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 Post subject: Solving The Helicopter Dodecahedron
PostPosted: Fri Apr 08, 2011 10:55 pm 
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Location: Marquette, MI, USA
continued from this thread.

I have basically found the proper center orientations by luck on the puzzle. I am sure there is a way to determine possible positions by observing the paths that the pieces can possible take. Anyone care to try and figure out some rules for this?

I am also curious to the number of possible positions without jumbling, but including the center orientation problem? I have no idea how to determine this.

Basically after finding the first three centers, placing the rest of the centers in corners is very basic and intuitive. My problem is that I do not know how to rotate the corners once they are in the proper spots. Anyone help with this?

I realize I have lots of questions and not a whole lot of answers. Hope you guys don't mind.

I have also come upon an interesting situation with the last two centers. Is there a way to fix this without jumbling? or did I mess up a corner swap?

Pictures of what I have.


Attachments:
File comment: just another picture showing that some corners are oriented properly, others are not.
SD533649.JPG
SD533649.JPG [ 2.06 MiB | Viewed 1554 times ]
File comment: two center pieces swapped
SD533648.JPG
SD533648.JPG [ 2.04 MiB | Viewed 1554 times ]
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 Post subject: Re: Solving The Helicopter Dodecahedron
PostPosted: Sat Apr 09, 2011 5:25 pm 
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The trick to re-orienting pieces on basically any puzzle is to first find a 3-cycle for those pieces. Then find a way to re-orient one of the pieces ignoring everything else. If you can get your 3-cycle and your re-orientation routine to overlap by only one piece then you can re-orient pieces pure.

For the Helicopter Dodecahedron, first some notation:
Attachment:
helicopter_dodeca_notation.png
helicopter_dodeca_notation.png [ 17.96 KiB | Viewed 1522 times ]

Now a [1,3] 3-cycle [A] [B C B] [A] [B C B]:
Attachment:
helicopter_dodeca_3-cycle.png
helicopter_dodeca_3-cycle.png [ 10.53 KiB | Viewed 1522 times ]

Now observe that [m n o p q] will re-orient the white-green-cyan piece in place:
Attachment:
helicopter_dodeca_dirty_re-orient.png
helicopter_dodeca_dirty_re-orient.png [ 10.49 KiB | Viewed 1522 times ]

Finally combine. Do 3-cycle, dirty re-orient, undo 3-cycle, undo dirty re-orient:
Attachment:
helicopter_dodeca_pure_3-cycle.png
helicopter_dodeca_pure_3-cycle.png [ 10.44 KiB | Viewed 1522 times ]

This construction is the same for all twisty puzzles. If you understand it than you can re-orient any pair of pieces on any twisty puzzle provided that you can find a 3-cycle and a appropriate dirty way to re-orient a piece.

As for your center triangles problem -- I spent some time trying to create this situation and was unable to. I'm pretty sure it requires jumbling moves. Even if you can use non-jumbling moves to fix that, if you don't see an easy way to do that then you should probably use jumbling moves instead.

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Prior to using my real name I posted under the account named bmenrigh.


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 Post subject: Re: Solving The Helicopter Dodecahedron
PostPosted: Sat Apr 09, 2011 6:18 pm 
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Wow thanks a lot for the great write up. As an update I have the puzzle completely solved, spare two corners and those two centerpieces. On my first solve I must have only half completed on of the corner swap moves that uses jumbling. So somehow I need to figure out how to arrange them so I can finish it, and then resolve the puzzle once more. I might make a video sometime to show the cycles that I came up with. I guess they are cycles. I don't know really.


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 Post subject: Re: Solving The Helicopter Dodecahedron
PostPosted: Sat Apr 09, 2011 9:24 pm 
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I have created an image showing the 6 different orbits:
Attachment:
helicopter_dodeca_orbits.png
helicopter_dodeca_orbits.png [ 46.94 KiB | Viewed 1485 times ]
For each face that you are looking at, there is an orbit around the equator of the puzzle (light green in the image) that never touches that face or the opposite face. Using that, you can look at a face and then by a process of elimination, determine that the face is one of two possible colors. You can use this trick to quickly determine the correct core orientation.

As schuma has said previously, if you allow jumbling moves then you can solve this puzzle regardless of the invisible core orientation.

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Prior to using my real name I posted under the account named bmenrigh.


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 Post subject: Re: Solving The Helicopter Dodecahedron
PostPosted: Sat Apr 09, 2011 11:08 pm 
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Joined: Wed Oct 01, 2008 2:39 pm
Location: Marquette, MI, USA
Also thank you again. I have spent more time trying to solve this thing than I have with playing with all of my puzzles combined in the past month. The picture shows as far as I possibly can go. I spent a half hour trying to unjumble those two center pieces, failed and had to redo half of the puzzle(you can see it is a different two pieces now).

I then decided to try and make the same situation with my helicopter cube. I must say it was actually a challenge. It seems to stay mostly with having 3 face pieces out of spot. I did however manage to get just two of them swapped finally. The problem is I still have no idea how to fix it. :lol:

So I have broken down and decided to just take apart those last two edges and fix it....tomorrow. Now it is time to sleep. I have spent too much time on puzzles these past two days. :lol:


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SD533652.JPG [ 2.16 MiB | Viewed 1478 times ]
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