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 Post subject: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Jul 06, 2013 3:21 pm 
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Hi Twisty Puzzles fans,

Alternating Cube was suggested by Bram Cohen at the Twisty Puzzles Forum as follows: "Hey Oskar, can you design a 2x2x2 which must alternate clockwise and counterclockwise quarter turns?". Bram suggested to use the core of Alex Black Hole as a basis. To this, Oskar added the concept of the Enabler Cube. One corner, the "enabler", moves through the core in a CW-CCW fashion. The mechanism inside the core pins one of the corners to the core opposite of the "enabler".

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

Enjoy!

Oskar

P.S. Here is a question for the solving specialists: how many moves does it take to make a 180-degrees turn?
Attachment:
Alternating Cube - view 1.jpg
Alternating Cube - view 1.jpg [ 51.09 KiB | Viewed 7205 times ]

Attachment:
Alternating Cube - view 2.jpg
Alternating Cube - view 2.jpg [ 51.09 KiB | Viewed 7205 times ]

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Alternating Cube - view 3.jpg
Alternating Cube - view 3.jpg [ 50.09 KiB | Viewed 7205 times ]

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Alternating Cube - view 4.jpg
Alternating Cube - view 4.jpg [ 53.17 KiB | Viewed 7205 times ]

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Alternating Cube - view 5.jpg
Alternating Cube - view 5.jpg [ 47.96 KiB | Viewed 7205 times ]

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Alternating Cube - view 6.jpg
Alternating Cube - view 6.jpg [ 37.44 KiB | Viewed 7205 times ]

Attachment:
Alternating Cube - view 7.jpg
Alternating Cube - view 7.jpg [ 47.27 KiB | Viewed 7205 times ]

Attachment:
Alternating Cube - view 8.jpg
Alternating Cube - view 8.jpg [ 44.13 KiB | Viewed 7205 times ]

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Last edited by Oskar on Sat Nov 02, 2013 7:12 am, edited 1 time in total.

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Jul 06, 2013 5:53 pm 
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I may or may not have an idea as to how the mechanism works from those pictures. As far as I can tell, I don't think that a single 180-degree turn is even possible.

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 07, 2013 2:27 am 
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Jorbs3210 wrote:
I don't think that a single 180-degree turn is even possible.
Colton,

You are correct. Hence my question "how many moves does it take to make a 180-degrees turn?". A related question is "how many moves does it take to solve the puzzle, bringing it from a right-turn-next solved state into a left-turn-next solved state?".

Oskar

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 07, 2013 11:51 am 
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The out-of-the-box creativity behind this idea/mechanism blows my mind! :shock:

I'm wondering if it would be possible to indicate the current right/left-turning state of this puzzle with little "indicator windows" that show a glimpse of the inner mechanism?

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 07, 2013 5:37 pm 
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Oskar wrote:
Jorbs3210 wrote:
I don't think that a single 180-degree turn is even possible.
Colton,

You are correct. Hence my question "how many moves does it take to make a 180-degrees turn?".
I understand the question but I'm taking Jorbs3210's statement to mean that the state where the puzzle appears to have made a 180 degree turn may no be reachable. Oskar, do you have proof that it is a reachable state?

I must confess that while I do see how the Enabler Cube plays a role in this puzzle the connection to the Alex Black Hole puzzle eludes me. This is one of those puzzles where I'm tempted to order it just so I can play with the mechanism until I understand it.

I'm also curious if this would make an interesting 3x3x3 if you used the HandiCube method to add in the slice players.

Carl

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 07, 2013 11:56 pm 
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Amazing!
Oskar wrote:
A related question is "how many moves does it take to solve the puzzle, bringing it from a right-turn-next solved state into a left-turn-next solved state?".
Unfair question. I wanted to ask something similar: "Is it designed so that in the solved state only a CW move is possible?"
Maybe I have to help myself ...


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Wed Jul 10, 2013 1:59 pm 
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KelvinS wrote:
The out-of-the-box creativity behind this idea/mechanism blows my mind!
Thank you. Even more mindblowing may be the number of intellectual contributions from different people: Bram, Alex, Oskar, Rubik, Verdes, Cutler, Coffin and probably some TP Forum contributors as well on design techniques and tolerances.
wwwmwww wrote:
I must confess that while I do see how the Enabler Cube plays a role in this puzzle the connection to the Alex Black Hole puzzle eludes me. This is one of those puzzles where I'm tempted to order it just so I can play with the mechanism until I understand it.
Carl,

As you know, there will be a good buying opportunity very soon.

The photo below shows the mechanism of Alex Black Hole. You see the same pattern at the core of Alternating Cube.

Oskar
Attachment:
Alex-Black-Hole---view-5.jpg
Alex-Black-Hole---view-5.jpg [ 30.96 KiB | Viewed 6713 times ]

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 14, 2013 9:55 pm 
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Turning a face is a 4-cycle, which is an odd permutation, ergo every move alternates the parity of the permutation. Since every move also flips which direction the next turn is in, the two of them are always correlated, so it's impossible to get back to what appears to be the start state but with the next type of move flipped.


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Jul 15, 2013 1:53 pm 
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Bram wrote:
Turning a face is a 4-cycle, which is an odd permutation, ergo every move alternates the parity of the permutation. Since every move also flips which direction the next turn is in, the two of them are always correlated, so it's impossible to get back to what appears to be the start state but with the next type of move flipped.

I guess that means there must be two (mirror image) versions of this puzzle?

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Jul 15, 2013 9:18 pm 
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KelvinS wrote:
I guess that means there must be two (mirror image) versions of this puzzle?


None of the pieces are any different. It has to do with the positioning of the core, which you can't see. Rotating the core 90 degrees flips between the two possibilities.


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Tue Jul 16, 2013 9:42 am 
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Bram wrote:
KelvinS wrote:
I guess that means there must be two (mirror image) versions of this puzzle?


None of the pieces are any different. It has to do with the positioning of the core, which you can't see. Rotating the core 90 degrees flips between the two possibilities.

Even more interesting. That means the puzzle can be assembled from the same parts into two different (mirror image) versions of this puzzle.

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Tue Jul 16, 2013 10:21 pm 
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Bram and Oskar, this is a genius design. It took me awhile to figure out how it works, but I think I understand now. Those interlocking yellow columns aren't nearly as complicated as I thought at first, but I do see why they are necessary. Well done! :mrgreen:

I've played with a simulated version of this puzzle for about an hour and I am quite convinced that the pieces can only be rearranged into 336 permutations (not counting orientation and leaving no piece fixed), 168 where a clockwise turn must be made and 168 where a counter-clockwise turn must be made. In those permutations that a clockwise turn must be made next I have observed that if 3 pieces are permuted correctly relative to eachother, then all 8 pieces are permuted correctly relative to eachother. Bram's observation is also helpful here - he correctly noted that each move flips the parity of the permutation of the pieces of the puzzle so every even permutation must require a clockwise move while every odd permutation must require a counterclockwise move. Note that there is no fixed orientation on a 2x2 so we must either consider one piece to be fixed or we must allow for 24 different solved states (the 24 orientations of the cube colors), fortunately all 24 of these have even permutation, so this does not refute Bram's point. Since a 180 degree move from solved is an even permutation, it requires a clockwise move next (or it means that last move performed was a counterclockwise move), and according to my previous observation, this permutation is not in the group. I therefore conclude that it is impossible to reach a state that is 180 degrees from solved. This is by no means a rigorous proof and I have not yet found a clever insight to help explain why this is - just the power of observation.

I'm not sure if you can represent the entire state space of the puzzle as a group, but you certainly can represent half of it. The generators of the group are every combination of making a clockwise move and then a counterclockwise move. Since rotating a right face clockwise versus a left face clockwise results in nothing more than a global orientation of the puzzle, I think the easiest representation would be considering the 3 axes: x, y, and z. Then there are 6 generators: xy, xz, yx, yz, zx, and zy. They each have order 9 which is certainly different, and the inverses of xy, yz, and zx are xz, zy, and yx respectively If someone who is adept at GAP could simulate this I would be interested in seeing how many elements are in this group (pick a piece and leave it fixed for all 6 moves to prevent global reorienations). This only hits the states of the puzzle where the next move must be clockwise, but doubling the result will give the total number of states of the puzzle.

After playing with it, I suspect that there are ((8*7*3)*(3^7))/24 = 15309 (holy cow odd number! :shock: ) elements in the group which means there are 30618 possible puzzle positions, but it could be a small multiple of that because I am assuming all 24 orientations are solvable, which may not be true.

If I have more time to play with it later I'll post what I find :)

REALLY interesting puzzle Bram & Oskar, congratulations!

Peace,
Matt Galla


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Tue Jul 16, 2013 11:53 pm 
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If that really is the number of permutations this puzzle has, I can't help but wonder if it corresponds to a sporadic simple finite group. That would be very strange indeed!


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Wed Jul 17, 2013 12:15 pm 
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GAP tells me:
15309 states
729 orientations

States is the size of the Group (see the GAP-file for details of the definition).
States in this context are the permutations multiplied with all the orientations possible within that permutation.
The orientations is the size of the group which is left after restricting all pieces to stay with in the solved permutation.


Attachments:
Alternate.txt [1.72 KiB]
Downloaded 45 times


Last edited by Andreas Nortmann on Sun Jul 21, 2013 5:33 am, edited 1 time in total.
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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Wed Jul 17, 2013 12:17 pm 
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Come to think of it it's unlikely that this one is sporadic because it generalizes to any number of dimensions. Of course that raises the question of the alternating pentultimate...


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Thu Jul 18, 2013 5:11 pm 
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Bram wrote:
alternating pentultimate


Oh, snap!


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Fri Jul 19, 2013 5:30 pm 
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Andreas Nortmann wrote:
GAP tells me:
15309 states
729 orientations


Sorry for my unfamiliarity, but what do 'states' and 'orientations' mean in this context? Is orientations the number of positions with the pieces in their current positions but only orientations different? Is states the total number of positions the cube can get into, not counting reorientations? That number of states is very bizarre, and implies that this is a much more novel solving experience than I expected.


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Fri Jul 19, 2013 6:16 pm 
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Allagem wrote:
I've played with a simulated version of this puzzle for about an hour and I am quite convinced that the pieces can only be rearranged into 336 permutations
Allagem wrote:
After playing with it, I suspect that there are ((8*7*3)*(3^7))/24 = 15309 (holy cow odd number! :shock: ) elements in the group which means there are 30618 possible puzzle positions, but it could be a small multiple of that because I am assuming all 24 orientations are solvable, which may not be true.
Is the second statement an update to the first? If you have defined "permutations" and "possible puzzle positions" to mean different things I'm lost as I don't see how both statements can be true.

If the second statement is an update, it appears to be correct based on Andreas' post, can you walk me though your calculation?

This puzzle just keeps getting more interesting...
Allagem wrote:
It took me awhile to figure out how it works, but I think I understand now.
Oh I wish I could say that. It sounds like you've had a chance to play with the actual puzzle... is that correct? I think I''ll get that chance at IPP but I doubt I'll get this understood before then.

Carl

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 21, 2013 5:38 am 
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Bram wrote:
Sorry for my unfamiliarity, ...
I updated my earlier post. I hope that explains something.


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 21, 2013 12:54 pm 
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What an excellent puzzle... I also tried to use some gap on this, and please
allow me to present it in a different way so that I can end up with an interesting conclusion. ;-)

Structure/numbering of the puzzle:

******------------
******| 1 2 |
******| 3 4 |
---------------------------
| 17 18 | 5 6 | 21 22 |
| 19 20 | 7 8 | 23 24 |
---------------------------
******| 9 10 |
******| 11 12 |
****** -----------
******| 13 14 |
******| 15 16 |
****** -----------

First we may assume that we start with a clockwise move. Then (as expected)
I took the 2x2x2 cube and used each pair of clockwise and anticlockwise move
to define a closed group

Stable elements (corresponding to the back-down-right corner): 1, 15, 17

Possible three clockwise moves and their three anticlockwise inverses:

Uc=(5,6,8,7)(3,21,10,20)(4,23,9,18)
Ua=(7,8,6,5)(20,10,21,3)(18,9,23,4)
Rc=(21,22,24,23)(8,4,16,12)(6,2,14,10)
Ra=(23,24,22,21)(12,16,4,8)(10,14,2,6)
Fc=(9,10,12,11)(7,23,14,19)(8,24,13,20)
Fa=(11,12,10,9)(19,14,23,7)(20,13,24,8)

then by defining each pair of moves, to allow the existence of a finite group, we get:

Uc*Fa = (5,6,8,7)(3,21,10,20)(4,23,9,18) (11,12,10,9)(19,14,23,7)(20,13,24,8)
Uc*Ra = (5,6,8,7)(3,21,10,20)(4,23,9,18) (23,24,22,21)(12,16,4,8)(10,14,2,6)
Fc*Ua = (9,10,12,11)(7,23,14,19)(8,24,13,20) (7,8,6,5)(20,10,21,3)(18,9,23,4)
Fc*Ra = (9,10,12,11)(7,23,14,19)(8,24,13,20) (23,24,22,21)(12,16,4,8)(10,14,2,6)
Rc*Ua = (21,22,24,23)(8,4,16,12)(6,2,14,10) (7,8,6,5)(20,10,21,3)(18,9,23,4)
Rc*Fa = (21,22,24,23)(8,4,16,12)(6,2,14,10) (11,12,10,9)(19,14,23,7)(20,13,24,8)

After simplifying the permutations and eliminate duplicate permutation numbers, we get:

Uc*Fa = (5,6,20,3,21,9,18,4,7)(8,19,14,23,11,12,10,13,24)
Uc*Ra = (5,10,20,3,23,9,18,8,7)(6,12,16,4,24,22,21,14,2)
Fc*Ua = (9,21,3,20,6,5,7,4,18)(10,12,11,23,14,19,8,24,13)
Fc*Ra = (9,14,19,7,24,13,20,12,11)(10,16,4,8,22,21,23,2,6)
Rc*Ua = (21,22,24,4,16,12,6,2,14)(23,3,20,10,5,7,8,18,9)
Rc*Fa = (21,22,8,4,16,10,6,2,23)(24,7,19,14,9,11,12,20,13)

GAP command:

cubeoneway := Group(
(5,6,20,3,21,9,18,4,7)(8,19,14,23,11,12,10,13,24),
(5,10,20,3,23,9,18,8,7)(6,12,16,4,24,22,21,14,2),
(9,21,3,20,6,5,7,4,18)(10,12,11,23,14,19,8,24,13),
(9,14,19,7,24,13,20,12,11)(10,16,4,8,22,21,23,2,6),
(21,22,24,4,16,12,6,2,14)(23,3,20,10,5,7,8,18,9),
(21,22,8,4,16,10,6,2,23)(24,7,19,14,9,11,12,20,13));
Size(cubeoneway);

Answer: 15309




Now, assume that we had the look of a 2x2x2 cube, this time, no clockwise/anticlockwise limitations,
but the generators were a bit "different", as expressed below.


Uc=(5,6,8,7)(3,4,21,23,10,9,20,18)
Ua=(7,8,6,5)(18,20,9,10,23,21,4,3)
Rc=(21,22,24,23)(8,6,4,2,16,14,12,10)
Ra=(23,24,22,21)(10,12,14,16,2,4,6,8)
Fc=(9,10,12,11)(7,8,23,24,14,13,19,20)
Fa=(11,12,10,9)(20,19,13,14,24,23,8,7)

Then, the group will have a cardinality (i.e. positions of the puzzle) in the 3x3x3's level, i.e.

25,545,471,085,854,720,000

I would call the puzzle that represents the above permutations as "tank cube",
because the pieces around of the face's pieces, move slightly slower (in fact,
half speed compared to a regular 2x2x2).

Observation: If we use the clockwise/anticlockwise method used by Oskar's puzzle above
to the tank cube, then we get the same number of positions:

Uc*Fa = (5,6,8,7)(3,4,21,23,10,9,20,18)(11,12,10,9)(20,19,13,14,24,23,8,7),
Uc*Ra = (5,6,8,7)(3,4,21,23,10,9,20,18)(23,24,22,21)(10,12,14,16,2,4,6,8),
Fc*Ua = (9,10,12,11)(7,8,23,24,14,13,19,20)(7,8,6,5)(18,20,9,10,23,21,4,3),
Fc*Ra = (9,10,12,11)(7,8,23,24,14,13,19,20)(23,24,22,21)(10,12,14,16,2,4,6,8),
Rc*Ua = (21,22,24,23)(8,6,4,2,16,14,12,10)(7,8,6,5)(18,20,9,10,23,21,4,3),
Rc*Fa = (21,22,24,23)(8,6,4,2,16,14,12,10)(11,12,10,9)(20,19,13,14,24,23,8,7));


cubetankoneway := Group(
(5,6,7)(8,20,18,3,4,21)(23,9,19,13,14,24)(10,11,12),
(5,8,7)(6,10,9,20,18,3)(4,23,12,14,16,2)(24,22,21),
(9,23,24,14,13,19)(10,12,11)(7,6,5)(8,21,4,3,18,20),
(9,12,11)(10,14,13,19,20,7)(8,6,4,2,16,24)(21,23,22),
(21,22,24)(23,4,2,16,14,12)(8,5,7)(6,3,18,20,9,10),
(21,22,23)(24,8,6,4,2,16)(12,9,11)(10,7,20,19,13,14));
Size(cubetankoneway);

Answer: 25,545,471,085,854,720,000


So this post has two questions:

1. Can the "tank cube" be constructed? (I am sure it can!)
2. Which puzzles give the same number of positions, if we apply to them the clockwise/anticlockwise mechanism?

:)


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 21, 2013 2:02 pm 
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kastellorizo wrote:
2. Which puzzles give the same number of positions, if we apply to them the clockwise/anticlockwise mechanism?


The megaminx is one, I believe.
If you take two adjacent faces, say A and B, and choose a third face C that is independent from them, then you can do A+ C- B+ A- C+ B-. This is equivalent to A+ B+ A- B- = [A,B]. So you can do the commutator of any two adjacent faces. As the puzzle has no odd permutations, that should enough to be able to solve any position.

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 21, 2013 2:09 pm 
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What is the 'tank cube'?

There's a much more elegant encoding in GAP than encoding all possible two-move sequences. You use three moves: Rotating either R, U, or F, followed by mirror imaging the whole cube on the plane going through the RFU, RFD, LBU, and LBD vertices.

Seriously, anybody want to analyze the alternating skewb?


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 21, 2013 5:13 pm 
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Bram wrote:
What is the 'tank cube'?



Based on the numbered cube, the corresponding generators show its three types of movement.
Instead of having three moves (F,U, & R) of order four, we have three similar moves of order eight.

Here is using an image:

Attachment:
Tank_Cube.jpg
Tank_Cube.jpg [ 151.27 KiB | Viewed 5768 times ]


Note that the front side numbers of the tank cube, do not move as if they were fixed with the
eight numbers surrounding them (which is what happens with a regular 2x2x2 cube). And to me,
it looks like the way a tank moves with the caterpillar system, where the outside belt can miss
some of the gears (and hence, move slower).



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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Jul 21, 2013 6:48 pm 
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Andreas Nortmann wrote:
GAP tells me:
15309 states
729 orientations

States is the size of the Group (see the GAP-file for details of the definition).
States in this context are the permutations multiplied with all the orientations possible within that permutation.
The orientations is the size of the group which is left after restricting all pieces to stay with in the solved permutation.
I'm still confused. I see 15309/729 is 21. However a cube has 24 possible orientations. You can pick any of 6 faces for up and any of 4 faces for the front. 6*4=24.

Also I looked at your GAP file and I would say that the LFU cubie never moves so I would think this model only allows for one solved "state".... not 24. Not sure I'm using state the same way you did or not.

Can you define "solved permutation"? If you restrict all the pieces to stay in the solved "state" then I'd expect the size of the group to be 1. So is a "solved permutation" a permutation where all the pieces are in their solved "positions" but maybe not their solved orientations (except for the FLU cubie which I believe is always in its solved position and orientation)?

If my understanding is correct we are saying there are 21 reachable states that the Alternating Cube has where all the pieces are in their solved position not counting their orientation... correct?

Even if I'm correct... I'm still not sure why we are looking at the puzzle this way. Have we ever counted the reachable states of a 3x3x3 in which all the pieces are in their solved position but not orientated.

I feel like I'm missing some basic understanding here and I really like this puzzle so I want to understand.

Thanks,
Carl

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Jul 22, 2013 2:08 am 
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Carl, the factor of 21 is because there are only 42 possible positions of this puzzle if you don't count corner orientations. A factor of 2 is lost because they're only looking at positions where a clockwise move comes next, and there are the same number where a counterclockwise comes next as well, then the 42 is 7*6, meaning in this puzzle if you hold one corner fixed in place and position two other corners the remaining 5 corners will all magically be where they're supposed to go. Yes, it's surprising to me as well.

This raises the obvious solving question: What's the shortest sequence which just rotates two adjacent corners?


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Jul 22, 2013 2:11 am 
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My intuition is telling me that most alternating puzzles have the same number of possible positions as their non-alternating counterparts. What I really want to know about is the alternating clockwork 4x4x4.


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Jul 22, 2013 2:25 am 
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wwwmwww wrote:
Can you define "solved permutation"? If you restrict all the pieces to stay in the solved "state" then I'd expect the size of the group to be 1. So is a "solved permutation" a permutation where all the pieces are in their solved "positions" but maybe not their solved orientations (except for the FLU cubie which I believe is always in its solved position and orientation)?
Correct.
wwwmwww wrote:
If my understanding is correct we are saying there are 21 reachable states that the Alternating Cube has where all the pieces are in their solved position not counting their orientation... correct?
Correct again.


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Jul 22, 2013 3:16 am 
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wwwmwww wrote:
Allagem wrote:
I've played with a simulated version of this puzzle for about an hour and I am quite convinced that the pieces can only be rearranged into 336 permutations

Allagem wrote:
After playing with it, I suspect that there are ((8*7*3)*(3^7))/24 = 15309 (holy cow odd number! :shock: ) elements in the group which means there are 30618 possible puzzle positions, but it could be a small multiple of that because I am assuming all 24 orientations are solvable, which may not be true.

Is the second statement an update to the first? If you have defined "permutations" and "possible puzzle positions" to mean different things I'm lost as I don't see how both statements can be true.

If the second statement is an update, it appears to be correct based on Andreas' post, can you walk me though your calculation?

It is not an update; I meant both statements (not sure if you have figured it out since as a few people have posted relevant information, but just in case). The 336 permutations are only permutations of the pieces, so I am not looking at any twisting of the corners. The reason my number here is slightly different is because I was not fixing a single corner anywhere - which, in hindsight was not the clearest thing to do, but the math is still consistent in the end result.

And sure I can walk you through my observations. I do not have a physical copy of it, I was merely simulating it on a normal 2x2x2 but being very careful about what moves I made. I found it easier to consider a single move to be a pair of moves, one CW followed by one counterclockwise, so I only looked at configurations that could be reached after making CW and CCW move pairs.
I should also say I was not fixing a piece so I was considering all moves: FBRLU and D. Yes L is equivalent to R plus a global reorientation of the puzzle, but I calculated on the puzzle assuming that these moves were not equivalent and then divided by 24 at the end to account for the fact that under moves of all 6 faces, you can solve the puzzle into 24 different solved states that really should all be considered the same.
I observed that a single piece can visit all 8 positions of the puzzle - this is actually a very uninteresting observation because it can't not be true if we are allowing global reorientations as I am - I am only mentioning it because it comes up in the math. I next observed that with one piece fixed, any given second piece could visit all 7 of the remaining positions on the puzzle - so this can basically be summarized as the permutation (think location only, not considering orientation) of any two pieces is independent. Now if we fix 2 pieces (for example fix two adjacent pieces) then the locations of the other pieces seemed to be restricted - I observed this by scrambling and then solving two adjacent pieces many times. In every case, the remaining 6 pieces were always in one of 3 cases. Basically, one of the pieces adjacent to the two fixed pieces was at some point in a 3-cycle: either currently correct, or one position off or two positions off. If the piece was in the right place, then every piece was in the right place. Hence the permutations of the remaining 5 pieces is dependent on the first 3 - if 3 pieces have the correct permutation, all 8 pieces have the correct permutation (again I mean location only).

Putting together the apparent options we have 8*7*3 = 168 permutations that can be reached if moves are always considered in pairs, first CW and then CCW, and WITHOUT fixing a piece. To get the number of possible permutations stopping after EITHER a CW or CCW move, we simply multiply by 2 to get my number 336. This number is not an accurate reflection of anything really :lol: and I apologize for not explaining further earlier. It has the glaring caveat that no piece was fixed and thus I was accounting for every permutation 8 times*Actually 24 times, see below over again - one with the fixed corner in each of the 8 possible positions. Dividing the permutations of the puzzle resulting from the pairs of moves by 8 we get 7*3=21, the same number you found from Andreas's work.

*Now TECHNICALLY, there is a little error here; there are actually only 7 unique permutations on this puzzle. By fixing a corner, we have also confined the orientation of THAT corner. For every permutation of the pieces we find around that corner, we find two more permutations that are actually the same permutation but rotated by 1/3 around the fixed corner but since we aren't considering orientations, we should not count these as different permutations. This is why I don't like fixing pieces and prefer to take out global reorientations at the very end instead :wink:

The rest of my math I think you probably understand - there are 8 corners that can each be rotated 3 ways. After playing with the puzzle for awhile, I ran into the case where 2 corners were rotated but the rest of the puzzle was solved so I knew there were no further restrictions on corner orientation. The orientation of the 8th corner is determined by the orientation of the other 7 corners. On the puzzle where moves are considering only in pairs, this gives (8*7*3)*(3^7) configurations, but remember I did not fix any piece so this math will overcount the number of configurations by a factor of the number of different ways we can get what appears to be a solved puzzle - and this is equal to the number of orientations of a cube, 24. Thus, ((8*7*3)*(3^7))/24 = 15309 configurations for the puzzle where moves are only considered in pairs, and double that for the puzzle where CW and CCW moves are considered individual moves (as they probably should be) = 30618 possible configurations for the Alternating Cube.

(My final hint of uncertainty was, in hindsight, foolish - of course all 24 orientations are solvable because they are only in my head - one piece on a 2x2x2 is physically attached to a corner so all 24 orientations really ARE all the same orientation - the duplicity only exists in the formalization in the math - and even then U D' is a legal pair of moves in my formalization so if one orientation is solvable, they are all trivially solvable... basically ignore my last little comment:) )

Bram wrote:
This raises the obvious solving question: What's the shortest sequence which just rotates two adjacent corners?


I found a 24 sequence: (or 12 CW-CCW pairs) that rotate two non-adjacent corners... Or you could add a single turn to put them adjacent and take it out in the end, making 26 moves for adjacent (you would have to do the mirror image of this sequence to preserve the CW-CCW rules)

RU'RF'RF'RF'RF'RF'RF'RU'RU'RF'RF'RF'
= (R U') (R F')^6 (R U')^2 (R F')^3

Not the DLB corner remains stationary for this entire algorithm - to keep things clear 8-)

Probably not optimal but at least it's an upper bound.

Using this algorithm, I can solve the entire puzzle. Permutation of the pieces is pretty easy once you observe that R U' creates two 3 cycles, affecting the position of 6 pieces but preserving the other two. Since the permutation of 3 pieces determines the permutation of the remaining 5, solving the permutation is as simple as getting 2 adjacent pieces adjacent (in the DBL and DFL positions) and then performing R U' until the permutation is correct. Then you only have to worry about orientations (btw (R U')^3 rotates 3 pieces CW and 3 CCW so if many orientations are wrong it might lead to a configuration in which more orientations are correct) Again, probably not optimal, but it does work

Peace,
Matt Galla


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Jul 22, 2013 7:01 pm 
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Thanks. I now understand the calculations and why they were done the way they were.

Next question... Is this a doctrinaire puzzle?

I think the answer is yes if the basic operation is considered a CW&CCW turn pair. But if the basic operation is a single turn then I'm not so sure.

Carl

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Jul 22, 2013 7:12 pm 
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wwwmwww wrote:
Next question... Is this a doctrinaire puzzle?


The question of whether alternating puzzles are doctrinaire has come up before - both the Spider Gear and Tracker Ball are alternating (okay, people didn't really get into discussion of that with the Tracker Ball, I think because hardly anybody understands it.) The general consensus seems to be that they are in fact doctrinaire because (a) their underlying group can be expressed using permutations, albeit with a mirror imaging trick like I explained before, and (b) they're vertex transitive (okay maybe that's just another way of reiterating the same point.)


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Jul 22, 2013 11:30 pm 
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Allagem wrote:
*Now TECHNICALLY, there is a little error here; there are actually only 7 unique permutations on this puzzle.
One more question and I think this is part of what threw me off initially. You make it quite clear in your post you are using permutations with regards to position only, not orientation. You have used "elements", "possible puzzle positions", and "configurations" as terms which also take into account orientation. But this isn't the typical use of this term is it? When one counts the permutations of the Rubik's Cube orientation is always considered isn't it? I see why you used the method you did to count the "states" another term which may have more then one meaning in this thread. But I think we may be using some of these terms incorrectly. Maybe this is a place Oskar's Wiki should step in an define these terms in the context of twisty puzzles but its certainly not clear to me that the term permutations doesn't take into account piece orientation yet the phrase "possible puzzle positions" does even though it specifically mentions "position" in the name.

I'm not really wanting to create new definitions of these terms as math and group theory I believe already have definitions we should be using however I just checked and actually surprised myself to learn maybe I've had the wrong understanding all along.

https://en.wikipedia.org/wiki/Permutation

Looking at the group theory section as I believe all (well maybe I should say most) twisty puzzles can be viewed as group theory problems I see:

Quote:
In group theory and related areas, one considers permutations of arbitrary sets, even infinite ones. A permutation of a set S is a bijection from S to itself. This allows for permutations to be composed, which allows the definition of groups of permutations. If S is a finite set of n elements, then there are n! permutations of S.
I'd view the Rubik's cube as having 27 elements and ok... I just checked 27! is a bigger number then the number of states of a Rubik's cube. But this number isn't taking into account orientation and many other things, i.e. that a corner and the core can't change positions... etc.

So is the definition on Wikipedia complete? If so maybe we really do need a twisty puzzle specific definition. Thoughts? I just know going into this thread I had a set of definitions in my head and this created a fair bit of confusion as that wasn't the way the terms were being used.

Carl

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Tue Jul 23, 2013 2:49 am 
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wwwmwww wrote:
I'd view the Rubik's cube as having 27 elements and ok... I just checked 27! is a bigger number then the number of states of a Rubik's cube. But this number isn't taking into account orientation and many other things, i.e. that a corner and the core can't change positions... etc.
If you sticker a 3x3x3 in a way that orientations vanish then it has 20 elements (core and faces can be ignored).
If you want to take orientations into account you consider permutations of stickers. In that case the 3x3x3 has 48 elements.
If you introduce slice moves you have to inlcude the faces as well => 54 elements.
If you want to implement the super 3x3x3 I see only one way to make the faces orientations visible => 72 elements.
Everything depends on what puzzle you want to simulate.


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Tue Jul 23, 2013 11:14 pm 
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After some fiddling I came up with the sequence (UF-)3(UR-)3(LD-)3 which rotates three corners counterclockwise. It's of course possible to optimize out the last R- and the first L. I'm not sure if it's possible to compose that basic sequence in three different ways to get a rotation of just two corners.


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Wed Jul 24, 2013 10:05 am 
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Bram wrote:
After some fiddling I came up with the sequence (UF-)3(UR-)3(LD-)3 which rotates three corners counterclockwise. It's of course possible to optimize out the last R- and the first L. I'm not sure if it's possible to compose that basic sequence in three different ways to get a rotation of just two corners.


You can, perform it twice on the same three pieces, then rotate the puzzle to perform it on two of those pieces and one other piece.


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 Post subject: Alternating Cube -new mech- by BRAM & OSKAR
PostPosted: Sat Nov 02, 2013 7:18 am 
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Hi Twisty Puzzles fans,

I built a new version of Alternating Cube. It has a toggle switch that can be in two states. In one state, any face can only make a clockwise quarter turn. Such a turn will bring the switch into the other state, where any face can only make a counter-clockwise (anti-clockwise) quarter turn. The mechanism is very stable, reliable and robust.

Watch the YouTube video of the new version.
Buy the puzzle at my Shapeways Shop.
Check out the photos below.

Enjoy!

Oskar
Attachment:
Alternating-Cube-v2---view-1.jpg
Alternating-Cube-v2---view-1.jpg [ 302.38 KiB | Viewed 4042 times ]

Attachment:
Alternating-Cube---view-2.png
Alternating-Cube---view-2.png [ 268.64 KiB | Viewed 4042 times ]

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Nov 02, 2013 8:29 am 
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That mechanism is so geniously simple!! :D

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Nov 02, 2013 10:50 am 
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Oskar wrote:
I built a new version of Alternating Cube.
Nice!!! May I ask if this is your second design or third design for the Alternating Cube? I remember you pointing me to this on Shapeways back in August and I had remembered it as being a closer copy of your Alternating Skewb which made use of gears. I now don't see gears. However when I look at this image:
Image
I see how I could have easily assumed those tabs on the bottom of the face centers were gear teeth. So I'm now not certain if this is what I saw in August or not.
Kattenvriendin wrote:
That mechanism is so geniously simple!! :D
Agreed... assuming geniously and ingeniously mean the same thing. Generally the "in" prefex means opposite as inhospitable and hospitable have opposite meanings. But then shouldn't ingeniously be the bad one? Not the biggest fan of the English language at times... though its the only one I know so I make do. But yes this mechanism is pure genius. If this is what I saw in August its so simple once you've seen it in motion its obvious. However in August working off the static picture above I simply had no clue how it worked.

Carl

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Nov 02, 2013 10:55 am 
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My language skills aren't superb haha, maybe I should spell it genius-ly, that fits :) Oskar is a genius for finding a solution to make this mechanism that is just.. so utterly simple.

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Nov 02, 2013 11:21 am 
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wwwmwww wrote:
May I ask if this is your second design or third design for the Alternating Cube?
This is the second prototype. You saw the first prototype in August. You can find photos of that mechanism in the first post of this thread.

Oskar

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Nov 02, 2013 11:41 am 
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Oskar wrote:
wwwmwww wrote:
May I ask if this is your second design or third design for the Alternating Cube?
This is the second prototype. You saw the first prototype in August. You can find photos of that mechanism in the first post of this thread.
I bought the first prototype at IPP and I took it apart and reassembled it in the other configuration. It is currently sitting on a shelf just over my monitor as I type this. So I'm very familiar with what you called the "original Bram-suggested mechanism". I was just curious if there was a design you made between these two which may never have been prototyped. I ask simply because I seem to recall an image which made use of gears and as I mention above I may simply have miss interpreted the image I posted. In fact I'm now almost certain that is exactly what happened.

Thanks,
Carl

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Nov 02, 2013 7:33 pm 
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That new mechanism should miniaturize well so it can be used at the center to make an alternating clockwork 4x4x4.


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Nov 02, 2013 7:52 pm 
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The new version seems based on the Alternating Skewb's design. Was that the inspiration?


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Nov 02, 2013 8:04 pm 
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I bet a gears mechanism could be used in unison with this mechanism to make an alternating 3x3x3. You would need 4 switches, each connected by a set of gears so they all rotated at the same time.

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sat Nov 02, 2013 8:20 pm 
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I love the simplicity and economy of this idea. Very, very neat.

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Sun Nov 03, 2013 1:29 pm 
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benpuzzles wrote:
... make an alternating 3x3x3.
How should opposite faces behave? By alternatingly turning opposite faces (e.g. B'FB'FB'FB'), one can make a turn in the opposite direction (viz. F'). Also, the mechanism does not prevent two opposite faces to be turned clockwise at the same time (e.g. B+F anti-slice). In any case, solving Alternating 3x3x3 would be not harder than a regular Rubik's Cube.

Oskar

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Nov 04, 2013 1:44 pm 
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Door - via PM wrote:
Subject: Alternating Cube by BRAM & OSKAR
Oskar wrote:
benpuzzles wrote:
... make an alternating 3x3x3.
How should opposite faces behave? By alternatingly turning opposite faces (e.g. B'FB'FB'FB'), one can make a turn in the opposite direction (viz. F'). Also, the mechanism does not prevent two opposite faces to be turned clockwise at the same time (e.g. B+F anti-slice). In any case, solving Alternating 3x3x3 would be not harder than a regular Rubik's Cube.

Oskar
Hi Oskar,

I've been thinking about an alternating 3x3x3, and unless I missed something, I think it is different/harder than a regular 3x3x3 puzzle.

In the example you gave, where by doing B'FB'FB'FB' you can achieve F', both sequences start with an anti-clockwise turn. If you wanted to end up with F, but had to start with an anti-clockwise turn, I don't think it is as simple.

As for the slices, the middle layer should move freely, as FB' will have the same effect as a slice turn. This could be implemented with a similar mechanism to the mixup cube, on top of the alternating cube core. This also brings up the idea of an alternating mixup cube. :wink:

Lastly, I must say the new mechanism for the Alternating cube is excellent, it's so simple and elegant! Well done!

Kind regards,
-Mark- :)

P.S: I thought I better send message this as a PM; didn't want the thread to go off topic. :P
Mark,

Thank you for the compliments. I do not think that your ideas are off-topic, so I am responding in the thread instead.

I like your idea of a HandiCube with an Alternating Cube core. And yes, you are right, every puzzle that has a 2x2x2 core could be turned into an alternating version.

Oskar

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Mon Nov 04, 2013 1:57 pm 
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benpuzzles wrote:
I bet a gears mechanism could be used in unison with this mechanism to make an alternating 3x3x3...

Actually...you wouldn't need gears at all to make an alternating 3x3. You would just need to extend the corners out from the alternating 2x2, and add grooves for the other pieces.

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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Thu Dec 12, 2013 3:18 am 
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Sorry for the bump, but I wanted to share some of my work with this puzzle. After the talks about bandaged puzzles, my thoughts kept drifting to this puzzle. I once thought that all puzzles for which every move is available in every state and the inverse of every move is also an available move meant the states of the puzzle always formed a group (the second point may or may not be necessary ). However, 3 puzzles that I am aware of challenge this statement: the Alternating Cube, the Alternating Skewb, and Even Less Gears (debatable, but I will include it for now), all by you-know-who :wink:

Now in a group structure formed by generators, every element is connected by composition with a single generator to exactly n other elements where n is the number of generators. This is not the only statement that can be made about groups, but it certainly is one of them. Depending on how you view this statement, it may be obvious, so I apologize if I insulted anyone's intelligence. The reason I bring it up is because this immediately refutes the possibility that any puzzle where moves can be blocked can possibly form a group generated by its moves. But what about these 3 puzzles? They each always have the same number of moves available in every state, so could it not be possible to pose a group structure in such a way that the moves on one of these puzzles correspond to the generators of the group? Can we find a group structure and corresponding generators isomorphic to the states of these puzzles, keeping in mind that global reorientations of each puzzle are not considered distinct states? For at least two of these puzzles, the answer is YES :shock:

I have found a way to bijectively map the moves and ALL of the states of the Alternating Cube (and Skewb!) to a set of generators and the group they form, creating an isomorphism which not only proves the states of the Alternating Cube (and Skewb!) form a group, but also allows me to analyze it via computer in a very powerful way. Before I give away how I did this, does anyone have any ideas on how to find such a mapping? :D

For the Alternating Cube
The number of states at each depth is as follows:
0 - 1
1 - 3
2 - 6
3 - 12
4 - 24
5 - 48
6 - 96
7 - 192
8 - 384
9 - 720
10 - 1232
11 - 2040
12 - 3314
13 - 5034
14 - 6344
15 - 6456
16 - 3862
17 - 804
18 - 46

Total number of configurations: = 30618

I can give the 46 hardest positions (or any positions in this list, really) explicitly if anyone is interested - you may have to suggest a format though (text or pictures?)
Bram wrote:
This raises the obvious solving question: What's the shortest sequence which just rotates two adjacent corners?
I can answer that now :)
This requires a minimum of 16 moves. There are 6 equally short answers given a specific 2 corners, a direction to rotate, the direction of rotation of the next move, and a corner to hold fixed (otherwise, the number grows exponentially :? ) (hmm.. I think 3 of these are inverses of the other 3 with multiple global reorientations throughout the sequence [not sure which! :lol: the global reorientations mean half of the moves get relabeled multiple times making it extremely difficult to detect inverses] so that means there are 3 "essentially" different solutions) :
U R' F U' R U' R U' R U' F U' F R' U R'
U R' F U' F U' F R' U F' U L' U F' U R'
R U' R U' R L' R F' U R' U F' U F' R F'
R U' F R' F U' R F' U F' U R' U F' R F'
F U' R U' R U' F U' F R' U R' F R' F U'
F U' R F' U F' R U' F R' F R' U R' F U'
Reorientations of these (map U to F and R to L) or simple inverses will generate the inverse operation and the mirror images of those will generate the same operation but starting with a CCW move first.

Any other questions? 8-)

Peace,
Matt Galla


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 Post subject: Re: Alternating Cube by BRAM & OSKAR
PostPosted: Fri Dec 13, 2013 10:46 am 
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Interesting Matt. Given that U and D are, essentially, the same thing, is there a way of phrasing those shortest sequences to rotate two corners which clarifies how/why they work?


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