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 Post subject: Slice 7x7x7 by OSKAR
PostPosted: Sat Sep 29, 2012 7:11 am 
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Hi Twisty Puzzles fans,

Slice 7x7x7 is a variation to the Rubik's Cube, in which layer 1 is rigidly connected to layer 7, 2 to 6, and 3 to 5. The opposite-layer connection is there in all three directions. The result is a puzzle that only makes nice symmetrical patterns, no matter how you scramble it.

The creation of this puzzle was triggered by email communication with Nersés Aramian. He suggested to build a puzzle based on the projective plane, i.e. pairs of pieces opposite from the origin always remain opposite. Slice 7x7x7 satisfies Nersés' requirement, although this puzzle may not be what he had in mind.

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

Enjoy!

Oskar
P.S. The contest to win yourself a Mixup Cube is still open.
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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Sat Sep 29, 2012 8:48 am 
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Babyface 3x3x3 is also a cube making symmetrical patterns only - actually subset of Slice 7x7x7.
Great Job, Oskar! :D

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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Sat Sep 29, 2012 9:15 am 
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NICE!!! I like this one. Looking at the pictures and the animated model on Shapeways I'm guessing there aren't any screws in this puzzle. Is that correct? My I ask do you glue the two halves of the core (which is also the 8 corners) together? Or do the other pieces hold to core together for you? It looks like assembly might be a little tricky but that's really hard to tell just by looking at pictures. Any chance you could make an disassembly/assembly video assuming its not too hard to take apart?

Carl

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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Sat Sep 29, 2012 11:58 am 
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I would much rather like to see an anti-slice version but that may in fact be impossible. ;)


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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Sat Sep 29, 2012 3:48 pm 
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Would love to see a 2x2x2 version of this. :lol: :shock: :( :roll:

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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Sun Sep 30, 2012 12:16 pm 
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wwwmwww wrote:
there aren't any screws in this puzzle. Is that correct?
That is correct. No screws.
wwwmwww wrote:
May I ask do you glue the two halves of the core (which is also the 8 corners) together?
Two-component epoxy glue.
wwwmwww wrote:
Or do the other pieces hold to core together for you?
No, they do not. If you pull the core apart slightly, then the large chunks start falling out.
wwwmwww wrote:
It looks like assembly might be a little tricky
It was not too hard. First assemble all central parts, such that you end up with 12 composite edges and 6 composite centers. Then apply glue to the core halves (connected to the corners) and put everything together.
wwwmwww wrote:
Any chance you could make an disassembly/assembly video ...?
I'd rather not. Most likely, I would damage the glued-together core.

Oskar

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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Mon Oct 01, 2012 10:40 am 
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Location: Koblenz, Germany
Beautiful.
I assume you designed this rounded shape to have flat surfaces and identically sized stickers. Didn't you?

This puzzle logically consists out of only 35 pieces.
@wwwmwww: Have you recognized that this puzzle consists solely out of NHPs (aka imaginary) pieces?


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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Mon Oct 01, 2012 11:09 am 
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Andreas Nortmann wrote:
This puzzle logically consists out of only 35 pieces.
@wwwmwww: Have you recognized that this puzzle consists solely out of NHPs (aka imaginary) pieces?


I may be counting wrong but I get more than 35 pieces.

(1) Core (the 8 corners are the core)
(15) Edge 4-groups (these pieces only have orientation)
(3) Center-bars
(12) Outer X-centers (opposite faces bandaged together)
(12) Inner X-centers (opposite faces bandaged together)
(12) Outer +-centers
(12) Inner +-centers
(24) Center-wings (left piece connected diagonally through the puzzle to a right piece)

This comes to 91 pieces.

Perhaps my analysis isn't correct?

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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Mon Oct 01, 2012 11:28 am 
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Location: Koblenz, Germany
I count like this:
1 combined pieces of 8 Corners
3 combined pieces of 8 outer Wings
3 combined pieces of 8 inner Wings
3 combined pieces of 4 Edges
3 combined pieces of 8 outer X-pieces
3 combined pieces of 8 inner X-pieces
6 combined pieces of 4 outer T-pieces
6 combined pieces of 4 inner T-pieces
6 combined pieces of 8 obLiques
1 combined pieces of 6 (central) Faces

Andreas


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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Mon Oct 01, 2012 4:58 pm 
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Location: Berkeley, CA, USA
This idea is very interesting. I simulated this puzzle on my 7x7x7. Solving is quite easy. The [1,1] commutator of every two slice moves forms a 3-cycle, unless the two slices are parallel. I would say solving-wise it's easier than a 3x3x3.

In terms of the number of pieces, my count is similar to Andreas, except I think there's only one outer X-piece, and one inner X-piece.

I'm not sure about the number of states yet, because I don't know how to deal with the "identical pieces". It'll be easier if all the stickers are super stickers.


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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Mon Oct 01, 2012 5:24 pm 
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Interesting. Presumably each opposite pair of pieces through the center always stay opposite each other. This fact (constraint) together with the 2x2 fold symmetry on each face may help to work out the number of states... :wink:

I'm guessing it will be comparable with a 4x4x4, as there are four independent "layers" around each axis.

But regardless of the number, it does look easy compared with the 3x3x3.

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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Mon Oct 01, 2012 5:39 pm 
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Andreas Nortmann wrote:
I count like this:
1 combined pieces of 8 Corners
3 combined pieces of 8 outer Wings
3 combined pieces of 8 inner Wings
3 combined pieces of 4 Edges
3 combined pieces of 8 outer X-pieces
3 combined pieces of 8 inner X-pieces
6 combined pieces of 4 outer T-pieces
6 combined pieces of 4 inner T-pieces
6 combined pieces of 8 obLiques
1 combined pieces of 6 (central) Faces

Andreas

I feel dumb. I was assuming that each slice was independent but I just watched the video and I see that the outer and inner slices are bandaged together. We were analyzing fundamentally different puzzles.

Andreas Nortmann wrote:
1 combined pieces of 6 (central) Faces

I'm really on the fence about this. I'd rather say this is 3 bar pieces that happen to show the orientation of the central core. The three pieces can not be permuted relative to each other's positions. The reason I want to call these three discrete pieces is that if you do a single middle-slide move, you move two of the bars without twisting the third. If they were one piece the third bar would twist in place.

I make this distinction because of the UD pieces on the Complex 3x3x3. There are 3 of them and they can't be permuted relative to each other (they show the orientation of the core) but they have orientation that is independent of each other.


But, following your analysis of calling them one piece (with 24 orientations), then:
Andreas Nortmann wrote:
3 combined pieces of 8 outer X-pieces
3 combined pieces of 8 inner X-pieces

Are also just one piece each. They behave exactly like the center piece(s).

The:
Andreas Nortmann wrote:
6 combined pieces of 4 outer T-pieces
6 combined pieces of 4 inner T-pieces

Also each seem to have a permutation restriction relative to each other so going by your analysis I think each of these should actually be 2 pieces.

What an odd puzzle. Looks super-easy to solve but understanding the pieces seems harder.

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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Mon Oct 01, 2012 5:43 pm 
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bmenrigh wrote:
What an odd puzzle. Looks super-easy to solve but understanding the pieces seems harder.
My thoughts exactly. :lol:

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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Mon Oct 01, 2012 6:00 pm 
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I'd really like somebody to code this puzzle up in GAP.

I'm inclined to say that this puzzle has:

(1) Core with no orientations
(1) Core with 24 orientations
(2) Core-like pieces each also with 24 orientations
(3) Edges-groups with 4 orientations.
(3) Inner-edge-wing-groups with 4 orientations
(3) Outer-edge-wing-groups with 4 orientations
(2)x3 Oblique pieces that also behave like a core with 24 orientations

So ignoring parity for a moment, the number of distinct positions would be:
? 24^9 * 4^9
% = 692533995824480256


There are definitely some restrictions on orientation. I can't work through all of them in my head but if I had to guess I'd say there are 3 factors of 2 in parity restrictions leaving:
? (24^9 * 4^9) / (2^3)
% = 86566749478060032


It doesn't get much more hand-wavy than that but hopefully I'm in the ballpark. If I'm only off by a factor of 2^N for small N I'll be pleased.

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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Tue Oct 02, 2012 9:55 am 
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bmenrigh wrote:
Andreas Nortmann wrote:
1 combined pieces of 6 (central) Faces

I'm really on the fence about this. I'd rather say this is 3 bar pieces that happen to show the orientation of the central core. The three pieces can not be permuted relative to each other's positions. The reason I want to call these three discrete pieces is that if you do a single middle-slide move, you move two of the bars without twisting the third. If they were one piece the third bar would twist in place.
I make this distinction because of the UD pieces on the Complex 3x3x3. There are 3 of them and they can't be permuted relative to each other (they show the orientation of the core) but they have orientation that is independent of each other.

But, following your analysis of calling them one piece (with 24 orientations), then:
Andreas Nortmann wrote:
3 combined pieces of 8 outer X-pieces
3 combined pieces of 8 inner X-pieces

Are also just one piece each. They behave exactly like the center piece(s).

The:
Andreas Nortmann wrote:
6 combined pieces of 4 outer T-pieces
6 combined pieces of 4 inner T-pieces

Also each seem to have a permutation restriction relative to each other so going by your analysis I think each of these should actually be 2 pieces.

You are right.
My treatment of the X-pieces and the faces was inconsistent.
Therefore I say: There are 3 combined pieces of 2 (central) Faces
That makes up for 37 pieces.

Parity restrictions for the T-pieces do not make them identical pieces.
Can you name these restrictions?

BTW: Implementing the 7x7x7 (Slice or not makes no difference) in GAP means writing down 900 numbers. For now there are other things to debug.


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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Tue Oct 02, 2012 12:00 pm 
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As always, very nice work Oskar. I particularly like this solution to keeping the faces flat, while allowing for all the stickers to be the same size. Your eye is fooled (at least for a moment) into thinking it's a truly proportional 7x7x7.

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 Post subject: Re: Slice 7x7x7 by OSKAR
PostPosted: Wed Oct 10, 2012 6:42 pm 
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Handicube 7x7x7? Probably not possible, but that would be boss.

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