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 Post subject: Tunnel Cube by OSKAR
PostPosted: Sat Oct 12, 2013 5:59 am 
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Hi Twisty Puzzles fans,

Tunnel Cube can be considered the inverse of Adam G. Cowan's Handlebar Cube. Whereas Handlebar Cube has three handles at the outside that bandage non-adjacent corners and edges together, Tunnel Cube has three tunnels that achieve the same effect. The holes enable you to look inside the puzzle and to see which corners and edges are bandaged through these tunnels.

Watch the YouTube video.
Buy the puzzle at my Shapeways Shop.
Read more at the Shapeways Forum.

Check out the photos below.

Enjoy!

Oskar
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Last edited by Oskar on Sun Oct 13, 2013 5:17 am, edited 1 time in total.

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Sat Oct 12, 2013 11:39 pm 
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If anybody knows the answer to his question at the end I would love the insight

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Sun Oct 13, 2013 12:11 am 
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TheCubingKyle wrote:
If anybody knows the answer to his question at the end I would love the insight
The answer is in the comments posted at the video. But in short, all the pieces in the red layer are moved only by the movement of the two red connected edges. If this pieces didn't extend down into that slice layer it could come misaligned with the rest of the puzzle and you'd have no easy way to get it back in alignment. I don't believe both of the faces in the slice layer needed to be attached but even if just one was attached then the opposite one would never move relative to the connected edges anyways so why not attach it.

Very nice design Oskar. You are making too many puzzles I'd love to play with.

Carl

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Sun Oct 13, 2013 4:01 am 
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This cube allows more moves than Handlebar Cube.
The Tunnel Cube allows the tunnel between edges to stand orthogonal to those of the corners. This does not work for the Handlebar Cube. Oskar demonstrates this in his video.

Oskar:
Can you post an image of that state?
How flexible is that design principle? How many bandaged cubes with handlebars can you implement?

Andreas


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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Sun Oct 13, 2013 5:20 am 
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Andreas Nortmann wrote:
The Tunnel Cube allows the tunnel between edges to stand orthogonal to those of the corners. This does not work for the Handlebar Cube.
Thank you for noticing. Adam's three handlebars have the same hight, so one cannot pass under another. My tunnels can go under each other.
Andreas Nortmann wrote:
Can you post an image of that state?
Done.
Andreas Nortmann wrote:
How many bandaged cubes with handlebars can you implement?
That depends on how deep you want to tunnel.

Oskar

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Last edited by Oskar on Sun Oct 13, 2013 12:58 pm, edited 2 times in total.

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Sun Oct 13, 2013 7:56 am 
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Why use tunnels/holes at all, instead of symbolic stickers to show which cubes are linked and bandaged together by an invisible handle?

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Sun Oct 13, 2013 12:56 pm 
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KelvinS wrote:
Why use tunnels/holes at all, instead of symbolic stickers to show which cubes are linked and bandaged together by an invisible handle?
Why symbolic stickers, instead of showing the mechanism for real?

Edit: Oh, now I understand. You want me to build a cube with invisible handles!

Oskar

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Sun Oct 13, 2013 2:34 pm 
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Oskar wrote:
KelvinS wrote:
Why use tunnels/holes at all, instead of symbolic stickers to show which cubes are linked and bandaged together by an invisible handle?
Why symbolic stickers, instead of showing the mechanism for real?

Edit: Oh, now I understand. You want me to build a cube with invisible handles!
Being a big fan of mechanisms I'm squarely in Oskar's camp on this one. If I come up with a novel mechanism then I too would want to show off as much of it as possible. I could understand the desire to hide the handles to make the puzzle harder but its already hard enough in my opinion. If you were interested in purchasing one without the holes I bet if you asked Oskar nicely he could make one available for you.
Oskar wrote:
Andreas Nortmann wrote:
How many bandaged cubes with handlebars can you implement?
That depends on how deep you want to tunnel.
Interesting question... if one looks at a normal Rubik's cube there is only a single piece hidden... the core. The Mixup Cube has a whole hidden layer inside... the fuzed cube. This puzzle has two hidden layers inside. The first is... using Andreas' notation (L, U) {or at least it would be if the 2 green corners weren't bandaged together} and the second layer could be considered to be (L,R,U). It isn't till the 3rd shell that you get the bulk of the external pieces. This makes me wonder... and I'm tempted to say Andreas has probably already performed this first calculation.

(1) How many uniquely different 3x3x3s are possible simply by bandaging/fuzing subsets of the original 27 cubies (counting the core) together such that they must be treated as one piece even if some of the cubies in that subset don't touch the others? By uniquely different I mean that if the stickers were removed that if two states which appear different could be scrambled from one one state to the other they are NOT uniquely different. Bridges(or tunnels) should never interfere with each other. To me that is another type of bandaging which the original Handlebar Cube had.

(2) What is the maximum number of shells which would be needs to handle any of the uniquely different 3x3x3s detailed in the first question?

(3) Assuming one were to make a kit which could be used to build any one of these uniquely different 3x3x3s how many pieces would be needed? Think of it as a generalized version of this and for neighboring cubies it could even be designed to use that same tile set. It's only for the fusing of non-adjacent cubies that one needs the tunnels.

Hmmm... In that thread Andreas did make a very interesting post that I see:
Andreas Nortmann wrote:
mixer wrote:
Has anyone designs of the 3563 variations?
1. Download this program I posted here: viewtopic.php?f=1&t=24406
2. Run "Move everything to left list" in the Edit-menu.
3. Run "Filter for surface-bandagings only" in the Edit-menu.
4. Run "Trivial paths" in the Edit-menu.
5. Run "Filter permutation-free variants" in the Edit-menu.

All variants you now can see in the left list are the ones which are
1. non-trivial
2. implementable without the need of face-core-bonds.
Andreas, am I correct to assume that most of those 3563 variations are NOT uniquely different? For example does this count each of the 440 "configurations" (aka signatures) of the Mefferts Bandaged Cube? Or does this count as just one possible bandaging? Now if one wanted to include face-core bonds and tunnel bonds what does the number become? Though I'd really like to be able to exclude different configurations (or signatures) of the same puzzle.

Carl

P.S. Just realized that two states could be mirror images of each other and it may not be possible to reach one from the other. Per the above definition these could be states of two "uniquely different" puzzles. Maybe the definition of "uniquely different" should be expanded to exclude one of these. If you had one you could just solve it while looking in a mirror and pretend you had the other so there really wouldn't be anything new here.

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Mon Oct 14, 2013 11:09 am 
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Oskar wrote:
Andreas Nortmann wrote:
How many bandaged cubes with handlebars can you implement?
That depends on how deep you want to tunnel.
Thank you for the additional image.
Does that mean you can implement an arbitrary number of tunnels? Face tunnels included?

@Carl:
All 3563 variants are different. The Mefferts Bandaged Cube is on that list with just ONE configuration. The program uses rotations AND reflections for filtering.

I already calculated the number of bandaged 3x3x3s including bridges.
Start that program and choose "Puzzles"->"3x3x3 with bridges"
For the sake of mathematical completeness that list includes the 3563 variants.


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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Mon Oct 14, 2013 12:07 pm 
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Andreas Nortmann wrote:
@Carl:
All 3563 variants are different. The Mefferts Bandaged Cube is on that list with just ONE configuration. The program uses rotations AND reflections for filtering.
Yes, but the 440 configurations of the Mefferts Bandaged Cube differ by more then just global rotations and reflections of the entire puzzle. Per Jaap, some of the configurations are "12 moves (or 15 if half turns are counted as two moves)" away from the solved configuration. Is your program really able to tell these are all assesable from a single puzzle? That sounds like an incrediblely complex problem to solve in general. It would seem one would need an general bandaged cube solver and have it test which states are reachable from each state that has the same type and number of bonds.

Carl

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Mon Oct 14, 2013 12:23 pm 
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wwwmwww wrote:
KelvinS wrote:
Why use tunnels/holes at all, instead of symbolic stickers to show which cubes are linked and bandaged together by an invisible handle?
If you were interested in purchasing one without the holes I bet if you asked Oskar nicely he could make one available for you.
Kelvin, would you be interested in commissioning the world's very first Invisible Handlebar Cube?

Oskar
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Invisible Handlebar Cube - view 1.jpg
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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Mon Oct 14, 2013 1:36 pm 
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Oskar wrote:
Kelvin, would you be interested in commissioning the world's very first Invisible Handlebar Cube?

Oskar
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Invisible Handlebar Cube - view 1.jpg

Very cool. Trust me, Oskar, if I was wealthy I would buy ALL of your puzzles! Until then, I can only dream and drool. :lol:

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Mon Oct 14, 2013 4:53 pm 
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Oskar wrote:
Kelvin, would you be interested in commissioning the world's very first Invisible Handlebar Cube?
Oskar... in hind sight you should have published this design first and not revealed the mechanism. Just think how something like this would have gone over:

Allow me to present the Invisible Handlebar Cube.

One handlebar is completed by the invisible force of attraction between the up and down quark which I've traped in the partial handles.
One handlebar is completed by the invisible force of attraction between the charm and strange quark which I've traped in the partial handles.
And the last handlebar is completed by the invisible force of attraction between the top and bottom quark which I've traped in the partial handles.

Fortunately these forces are non-interacting which allows these Invisible Handlebars to pass through each other and allow this puzzle to obtain states that a normal Handlebar Cube can not reach.

The model is on shapeways. Contact me to obtain the isolated quark pairs which need to be placed (very carefully) into the handles.

Enjoy...

Now that I'm sure would have gotten some very interesting reactions and I would have LOVED to have seen them.

LOL,
Carl

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Mon Oct 14, 2013 8:30 pm 
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Oskar wrote:
Hi Twisty Puzzles fans,

Tunnel Cube can be considered the inverse of Adam G. Cowan's Handlebar Cube. Whereas Handlebar Cube has three handles at the outside that bandage non-adjacent corners and edges together, Tunnel Cube has three tunnels that achieve the same effect. The holes enable you to look inside the puzzle and to see which corners and edges are bandaged through these tunnels.

Watch the YouTube video.
Buy the puzzle at my Shapeways Shop.
Read more at the Shapeways Forum.

Check out the photos below.

Enjoy!

Oskar



Very cool! It is like the female version of the Handlebar cube. We really need to get these together sometime...

Adam

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Tue Oct 15, 2013 11:39 pm 
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wwwmwww wrote:
Yes, but the 440 configurations of the Mefferts Bandaged Cube differ by more then just global rotations and reflections of the entire puzzle. Per Jaap, some of the configurations are "12 moves (or 15 if half turns are counted as two moves)" away from the solved configuration. Is your program really able to tell these are all assesable from a single puzzle? That sounds like an incrediblely complex problem to solve in general. It would seem one would need an general bandaged cube solver and have it test which states are reachable from each state that has the same type and number of bonds.
Just choose one variant from the program and hit CTRL+P.
The test you mention is rather easy: The program calculates all configurations which exists for that specific variant => No need for a general solver. This brute force approach works fine for the 3x3x3 because the maximal number of configurations is no higher than 97176. For the 4x4x4 this number is easily exceeded.
The Mefferts Bandaged Cube is just ONE variant out of 3563 (or 5844 if you allow face-core-bonds).

BTW: This is the variant with 97176 configurations:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=2682


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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Wed Oct 16, 2013 12:04 pm 
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Andreas Nortmann wrote:
Just choose one variant from the program and hit CTRL+P.
The test you mention is rather easy: The program calculates all configurations which exists for that specific variant => No need for a general solver. This brute force approach works fine for the 3x3x3 because the maximal number of configurations is no higher than 97176.
I was already very impressed with your program. Now even more so. Clearly I need to play with it more. Even brute for this sounds complicated.
Andreas Nortmann wrote:
For the 4x4x4 this number is easily exceeded.
The Mefferts Bandaged Cube is just ONE variant out of 3563 (or 5844 if you allow face-core-bonds).
Neither of those numbers include non-adjacent bandinging which would require tunnels or bridges... correct?
Andreas Nortmann wrote:
BTW: This is the variant with 97176 configurations:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=2682
Did you include tunnel or bridge bandaging when you looked for this maximal number of configurations case? And what is the minimum number of turns (either metric: quarter turn or face turn) to move between the two furthest configurations in the worst case. Can I assume that is also on this specific variant?

And to bring this back to Oskar's Tunnel Cube I had initially though this idea was general enough to allow the construction of any bandaged cube formed from fuzing cubies (adjacent or non-adjacent) if one simply used enough shells. Now I'm not so sure. Let's start with this Tunnel Cube. If I wanted to add just one more bond to this puzzle, say bond the red face center to the core such that it couldn't rotate could that be done? I believe it can be made with bridges but I don't see a way to do it with tunnels. Oskar... please prove me wrong.

Hmmm... now having said that I think I see a way. Only the innermost shell has access to the core so the red face center fuzed to the core would be the innermost shell. You'd then have a shell which contained the fuzed corners. Both would look like the green corners here as neither would be fuzed to the core. Then there would be a shell or the red fuzed edges. And then finally the rest of the pieces on the surface.... at least I think that would work.
[UPDATE: Nope I already see a problem. The shell which contains the bandaged corners has pieces which could get out of alignmnt. Need to think some more.]
[UPDATE #2: Just realized this idea has an even bigger problem. The red face center could still be rotated. Do you see why? ARG!!!]

Carl

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Wed Oct 16, 2013 12:34 pm 
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wwwmwww wrote:
Andreas Nortmann wrote:
For the 4x4x4 this number is easily exceeded.
The Mefferts Bandaged Cube is just ONE variant out of 3563 (or 5844 if you allow face-core-bonds).
Neither of those numbers include non-adjacent bandinging which would require tunnels or bridges... correct?
If you had looked again into that program and had took a look at the numbers in the bar at the bottom then you could have recognized that the numbers belong to the 3x3x3 without bridges.
wwwmwww wrote:
Andreas Nortmann wrote:
BTW: This is the variant with 97176 configurations:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=2682
Did you include tunnel or bridge bandaging when you looked for this maximal number of configurations case? And what is the minimum number of turns (either metric: quarter turn or face turn) to move between the two furthest configurations in the worst case? Can I assume that is also on this specific variant?
97176 is the maximum for the 3x3x3 without bridges.
I have no idea about the number of moves.
Your third question misses a word.


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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Wed Oct 16, 2013 12:50 pm 
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Andreas Nortmann wrote:
If you had looked again into that program and had took a look at the numbers in the bar at the bottom then you could have recognized that the numbers belong to the 3x3x3 without bridges.
I have your program installed on my home PC. I'm at work at the moment. I'll check when I get home.
Andreas Nortmann wrote:
97176 is the maximum for the 3x3x3 without bridges.
I have no idea about the number of moves.
I assumed your brute force method was making moves until no new configurations were being created. If I'm correct could it count the moves? If its doing something else how does it know when all configurations of a variant have been found?
Andreas Nortmann wrote:
Your third question misses a word.
I was just curious if the variant which contained two configurations that were as far apart move wise as possible was the same variant which contained the highest number of configurations. It seems possible that a puzzle that was very bandaged could behave like a maze and their could potentially be a very long move sequence (say greater then god's number) needed to reach two distinct states of some of these puzzles.

Carl

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 Post subject: Re: Tunnel Cube by OSKAR
PostPosted: Thu Oct 17, 2013 11:32 am 
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wwwmwww wrote:
I assumed your brute force method was making moves until no new configurations were being created. If I'm correct could it count the moves? If its doing something else how does it know when all configurations of a variant have been found?
It does this:
1. Enter the original configuration into a search structure.
2. Create 18 configurations out of configuration N. (one if possible obviously)
3. Enter the NEW configurations into the search structure.
4. If there are configurations left then switch to N+1 and GOTO 2.
More things are going on in the algorithm but this is the general idea.
I could easily change the algorithm to identify the layer of each configuration but this number does not necessarily equals the number you ask for.
wwwmwww wrote:
Andreas Nortmann wrote:
Your third question misses a word.
I was just curious if the variant which contained two configurations that were as far apart move wise as possible was the same variant which contained the highest number of configurations. It seems possible that a puzzle that was very bandaged could behave like a maze and their could potentially be a very long move sequence (say greater then god's number) needed to reach two distinct states of some of these puzzles.
There are many variants with configuration trees which look like mazes. The filtering program filters for bonds which change nothing in the tree but leaving open dead ends. B5A005AD is the best example of a variant which is deleted because of dead ends. It is still possible that there are some variants which trees look like mazes but I have no idea of them.


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