wound 3 tiles, I thought for the pentagon, it would have 5 sides, and wrap 4 tiles. I'll have to give it another go sometime soon, but I'm not set up to do anything along these lines right now :(

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 Post subject:
PostPosted: Sat Sep 09, 2006 11:27 pm 
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Note that the above diagrams are for stringing pentagons so that flips move to adjacent edges. There's an alternate way of cutting the grooves which would make flips go to one non-adjacent edges, although I'm not sure if the overlaps of strings would make the thing not work in that case.


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 Post subject:
PostPosted: Mon Sep 11, 2006 1:01 am 
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kwm wrote:
I'm no expert, but I don't think that will work.


I am also not yet convinced this will work nicely. You can see on the drawings that the strings cross each other on the same side of the pentagons. The long red sections have crossings. Things get worse when you draw in the string in the other groove. This crosses itself, and the first string too. These crossings must at least all go the same way so that when you try to flip a tile over to another edge, either both strings are free to move or neither.

Note that if a string goes underneath another, after a tile flip it will still go underneath the other (even if it is now another bit of string that crosses it). Therefore, any bit of string that is crossed by another will be useless.

If you start with just two square tiles strung together, you can move one tile round and round the other continuously, moving from one edge to the next. When you do this, every section of string will have been on one tile at some point and moved to the other tile later. In fact, after one round the metal bits of the strings will be in different places - the whole string has been shifted.

This will not be possible with pentagonal tiles because of the covered sections of string. When you have two tiles and move one around the other from edge to edge, eventually it will get stuck because it needs a bit of string that is crossed by another.

Jaap

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PostPosted: Mon Sep 11, 2006 6:02 pm 
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Thanks for the info jaap. I took some yarn to my haphazardly cut pentagons and couldn't get any progress. I may have to revisit the stringing pattern and maybe also the the groove pattern.

Any suggestions?

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