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 Post subject: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 6:35 am 
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Hi Twisty Puzzles fans,

Some time ago, Alex Polonsky showed me a new design of his with seven cubes tumbling in an unusual fashion inside an exoskeleton. I asked his permission to reuse the movement, but using an endoskeleton instead.

The result is a 3D sliding puzzle where seven cubes turn around a sphere. Because of the cube reorientations similar to Bram's Black Hole, I called this one Alex Black Hole. The mechanism is similar to PantaCube. I expect this puzzle to be quite confusing to solve.

Watch the YouTube video.
Buy the puzzle from my Shapeways Shop.
Read more at the Shapeways Forum.
Check out the photos below.

Enjoy!

Oskar
Attachment:
Alex Black Hole - prototype - view 1.jpg
Alex Black Hole - prototype - view 1.jpg [ 33.44 KiB | Viewed 5586 times ]

Attachment:
Alex Black Hole - prototype - view 2.jpg
Alex Black Hole - prototype - view 2.jpg [ 38.23 KiB | Viewed 5586 times ]

Attachment:
Alex Black Hole - prototype - view 3.jpg
Alex Black Hole - prototype - view 3.jpg [ 38.94 KiB | Viewed 5586 times ]

Attachment:
Alex Black Hole - prototype - view 4.jpg
Alex Black Hole - prototype - view 4.jpg [ 40.36 KiB | Viewed 5586 times ]

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 7:30 am 
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WOW! Looks awesome. It looks like the rotation would interfere with the pieces movement, but I'm sure that it work great.

Good job.

EDIT: Now that I have seen a video, I understand the mechanism. Very cool concept.

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Last edited by Adman234 on Fri Sep 17, 2010 7:32 am, edited 1 time in total.

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 7:32 am 
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That looks really great!

Would you consider this a twisty or a sliding block puzzle? It reminds me a bit of the CrossTeaser (but now in 3D).

Can we play with it at DCD?

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 9:01 am 
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I guess you can just do continous circulation of one piece to get its desired orientation. so whats the idea of the one way direction?

would it be possible to desing this puzzle with "free will direction" instead?

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 10:06 am 
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Nice one Oskar!

This puzzle is to the Octedron, what the Pandacube is to the Cubedron.
So unlike the Polo Cube, the groups generated by this puzzle and the
Octedron, are directly related, and based on rotational movement of eight
pieces with respect to the center of the puzzle. (In fact, by using a "dexterity
rotation" of three pieces at at time of the Octedron, someone could also simulate
the Polo Cube!).

And in a similar way, the Icosahedron version of this puzzle, has been made
in the form of Rubik's Brain Racker (which corresponds to the Krystalledron).

What I had found is that the problem with puzzles based on platonic solids
which are made of odd-numbered polygons, is that by just swapping one
piece back and forth, the orientation can be easily corrected, shifting the
difficulty to the placement of the pieces, as orientation is something easy to fix.

So Oskar, in the puzzle you presented, is it easy to rotate those pieces, or is there
something internal to prevent it, essentially bandaging the "orientation trick"?

:)


Pantazis

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Last edited by kastellorizo on Fri Sep 17, 2010 10:14 am, edited 1 time in total.

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 10:12 am 
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kastellorizo wrote:
This puzzle is to the Octedron, what the Pan[t]acube is to the Cubedron. ... is it easy to rotate those pieces, or is there something internal to prevent it, essentially bandaging the "orientation trick"?
Pantazis,

You have guessed it right. Yes, there is a "bandaging trick". The internal groove structure provides only one way of moving. So no easy orientation for this one. Please inspect grooves in the sphere closely closely in the picture below. Do you see the curvature of the grooves? This makes this puzzle much harder than you would expect, and unequal to Octedron.

Oskar

Attachment:
Alex Black Hole - view 6.jpg
Alex Black Hole - view 6.jpg [ 37.54 KiB | Viewed 5458 times ]

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Last edited by Oskar on Fri Sep 17, 2010 10:17 am, edited 1 time in total.

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 10:16 am 
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Oskar wrote:
kastellorizo wrote:
This puzzle is to the Octedron, what the Pan[t]acube is to the Cubedron. ... is it easy to rotate those pieces, or is there something internal to prevent it, essentially bandaging the "orientation trick"?
Pantazis,

You have guessed it right.Yes, there is a "bandaging trick". The internal groove structure provide only one way of moving. Please inspect the pieces closely and see the curvature of the grooves. This makes this puzzle much harder than you would expect, and unequal to Octedron.

Oskar



Yeap, that would *surely* make the puzzle harder! Like going to a point of no return!!! LOL
(Such bandaging can also be applied in an equivalent way on loose pyramids by adding
small holes/grooves and extensions to predetermined positions).

:)


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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 12:24 pm 
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Very nice puzzle. I like it a lot.

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 12:28 pm 
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Im more into twisty puzzles, but it still looks great.

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 12:57 pm 
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Hmmm.... The 8th piece of this puzzle is there. If you color it too does that make it any harder?

Attachment:
ABH2.png
ABH2.png [ 334.22 KiB | Viewed 5362 times ]


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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 1:00 pm 
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That seems like a very good idea. That way, the core has to be in the right position and orientated correctly.

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 1:02 pm 
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wwwmwww wrote:
The 8th piece of this puzzle is there. If you color it too does that make it any harder?
Carl,

That is an interesting question. Maybe it becomes harder, as you have an 8th piece to take care of. But maybe it becomes easier, as the core orientation may help you avoid a parity trap. Compare Rubik's Cube versus Void Cube.

Oskar

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 1:23 pm 
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Oskar wrote:
This makes this puzzle much harder than you would expect, and unequal to Octedron.


Agreed, since now it is confirmed that there is some extra internal mechanism,
and, as a puzzle, the Octedron is unequal to Alex's Black Hole.

Now, The Octedron's group is related to the groupoid (not group) of Alex's Black Hole,
in a similar restrictive way that the Rubik's 3x3x3 group is related to a bandaged 3x3x3's groupoid.
And what makes Alex's Black Hole harder is that lack of symmetry in those moves,
and the need to use larger "pattern preservative" (in this case "pattern" = "core position")
sequences for its solving algorithms (which is why I stated "point of no return! :) )



Oskar wrote:
But maybe it becomes easier



Oskar is right, as this will give a hint to where the initial positions of the puzzles were.
And even if you end up with some misplacement of a couple of pieces, it shouldn't be
too hard before finding an appropriate sequence to fix it up.

Something like adding colors to the ghost cube... without them, I have seen some of
the most known speedsolvers fail to solve it.

;)


Pantazis


PS. Apologies for misspelling the Panta-Cube!!! :oops:

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Fri Sep 17, 2010 11:51 pm 
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I think this one might have some commercial potential.


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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sat Sep 18, 2010 12:42 pm 
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kastellorizo wrote:
Oskar wrote:
But maybe it becomes easier

Oskar is right, as this will give a hint to where the initial positions of the puzzles were.

But I'm still not sure that "hint" will make things easier.

For example... can this puzzle be solved with the core in any orientation? If yes, then I think coloring the core would make it harder. If the core can only exist in one orientation in the solved state then yes... if you color it that just gives you more information and makes things easier. If the core can just be in 2 or 3 orientation when the other 7 pieces are solved then I'm still not sure.

I take it everyone is just guessing at this point. If I'm looking at it correctly the core has 3*8 or 24 possible orientations that the puzzle could be assembled in. If you assembled the cube such that the other 7 pieces are solved... how many of the other 24 states can you reach where the 7 outer pieces can appear solved? Using symmetry I think I can rule out the possibility of there being only 2 states. I'm tempted to say the answer is probably one of these:

1 - and coloring the core makes the puzzle easier
3 - the colored corner of the core can appear rotated on its axis but it can't be translated. And I'm thinking adding colore to the core make the puzzle harder.
8 - the colored corner of the core can appear translated but can't be rotated on its axis. Adding color makes the puzzle harder.
24 - all states are allowed and adding color makes the puzzle harder.

And thinking a bit more... I can also rule out 8 as a solution because 3 translations can be used to rotate the inner corner.

So I'm pretty sure its one of these...

1 allowed state - easier
3 allowed states - easier or harder?
24 allowed states - harder

Anyone know how to prove one way or the other? Oscar... as you are the only one with the puzzle at the moment I say color the core and then solve the other 7 pieces without regard to the core and let us know if the core solves itself in the process.

Carl

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sat Sep 18, 2010 12:51 pm 
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For whom it may be interesting - attached several pic's from the previous versions of this puzzle. (ver1,ver2)
(My versions)

Alex.


Attachments:
v1A.jpg
v1A.jpg [ 120.98 KiB | Viewed 5115 times ]
v1B.jpg
v1B.jpg [ 146.82 KiB | Viewed 5115 times ]
v2A.jpg
v2A.jpg [ 135.91 KiB | Viewed 5115 times ]
v2B.jpg
v2B.jpg [ 138.82 KiB | Viewed 5115 times ]

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sat Sep 18, 2010 2:11 pm 
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wwwmwww wrote:
can this puzzle be solved with the core in any orientation?


Ok, that is a different question. I assumed we know the exact initial state, and the goal
was to return to that state, and only that state!). Having solved configurations based on
differently oriented center and pieces, can lead to interesting new questions.


And Lexi, good to see you again here. But I need to inform you that the latest photos
correspond to something too close to my Octedron's mechanism. That is, the patent
I filed some time ago, covers blocks similar in a topological way (not just pyramids, which may
be bandaged, gear-like, i.e. blocks with extensions and holes) moving with the aid of hands,
gravity etc (that is, holes *can* exist), based on all platonic solids. In this case, it is the Octahedron
(i.e. eight block pieces minus one to free space, plus the housing).

In all cases of the platonic solids, the movement resembles rotational symmetry with respect
to the center of the puzzle. The mechanism of the filed patent is "caged".

Have you seen my video yet?


Pantazis

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sat Sep 18, 2010 2:46 pm 
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kastellorizo wrote:
And Lexi, good to see you again here. But I need to inform you that the latest photos correspond to something too close to my Octedron's mechanism. That is, the patent I filed some time ago...
You do see that this is the puzzle Oskar reference in the original post right? Does your patent really cover such a huge spectrum?
Oskar wrote:
Some time ago, Alex Polonsky showed me a new design of his with seven cubes tumbling in an unusual fashion inside an exoskeleton. I asked his permission to reuse the movement, but using an endoskeleton instead.

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sat Sep 18, 2010 3:04 pm 
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GuiltyBystander wrote:
Does your patent really cover such a huge spectrum?


It is a super simple idea with very few number of pieces which behave in a similar rotational way
for a very limited number of solids. Why shouldn't it? (Paying extra for extra claims is not a bad thing)


GuiltyBystander wrote:
You do see that this is the puzzle Oskar reference in the original post right?


The Polo Cube is very different, as it also has 2x2x2 turns, so it is not related to the patent.
The Black Hole has an internal mechanism, so again, there is no relation. I was clearly only referring
to the new photos, which are of a different puzzle with a non-changeable external housing.


*** EDIT ***

Matter has been resolved between good friends (as expected). Thanks Alex!


Pantazis


PS. Apologies to Oskar for having such an off-topic intervention.

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sat Sep 18, 2010 6:33 pm 
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kastellorizo wrote:
GuiltyBystander wrote:
You do see that this is the puzzle Oskar reference in the original post right?
The Polo Cube is very different, as it also has 2x2x2 turns, so it is not related to the patent. The Black Hole has an internal mechanism, so again, there is no relation. I was clearly only referring to the new photos, which are of a different puzzle with a non-changeable external housing.
There seems to be a miscommunication here. I'm saying that these two puzzles are essentially the same.
Image
Image
And I don't think the 2nd one allows 2x2x2 movements.

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sat Sep 18, 2010 7:10 pm 
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kastellorizo wrote:
wwwmwww wrote:
can this puzzle be solved with the core in any orientation?

Ok, that is a different question. I assumed we know the exact initial state, and the goal
was to return to that state, and only that state!). Having solved configurations based on
differently oriented center and pieces, can lead to interesting new questions.

But when the puzzle is "solved" how do we know we are back in the "exact" initial state if the core is uncolored? There are certainly reasons to leave things uncolored though and Oskar's mentioning of the Void Cube is a perfect example. There it creates a parity trap that doesn't exist on a Rubik's Cube. Here I stongly suspect the coloring of the core gives you an 8th piece to solve and makes the puzzle harder... but short of being able to solve this puzzle with a colored core and showing that all states of the core are possible with the other 7 pieces solved I don't know how to prove that. So as I see it... its the same question.

Carl

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sun Sep 19, 2010 2:16 am 
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GuiltyBystander wrote:
There seems to be a miscommunication here. I'm saying that these two puzzles are essentially the same.


Indeed there is a miscommunication, because you need to understand the difference.
Mechanically, there are *not* the same. The first one uses no cage with sliding, and
the second one uses loose pieces within a cage. My concern is only about a tiny family of
caged puzzles with a non changeable/non movable cage.


GuiltyBystander wrote:
And I don't think the 2nd one allows 2x2x2 movements.


Of course not. No one said it did. Had you noticed, I mentioned it because it is caged.
(but it is different because the cage twists, i.e. it is movable).


wwwmwww wrote:
But when the puzzle is "solved" how do we know we are back in the "exact" initial state if the core is uncolored? There are certainly reasons to leave things uncolored though and Oskar's mentioning of the Void Cube is a perfect example. There it creates a parity trap that doesn't exist on a Rubik's Cube. Here I stongly suspect the coloring of the core gives you an 8th piece to solve and makes the puzzle harder... but short of being able to solve this puzzle with a colored core and showing that all states of the core are possible with the other 7 pieces solved I don't know how to prove that. So as I see it... its the same question.

Carl


Back to talking about the main topic!

In the beginning of the above paragraph, we are saying the same thing. I agree that an uncolored
core will assist to a parity. Now, coloring the core to provide an 8th piece would only work if the goal
was to give to us a "2x2x2 illusion", that is if you moved three pieces in one cycle (four moves in total
to complete what is equivalent to a 90 degree twist of the 2x2x2). Here you can move one piece at
a time (i.e. the algebraic group of this puzzle contains the one obtained from the 2x2x2 puzzle).

This is because on a 2x2x2 we are allowed to "get away with it" by fixing one piece on the core.
This is not the case here, as there is much more freedom of movement *and* the core position
does matter whenever its orientation is changed.

In other words, it is identical to have an 8th piece if we tried to solve a 2x2x2 within this
puzzle, yet if we tried to do the same "varikon one-by-one piece" way (of course, always
including piece orientation), then the core acts in a more revealing way. The ""varikon one-by-one
piece" way, breaks the 2x2x2 twist into two components: (a) change of position between the normal
pieces, (b) change of position of *all* pieces (not just the seven of the its corresponding 2x2x2)
with respect to the core.

I hope I am making sense!

:)


Pantazis

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Last edited by kastellorizo on Sun Sep 19, 2010 3:00 am, edited 2 times in total.

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sun Sep 19, 2010 2:53 am 
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You make a good point on the relationship to the 2x2x2.

My friend Dario suggested a similar puzzle, which I built as Giant Pearl. It's a moving-hole puzzle in 3 dimensions with 7 exterior parts. It permits 2x2x2 turns, so it allows the previously-mentioned rotation trick. But that's not a 'cheat' with this puzzle because it was intended that the solver could do any desired sliding move because the puzzle is solved by feel, not by sight.

Although there's a little kinship between the puzzles (both are 7-piece sliding-hole puzzles) I'm really impressed by the movement restrictions imposed by Alex Black Hole and the elegant method of implementing it.


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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sun Sep 19, 2010 2:55 am 
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That looks very cool, VeryWetPaint. It looks very fun!

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sun Sep 19, 2010 3:02 am 
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Indeed, the Alex Black hole, with all those forced paths, is an extremely elegant puzzle.
I also love the way the Giant Pearl works. And what an accommodating name!!!

:D


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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sun Sep 19, 2010 10:18 am 
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kastellorizo wrote:
In the beginning of the above paragraph, we are saying the same thing. I agree that an uncolored core will assist to a parity.

I'm not saying it does. I'm saying it may. I'm taking that to mean leaving the core uncolored will make the puzzle harder due to a parity issue. If you are saying something else I may be lost already. To me that point is still unproven and at this stage I'm inclined to doubt it. Here is why... on the Void Cube there are 6 face centers that are left uncolored and there is a parity issue that ties the edges to the orientation of the faces. Here we only have 1 piece that is uncolored... the core and I suspect the puzzle can be solved with the core in any of the 24 allowed orientations. If that is true I don't see how the core could be hiding any parity information.

kastellorizo wrote:
Now, coloring the core to provide an 8th piece would only work if the goal was to give to us a "2x2x2 illusion", that is if you moved three pieces in one cycle (four moves in total to complete what is equivalent to a 90 degree twist of the 2x2x2). Here you can move one piece at a time (i.e. the algebraic group of this puzzle contains the one obtained from the 2x2x2 puzzle).

I think I'm lost. Sure this puzzle looks like a 2x2x2 but I'm not suggesting that coloring the core makes it any more 2x2x2-like. I'm simply saying it is an 8th piece of this puzzle. It's not the same type of piece as the other 7 so I suspect it could add to the solving challenge. On a 2x2x2 all 8 pieces are the same. To me this is akin to adding face-centers to a 2x2x2 so you can see the core inside. I know I've seen this done already but I can't think of what the puzzle was named.

kastellorizo wrote:
This is because on a 2x2x2 we are allowed to "get away with it" by fixing one piece on the core. This is not the case here, as there is much more freedom of movement *and* the core position does matter whenever its orientation is changed.

I'm not sure what the 2x2x2 is getting away with? On this puzzle the important information about the orientation of the core is already visible as you can see the arcs that allow the movement. If the core can be in different orientations while the other 7 pieces appear solved than I think coloring it would add to the difficulty of the puzzle.

kastellorizo wrote:
In other words, it is identical to have an 8th piece if we tried to solve a 2x2x2 within this puzzle, yet if we tried to do the same "varikon one-by-one piece" way (of course, always including piece orientation), then the core acts in a more revealing way. The ""varikon one-by one piece" way, breaks the 2x2x2 twist into two components: (a) change of position between the normal pieces, (b) change of position of *all* pieces(not just the seven of the its corresponding 2x2x2) with respect to the core.

Ok... now I'm sure I'm lost. What is varikon? Ok google answers that question... But I'm still not seeing the relevance.

kastellorizo wrote:
I hope I am making sense!

I think you are to some...

VeryWetPaint wrote:
You make a good point on the relationship to the 2x2x2.

But aside from a visual appearance commonality they appear to be two totally different puzzles to me.

Carl

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sun Sep 19, 2010 11:23 am 
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wwwmwww wrote:
But aside from a visual appearance commonality they appear to be two totally different puzzles to me.


Ok this is a critical point and it must be cleared up before reviewing the other points. I will try again.

Let's go back to the Octedron (which has a group equivalent to a non-bandaged Alex Black Hole).
The Octedron's group contains the 2x2x2 group. In other words, if we assume that the empty
space is the Octedron's 8th piece, then we can simulate one 90 degree twist of the 2x2x2 cube
by moving (around in cycle) four pieces of the Octedron.

The heart of the issue here is "Why can't we color the core to be the eighth piece of the Octedron?"
We cannot, because then we will be omitting many other moves by fixing this invisible piece to the core.

In the 2x2x2 we can glue one external piece to the core as we are dealing with the smallest subgroup
of the Octedron where we are allowed to do this (which is what I meant by saying "we can get away with it").

Things become more complex when use the Octedron's full group, and fixing one piece
to the core is not even part of the puzzle any more. It then gets even more gruesome (in a
bandaged way) when we have to deal with the Alex's Black Hole groupoid.


Pantazis

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sun Sep 19, 2010 12:36 pm 
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kastellorizo wrote:
Let's go back to the Octedron (which has a group equivalent to a non-bandaged Alex Black Hole).

By "has a group" I take that to mean they are the same group. We are considering the puzzle this thread is about a "bandaged" Alex Black Hole? Which would then be a sub-group on the Octedron. Correct?

kastellorizo wrote:
The Octedron's group contains the 2x2x2 group. In other words, if we assume that the empty
space is the Octedron's 8th piece, then we can simulate one 90 degree twist of the 2x2x2 cube
by moving (around in cycle) four pieces of the Octedron.

Ok... I follow that. But I'm not saying the core of Alex Black Hole is equivalent to the empty space of the Octedron. Maybe it is but I haven't seen anything that proves that. And if we consider the Alex Black Hole bandaged then it can't simulate one 90 degree twist of a 2x2x2. Well maybe it can... but it would be more involved then just moving 3 pieces and a void around in a cycle.

kastellorizo wrote:
The heart of the issue here is "Why can't we color the core to be the eighth piece of the Octedron?"
We cannot, because then we will be omitting many other moves by fixing this invisible piece to the core.

Why can't the core be a 9th piece of this puzzle in addition to the invisible 8th piece? Why do we assume they must be fixed to each other? Hold Alex Black Hole in such a way that the core can't move. Now if you scramble the puzzle you can see the invisible 8th piece moves around the core too. The void isn't always in the same spot.

kastellorizo wrote:
In the 2x2x2 we can glue one external piece to the core as we are dealing with the smallest subgroup
of the Octedron where we are allowed to do this (which is what I meant by saying "we can get away with it").

Ok... I agree the 2x2x2 is a subgroup on the Octadron assuming you have a way to track the orientation of the invisible 8th piece. But the 2x2x2 itself has smaller subgroups (the 1x2x2 is one) so I don't think I'd call it the smallest.

Actually since a corner of a 2x2x2 can't be rotated by itself I guess you don't need to track the orientation of the invisible 8th piece. This would be the same as leaving one corner of a 2x2x2 unstickered.

kastellorizo wrote:
Things become more complex when use the Octedron's full group, and fixing one piece
to the core is not even part of the puzzle any more. It then gets even more gruesome (in a
bandaged way) when we have to deal with the Alex's Black Hole groupoid.

I'm not convinced coloring the core is equivalent to fixing one piece to the core. Notice the colored corner of the core can occupy the same position as one of the 7 corners on its surface. And I'm still not sure if coloring the core makes this puzzle harder or easier.

But I do see the relation to the 2x2x2 now. Note in a 2x2x2 with exposed face centers you no longer need to fix one of the corners to the core. The core does become a 9th piece. I wish I could find a link to that puzzle. I know Andreas has posted about it on one of his threads.

Carl

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sun Sep 19, 2010 12:50 pm 
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Carl I agree with all your points, i.e. pointing two of them:

1. Yes we do not have to fix a piece to the core of the 2x2x2, but the 2x2x2
corresponds to a puzzle with the smaller group where we can fix the center
such that we have eight external pieces. This is not true with the 2x2x1.

2. Yes, you are correct and I agree 100%, the core may certainly be regarded
as a ninth piece, in all of the puzzles described above! :)



Also, Alex's Black Hole (just like a bandaged 3x3x3) does not have a group.
It is based on a groupoid (they are related, but none is contained in the other).
A groupoid acts in a not so symmetric way, and it does no satisfy the properties
of a group. We can safely assume though, that a fiixed number of groupoid moves
can form a subgroup of our main group.

The point I am making, is that by knowing the coloring of the core, as well as its
initial position, works as a guide to place the pieces around the core and avoid some
parities. If we did not know the initial position (best described by orientation) with respect
to the rest of the pieces, then the core's new position may lead to "unsolvable" states,
just like with the Void Cube that you correctly mentioned.

:)


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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sun Sep 19, 2010 1:18 pm 
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kastellorizo wrote:
1. Yes we do not have to fix a piece to the core of the 2x2x2, but the 2x2x2
corresponds to a puzzle with the smaller group where we can fix the center
such that we have eight external pieces. This is not true with the 2x2x1.

What isn't true of the 2x2x1? I agree it doesn't have 8 external pieces. But one of its 4 external pieces still is usually attached to the core. Best example is that 4 piece clear 2x2x1 that was shown here not that long ago and again I can't find at the moment.

kastellorizo wrote:
The point I am making, is that by knowing the coloring of the core, as well as its
initial position, works as a guide to place the pieces around the core and avoid some
parities. If we did not know the initial position (best described by orientation) with respect
to the rest of the pieces, then the core's new position may lead to "unsolvable" states,
just like with the Void Cube that you correctly mentioned.

Ok... I think we are on the same page. To me the operative word here is "may". And I agree... it may. I want to know if it does or doesn't and I don't think anyone is sure of the answer to that question at the moment.

Carl

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 Post subject: Re: Alex Black Hole by ALEX & OSKAR
PostPosted: Sun Sep 19, 2010 7:21 pm 
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wwwmwww wrote:
kastellorizo wrote:
1. Yes we do not have to fix a piece to the core of the 2x2x2, but the 2x2x2
corresponds to a puzzle with the smaller group where we can fix the center
such that we have eight external pieces. This is not true with the 2x2x1.

What isn't true of the 2x2x1? I agree it doesn't have 8 external pieces. But one of its 4 external pieces still is usually attached to the core. Best example is that 4 piece clear 2x2x1 that was shown here not that long ago and again I can't find at the moment.


We are talking about puzzles which have eight external pieces. So we agree. :)


wwwmwww wrote:
kastellorizo wrote:
The point I am making, is that by knowing the coloring of the core, as well as its
initial position, works as a guide to place the pieces around the core and avoid some
parities. If we did not know the initial position (best described by orientation) with respect
to the rest of the pieces, then the core's new position may lead to "unsolvable" states,
just like with the Void Cube that you correctly mentioned.

Ok... I think we are on the same page. To me the operative word here is "may". And I agree... it may. I want to know if it does or doesn't and I don't think anyone is sure of the answer to that question at the moment.

Carl


Now here is where the "juicy part" is. A 2x2x2 may be considered as a 3x3x3 without edges and centers.
And if the center plays the partial role of an edge or a center, then it is possible to see that the
restrictions imposed by a bandaging, can lead to some interesting answers.

So the answer to your question, is that the type bandaging dictates the possibility of
having or not having such cases, but both cases are possible. Proof:

- In the Octedron case, such case exists, by simulating a Void Cube with no edges.
So a good example of a bandaged Octedron for this to occur, is when the top four pieces
cannot interact with the bottom pieces, resulting to only two cycles (Boob Cube simulation).
The core (most of the times) is horizontally rotated with respect to the top or bottom pieces
(or both).

- A good example of a bandaged case where this does not occur, is when one piece is
fixed to the core.


For the Alex's Black Hole, from what I can see in the core grooves, the orientation
of the core can already change by moving the top pieces in two cycles (equivalent
to 2x2x2 90 degree twists) and then doing the same for the bottom pieces.
(i.e. two cycles of the Alex Black Hole preserve the external coloring).

Finally, what is *not* answered here (yet), is the existence of additional such states
for the rest of the moves which do not correspond to the 2x2x2, but to the groupoid
of Alex's Black Hole or to the group in the case of Octedron.

An interesting observation here, is that in the Octedron case, the external sphere
takes the role of the core(!), which is why the position of the pieces in gravity rotational
puzzles *does* matter when calculating its total combinations (and yes, dexterity *can*
be used, but that does not cancel out this fact).

:)


Pantazis

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