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 Post subject: Puzzles which appear to be the Rubik's CubePosted: Sun Feb 26, 2012 7:47 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Since a few of them have been built lately, I was wondering: Just how many doctrinaire subsets of the motion of the Rubik's Cube are there which could be built?

The answer, it turns out, is far more than anybody's ever going to.

Let's start out with just the symmetric ways a single axis could be set up, they are:

regular
half turn only
center slice
half turn only center slice (meaning only the center slice can turn and it can only do half turns)
half turn only plus center slice (meaning the faces can do half turns and the center slice can do quarter turns)
geared
blocked

(I *think* I got them all, but might have missed one. Let me know if you can think of another one!)

That's seven possibilities for each axis. To my knowledge the half turn plus center slice and half turn center slice possibilities have never been done for even a single axis on a built puzzle ever, although half turn plus center slice on all three axis would be a straightforward build-up of a half turn only 2x2x2. Just combining the above results in over 50 possible puzzles, and I glossed over the subtlety that there's a difference between a center slice moving and the outer parts moving, which results in completely different behavior if the other two axis behave differently, because it changes which does what after the move. Even so, combining the above gives a total of 84 possibilities, and even excluding the ones which are nonfunctional or uninteresting there's probably still over 50.

Getting into asymmetric axes results in far more possibilities, and makes the issue of whether the core rotates with particular moves far more pervasive and involved. The vast majority of the possibilities there are complicated, counterintuitive, and inelegant, but there are some strange and interesting beasts, perhaps most notably that one could have an asymmetrically geared axis where when one face does a quarter turn the opposite face does a half turn. A puzzle which was asymmetrically geared on all three axes would be quite something!

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 Post subject: Re: Puzzles which appear to be the Rubik's CubePosted: Sun Feb 26, 2012 9:08 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Bram wrote:
Let's start out with just the symmetric ways a single axis could be set up, they are:

regular
half turn only
center slice
half turn only center slice (meaning only the center slice can turn and it can only do half turns)
half turn only plus center slice (meaning the faces can do half turns and the center slice can do quarter turns)
geared
blocked

(I *think* I got them all, but might have missed one. Let me know if you can think of another one!)
By "blocked", do you mean nothing rotates on that axis? Such that if all 3 axes were "blocked" you'd just have a big 1x1x1?

And would you consider the Mixup Cube to be a doctrinaire puzzle which appears to be a Rubik's Cube? If so you could add 45 degree slice turns to your list.

And the Fused Cube is certainly what I would consider a doctrinaire subset of the motion of the Rubik's Cube. Hmmm... maybe one axis of this puzzle fits your meaning of "blocked". Though if that is the case there are at least 2 ways an axis could be "blocked".

Carl

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 Post subject: Re: Puzzles which appear to be the Rubik's CubePosted: Sun Feb 26, 2012 10:31 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
wwwmwww wrote:
By "blocked", do you mean nothing rotates on that axis? Such that if all 3 axes were "blocked" you'd just have a big 1x1x1?

Yes.

wwwmwww wrote:
And would you consider the Mixup Cube to be a doctrinaire puzzle which appears to be a Rubik's Cube? If so you could add 45 degree slice turns to your list.

There are already too many variants which have never been built without getting into the 45 degree ones.

wwwmwww wrote:
And the Fused Cube is certainly what I would consider a doctrinaire subset of the motion of the Rubik's Cube. Hmmm... maybe one axis of this puzzle fits your meaning of "blocked". Though if that is the case there are at least 2 ways an axis could be "blocked".

The fused cube has each axis fused on one side and regular on the other. It's one of the asymmetric (or at least less symmetric) variants I mentioned.

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 Post subject: Re: Puzzles which appear to be the Rubik's CubePosted: Mon Feb 27, 2012 1:13 pm

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Lucky for me I reread this half-sentence:
Bram wrote:
and I glossed over the subtlety that there's a difference between a center slice moving and the outer parts moving,
I have always "glossed" (I had to look up the meaning for this case) over this differences as well. I really want to see a puzzle where this difference does really matter.
Anyway: I always thought something on every puzzle has to stay fixed in space. Usually this is the core but sometimes (e.g. on a Siamese Cube) it can be useful to fix a different piece in space. If I were allowed to move the center slice AND the parallel outer layers I would always replace a slice move with a combined move of the two outer layers. With measures like this I can always force the core (or any other piece) to stay fixed in space. Therefore something like "half turn only plus center slice" makes no sense to me.
Bram wrote:
Even so, combining the above gives a total of 84 possibilities, and even excluding the ones which are nonfunctional or uninteresting there's probably still over 50.
This is the result of Burnside's Lemma for several symmetric possibilities. If you reduce the allowed set to six possibilities there are 56 possible subsets.

The asymmetric cases I have calculated some subsets in the past.
viewtopic.php?f=14&t=15016
I presented all 35 were an axis can either have
* two layers moveable
* one layer and one slice moveable
* less then above
And I enumerated all cases when some moves are additionally restricted to half-turns.
In the latter case the number increases from 35 to 366.

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 Post subject: Re: Puzzles which appear to be the Rubik's CubePosted: Mon Feb 27, 2012 6:15 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Andreas Nortmann wrote:
Lucky for me I reread this half-sentence:
Bram wrote:
and I glossed over the subtlety that there's a difference between a center slice moving and the outer parts moving,
I have always "glossed" (I had to look up the meaning for this case) over this differences as well. I really want to see a puzzle where this difference does really matter.

Do you mean you don't believe that such a distinction can happen, or that you'd like a concrete example? There are many, many possible examples, most of them horribly counterintuitive and inelegant. Probably the simplest one is slice on one axis, blocked on another, and half turn only on the third. You make the center slice pieces be floating on the outside, and either enforce the blocking of the other faces on the core or on the floating centers, which results in very different behaviors for the two variants. In one of the two variants a whole bunch of pieces might as well be glued together, the other is much less trivial.

Andreas Nortmann wrote:
Anyway: I always thought something on every puzzle has to stay fixed in space. Usually this is the core but sometimes (e.g. on a Siamese Cube) it can be useful to fix a different piece in space. If I were allowed to move the center slice AND the parallel outer layers I would always replace a slice move with a combined move of the two outer layers. With measures like this I can always force the core (or any other piece) to stay fixed in space. Therefore something like "half turn only plus center slice" makes no sense to me.

Heh. In the case of half turn plus center slice, then if you 'cancel out' the center slice moves, you can turn the right 180 degrees, or the left 180 degrees, or both the right and the left 90 degrees, but if you try rotating just the right 90 degrees then the whole thing will be blocked, just like it is in the half turn only cube. Like I said before, you could build a puzzle which worked that way on all axis by starting with a half turn only 2x2x2 as the core and then adding the 'centers' and 'edges' to the outside in sort of the style of the mozaika.

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 Post subject: Re: Puzzles which appear to be the Rubik's CubePosted: Tue Feb 28, 2012 1:34 pm

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Bram wrote:
Andreas Nortmann wrote:
I have always "glossed" (I had to look up the meaning for this case) over this differences as well. I really want to see a puzzle where this difference does really matter.
Do you mean you don't believe that such a distinction can happen, or that you'd like a concrete example? There are many, many possible examples, most of them horribly counterintuitive and inelegant. Probably the simplest one is slice on one axis, blocked on another, and half turn only on the third. You make the center slice pieces be floating on the outside, and either enforce the blocking of the other faces on the core or on the floating centers, which results in very different behaviors for the two variants. In one of the two variants a whole bunch of pieces might as well be glued together, the other is much less trivial.
I doubted that such a distinction can happen because I never came across a concrete example.
Lets deal with your example. If I understood it right you want to describe the following two subgroups:
1. Version: <Fs, U2, D2> Fs means a slice parallel to F. This variant can be simulated with some bandaged parts and one bridge. See the following image:
Attachment:

BKFK.png [ 4.35 KiB | Viewed 1258 times ]
The rhombi indicate that the piece is connected to its opposite piece with a bridge.
2. Version: <FB' U2, D2> FB' means, that F and B can only be turned together. This variant can can be simulated with bandaging and bridges as well. See the following image:
Attachment:

BF111.png [ 4.77 KiB | Viewed 1258 times ]
The X indicates that the piece is connected to the core.

Both variants can be viewed with at least one piece of the cube as fixed in space.
Bram wrote:
Heh. In the case of half turn plus center slice, then if you 'cancel out' the center slice moves, you can turn the right 180 degrees, or the left 180 degrees, or both the right and the left 90 degrees, but if you try rotating just the right 90 degrees then the whole thing will be blocked, just like it is in the half turn only cube. Like I said before, you could build a puzzle which worked that way on all axis by starting with a half turn only 2x2x2 as the core and then adding the 'centers' and 'edges' to the outside in sort of the style of the mozaika.
This convinced me.
Even with the core fixed in space there are still the 7 possible variants you already mentioned.
To get back to your initial question: There are only the 84 variants you mention when you allow only symmetric allowances.
Bram wrote:
To my knowledge the half turn plus center slice and half turn center slice possibilities have never been done for even a single axis on a built puzzle ever
Do you count this one? The double Barbell Cube impements <Us, D2, F, B>. One of the blocks stays fixed. The slice can move freely. For the opposite block only halfturns make sense.
Attachment:

DoubleBarbell.png [ 4.39 KiB | Viewed 1258 times ]

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 Post subject: Re: Puzzles which appear to be the Rubik's CubePosted: Tue Feb 28, 2012 4:36 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Andreas Nortmann wrote:
If I understood it right you want to describe the following two subgroups:
1. Version: <Fs, U2, D2> Fs means a slice parallel to F. This variant can be simulated with some bandaged parts and one bridge.

2. Version: <FB' U2, D2> FB' means, that F and B can only be turned together. This variant can can be simulated with bandaging and bridges as well.

That is correct. Since there's only one bridge, it could go right through the core, although the point of these two is more illustrative than that they specifically are interesting.

Your pictures are in a different orientation than your notation, which is a bit confusing.

Other examples of whether the core moves are far weirder. For example, <UD2, FB2, RL2> and <UD2, FB2, Rs2R> are subtly and bizarrely different.

Andreas Nortmann wrote:
The double Barbell Cube impements <Us, D2, F, B>. One of the blocks stays fixed. The slice can move freely. For the opposite block only halfturns make sense.

Yep, that works!

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 Post subject: Re: Puzzles which appear to be the Rubik's CubePosted: Wed Feb 29, 2012 11:13 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Bram wrote:
<UD2, FB2, RL2> and <UD2, FB2, Rs2R>
What does Rs2R in this case? I assume you meant a wide move: RS and R must always move together => The R-face behaves connected to the core. In that case one can redefine it as <UD2, FB2, L>

My point here is that it seems to me you can always redefine the possible moves in a way that one piece of the 3x3x3 stays fixed in space. That again allows us to enumerat all possible definitions. GAP can then calculate the sizes of these groups and we (or maybe GAP) can delete the isomorphic ones.
In this case face-orientations (visible or not?) make a difference.

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 Post subject: Re: Puzzles which appear to be the Rubik's CubePosted: Wed Feb 29, 2012 8:08 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Andreas Nortmann wrote:
Bram wrote:
<UD2, FB2, RL2> and <UD2, FB2, Rs2R>
What does Rs2R in this case? I assume you meant a wide move: RS and R must always move together => The R-face behaves connected to the core. In that case one can redefine it as <UD2, FB2, L>

It's a wall move plus a center slice, a wall move is RsR, this is Rs2R. really I should have said Rs2R'. While it's true that you can make that look superficially similar to L2R by doing the move and rotating the whole cube, that results in the other two moves swapping from UD2 and FB2 to U2D and F2B. It's a similar distinction to the example above, where <Fs R L> and <FB' R L> are two different things despite having the same available first moves.

There are a *lot* of examples like this. Once you get into random asymmetric subsets this distinction matters more often than not.

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