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 Post subject: Clockwork 4x4x4 Cube by Timur
PostPosted: Mon Nov 12, 2012 6:48 pm 
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Hi!

This puzzle is a 4x4x4 cube with geared bandaging where all layers turn simultaneously, but with different speeds. If we consider bottom layer fixed, lower middle layer will turn with speed A, upper middle layer - with the speed 2A, and top layer - with speed 3A. A single move is finished when the layers have turned by 1/4 turn, 2/4 turn and 3/4 turn, respectively. So the principle is similar to Bram and Oskar's Geary Cube (3x3x3), but the mechanism is vastly more complex.

I dreamed to build this puzzle ever since I've made the Clockwork Master Pyraminx. Because it's not really comfortable to play with the latter (you have to turn a corner 7 times around to finish a turn), the main and firm requirement for Clockwork 4x4x4 Cube was minimum turning job - only 270° outer layer's turn had to be made for one move, so that all layers turn by 90° with respect to its neighbors.

Throughout a year or so I made three or four attempts to develop the mechanism, but always failed. Each time I felt that I'm close to the solution, I would discover some conflict ruining the whole idea. Still I was sure that it's possible and it became a challenge to myself. Meanwhile, I worked with other projects, but never forgot about the 4x4x4 cube.

Finally, it cracked and I'm glad that it works decently from the first prototype. The mechanism contains 156 3D-printed parts, 21 screws and 18 nuts (nuts could actually be omitted). I think it's the most complex thing I've designed (Emperor Pyraminx goes next :) ). The mechanism will be revealed in one of my next posts.

Can anyone calculate the number of possible states of this puzzle? I guess it's smaller than that of a normal 4x4x4 cube, right?

Anticipating question of Andreas: edge length is 62mm :D

YouTube Video

Attachment:
File comment: Solved
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ALIM5051.JPG [ 113.4 KiB | Viewed 8802 times ]


Attachment:
File comment: Starting to turn - 1
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ALIM5052.JPG [ 174.46 KiB | Viewed 8802 times ]


Attachment:
File comment: Starting to turn - 2
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ALIM5057.JPG [ 150.81 KiB | Viewed 8802 times ]


Attachment:
File comment: After 1 move
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ALIM5053.JPG [ 124.93 KiB | Viewed 8802 times ]


Attachment:
File comment: After 2 moves
ALIM5054.JPG
ALIM5054.JPG [ 153.93 KiB | Viewed 8802 times ]

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Mon Nov 12, 2012 6:56 pm 
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I've always thought of solving a 4x4 like this, but never gave any thought to a mod. Well done! It is extremely elegant in its seeming simplicity. Have you tried solving it yet?

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Mon Nov 12, 2012 9:36 pm 
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Brilliant puzzle!

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 12:39 am 
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Wow, Timur! Just thinking about how you accomplished this in the mechanism amazes me.
Congrats.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 8:05 am 
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Wow, it turns so smoothly. I simply can't wait to see the mechanism.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 8:29 am 
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That is incredible! I'd love this puzzle, it seems like it could be great fun to solve!

Great work as always Timur. I'm really looking forward to seeing your mechanism design.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 10:57 am 
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Absolutely brilliant!!! Take an Order=3 puzzle and turn it into an Order=1 puzzle... I love it! It's puzzles like this that really make me miss the puzzle of the year competition we used to have here.
io wrote:
Just thinking about how you accomplished this in the mechanism amazes me.
I've actually thought about this puzzle before but never got much further then that as I had no clue were to start with a mechanism. So I'm very eager to see what you came up with.
Timur wrote:
Can anyone calculate the number of possible states of this puzzle? I guess it's smaller than that of a normal 4x4x4 cube, right?
That should be an easy question for GAP to answer. But its certainly no more states then a normal 4x4x4. I'm tempted to think this may be possible to map to the problem of having 3 side-by-side 2x2x2's where a 90 degree turn on the first puzzle causes a 180 degree turn on the second puzzle and a 270 degree turn on the third puzzle. Though I'm far from certain of that and I don't know how many states that set of linked puzzles has either. How does the cube (^3) of the number of permutations of the 2x2x2 compare to the number of permutations of the 4x4x4? I need to check...

Carl

P.S. Looked up some numbers. The 4x4x4 has about 7.40*10^45 permutations. The set of linked 2x2x2's I describe above has at most 4.56*10^19 permutations. So if my guess is correct this puzzle has MANY MANY less states then the 4x4x4. I'd need to put this in GAP to be sure and even if I were able to prove this puzzle and the set of linked 2x2x2's had the same number of states its not obvious how to do the mapping from one to the other. My gut just tells me they should be related.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 11:14 am 
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Timur wrote:
Can anyone calculate the number of possible states of this puzzle? I guess it's smaller than that of a normal 4x4x4 cube, right?

Anticipating question of Andreas: edge length is 62mm :D

Thank you for answering one (!) of my questions before I can ask them.
And thank you for making such good images.
But I have another question: What does it weigh?

For Timur and Carl:
GAP tells me the number of permutations is 10616832
Those of you who are interested might look into the attachment


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Clockwork 4x4x4.txt [2.17 KiB]
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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 11:41 am 
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Andreas Nortmann wrote:
For Timur and Carl:
GAP tells me the number of permutations is 10616832
Those of you who are interested might look into the attachment
Nice!!! I knew you'd beat me to the punch on that one. And after looking at that GAP code you clearly know a few GAP tricks I have yet to learn. Judging from the code that is the number of permutations of the super-sitckered version of this puzzle... correct? Using your bag of tricks can you easily get the number for the normal-stickered version? Or if that is what this is... can you get the number for the super-stickered version? Considering just how small that number is compared to the 4x4x4 I suspect they may be the same number and its just not possible to mix up face centers of the same color with each other.

And as this is about 2.8896 times the number of permutations of a 2x2x2 my hypothesis that this could be mapped to a set of linked 2x2x2's seems blown out of the water. If any mapping were possible I believe that ratio would have to be a whole number... and its not. :( So much for my gut.

Ohhh... and Timur PLEASE tell me you have plans to offer this on Shapeways. Its at the TOP of my wish list.

Thanks,
Carl

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Last edited by wwwmwww on Tue Nov 13, 2012 4:10 pm, edited 1 time in total.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 12:27 pm 
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Andreas Nortmann wrote:
For Timur and Carl:
GAP tells me the number of permutations is 10616832
Those of you who are interested might look into the attachment

And Brandon :D

Can you double-check this number? I don't think it's right.

Here's why: Take all but the corner stickers off the Clockwork 4x4x4 and you can see it behaves like an odd 2x2x2 and can easily be seen to have the same number of states as a 2x2x2.

Here are the prime factors for the 2x2x2 group size:
? factor(((8!) * ((3^8) / 3)) / 24)
% = 2^4 * 3^8 * 5^1 * 7^1

Now the number GAP gave you:
? factor(10616832)
% = 2^17 * 3^4

It's missing four factors of 3, one factor of 5, and one factor of 7. It can't be a superset of the 2x2x2...

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 12:38 pm 
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The thought of trying to solve that one is terrifying, but it undoubtedly makes many pretty patterns.


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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 1:00 pm 
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Bram wrote:
The thought of trying to solve that one is terrifying, but it undoubtedly makes many pretty patterns.

I have a sneaking suspicion that the extra, non-corner pieces are linked up and can probably be (easily) manipulated without changing the corners.

Here is a toy example.

If you take [R, U'2]x6 which is equivalent to the identity for the corners, translate it to the Clockwork 4x4x4 and you get:

[[R'&4, R'2&2, R], [U'2&4, U'2]]x6

It changes some non-corner pieces.


If you compose it a few times you end up with more interesting sequences like:
[L&2, R'2&2, R, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2, R&2, L'2&2, L, D'2&2, U'2, R&2, L'2&2, L, D'2&2, U'2, R&2, L'2&2, L, D'2&2, U'2, R&2, L'2&2, L, D'2&2, U'2, R&2, L'2&2, L, D'2&2, U'2, R&2, L'2&2, L, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2, L&2, R'2&2, R, D'2&2, U'2]

My point is that I suspect a handful of repetitive sequences that do nothing to the corners are enough to solve all of the non-corner pieces.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 1:16 pm 
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bmenrigh wrote:
Here's why: Take all but the corner stickers off the Clockwork 4x4x4 and you can see it behaves like an odd 2x2x2 and can easily be seen to have the same number of states as a 2x2x2.
Boy it sure helps to read things correctly. The first two times I read this I missed the word "but" so mentally I took the corner stickers off and I was asking myself HOW were you easily seeing an odd 2x2x2 in what was left. However once read correctly, yes I see your point and I agree. Looks like one of Andreas's GAP tricks... tricked him. I don't know enough GAP to be able to see his error.

Maybe there is hope for my hypothesis yet. Looking closer at the 3 linked 2x2x2 puzzle idea... say a player makes a 90 turn on the first puzzle such that that same face makes a 180 degree turn on the second puzzle and a 270 degree turn on the third puzzle. I don't believe the third puzzle adds anything new as it would appear to be an odd mirrored image of the first puzzle (just mirrored in all 3 axes). Also the second puzzle is a half-turn 2x2x2 which I'm sure has many less states then the normal 2x2x2. And I'm not sure its possible to solve the first puzzle without the second puzzle also being solved. If that is the case my 3 linked 2x2x2 puzzles would have the same number of permutations as just a single 2x2x2 and it doesn't really add anything new. Just not sure about the 2x2x2 linked to a half-turn 2x2x2. If there is anything new there I should ask Oskar to make a Dumbbell 2x2x2 similiar to his Dumbbell Cube.

Carl

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 1:25 pm 
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You absolutely have to show the mechanism on this. This is incredible!

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 1:41 pm 
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darryl wrote:
You absolutely have to show the mechanism on this. This is incredible!
Oh and that prompts me to ask a mechanism question... could this mechanism be used to make a Circle Clockworks 4x4x4 Cube? Think of this puzzle AND being able to see the 8 hidden cubies on the inside. On the outside it would look something like this:

Image

Hmmm... if this is possible does it really even add any new challenge to the puzzle? Something tells me that inner 2x2x2 is just the same odd mirror image of the outer already visable 8 corners.

Carl

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 1:54 pm 
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wwwmwww wrote:
Hmmm... if this is possible does it really even add any new challenge to the puzzle? Something tells me that inner 2x2x2 is just the same odd mirror image of the outer already visable 8 corners.

I'm pretty sure this isn't just two or three 2x2x2s. The outer 2x2x2 is obvious but I've done some testing with Gelatinbrain's program and it is possible to move 4x4x4 edge-wings without moving corners and it's possible to move 4x4x4 X-centers without moving edge-wings or corners.

I need to do more thinking and more virtual twisting to say anything constructive about the puzzle. The more I play with it though the more I think the inner pieces aren't related to a 2x2x2.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 2:07 pm 
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My vote for puzzle of the year!

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 2:09 pm 
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Nice puzzle, and interesting discussion guys. If I didn't have some more important code to look at before a test this week I'd try and have a shot at the GAP for this myself. Hope to see some interesting results soon, and if I somehow get some time I might try to contribute to them.


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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 5:55 pm 
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Congratulations Timur. :)
Great creation!
I love gear puzzles, so I'm happy to see you moving in that direction. :D

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 6:14 pm 
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Oskar wrote:
My vote for puzzle of the year!


But the year isn't even over yet!

I love this thing, and I'd also like to see a "real" version of this as well.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 6:30 pm 
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Here is a pure 3-cycle of the X-centers:
[[R, R'2&2, R'&4, U, U'2&2, U'&4]x15, B'2, F'2&2, [U', U2&2, U&4, R', R2&2, R&4]x15, F2&2, B2]x3, R2, R2&4, [[R, R'2&2, R'&4, U, U'2&2, U'&4]x15, B'2, F'2&2, [U', U2&2, U&4, R', R2&2, R&4]x15, F2&2, B2]x3, R'2, R'2&4
(1108 moves on a regular 4x4x4, 374 on the Clockwork 4x4x4)

I think I'm getting close to a 3-cycle for the edge-wings too. If I can find one, this puzzle would have half the states of the 4x4x4, since I think the parity of the edge-wings is equal to the parity of the corners.

My best guess at this point is that the number of states is:
? (((8! * 24!) / 2) * (24! / (4!^6)) * (3^8 / 3) ) / 24
% = 3700598420782450934937046987249287168000000000


Which is the same calculation as the 4x4x4 but it assumes the parity of the corners and edge-wings are linked which halves the states.

EDIT:

Here is a 3-cycle for the edge-wings:
[F'2, F'2&4, U, U'2&2, U'&4, F'2, F'2&4, U', U'2&2, U&4]x10
(100 moves on a regular 4x4x4, 40 moves on the Clockwork 4x4x4)

I'm more confident of my above position count now.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Nov 13, 2012 9:55 pm 
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Incredible! A mechanism for this completely escapes me, I must say.
Do you plan to have this available on your shop?

-Doug

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Wed Nov 14, 2012 12:16 pm 
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Thank everyone for nice comments!
mitchblahman wrote:
Have you tried solving it yet?
No, I didn't scramble it :)
Andreas Nortmann wrote:
But I have another question: What does it weigh?
I've weighed it today, it is about 90g.
wwwmwww wrote:
Ohhh... and Timur PLEASE tell me you have plans to offer this on Shapeways. Its at the TOP of my wish list.
No problem, I will, but the model has about 2.8mln polygons, so it can be ordered only through a Shapeways backdoor for high poly models. Else I have to divide it into 2 or 3 parts.
wwwmwww wrote:
could this mechanism be used to make a Circle Clockworks 4x4x4 Cube?
Well I don't see a straightforward way of doing this, but it should be possible with a little more complicated mech :)

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Wed Nov 14, 2012 12:51 pm 
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bmenrigh wrote:
Can you double-check this number? I don't think it's right.
You are right, that it isn't right. :oops:
My next shot uses this piece of code:
Code:
Super5x5x5:=Group(U2*D2^-2*D1^-3,L2*R2^-2*R1^-3,B2*F2^-2*F1^-3);
Orbits4x4x4:=Concatenation(List([21,2,4,7],i->Orbit(Super5x5x5,i)));
Clockwork4x4x4:=Action(Super5x5x5,Orbits4x4x4);
and with it I get 1/2 of the number for the 4x4x4.

@ Carl: You are right. This is the number for the Super-variant.
The number for the non-super variant can be calculated by dividing by 95551488=24^5*12

Does anybody want to proove me wrong again?
Timur wrote:
I've weighed it today, it is about 90g.
Thank you.


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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Wed Nov 14, 2012 12:56 pm 
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Andreas Nortmann wrote:
bmenrigh wrote:
Can you double-check this number? I don't think it's right.
You are right, that it isn't right. :oops:
My next shot uses this piece of code:
Code:
Super5x5x5:=Group(U2*D2^-2*D1^-3,L2*R2^-2*R1^-3,B2*F2^-2*F1^-3);
Orbits4x4x4:=Concatenation(List([21,2,4,7],i->Orbit(Super5x5x5,i)));
Clockwork4x4x4:=Action(Super5x5x5,Orbits4x4x4);
and with it I get 1/2 of the number for the 4x4x4.

@ Carl: You are right. This is the number for the Super-variant.
The number for the non-super variant can be calculated by dividing by 95551488=24^5*12

Does anybody want to proove me wrong again?

This is exactly what I get although my method is highly manual and error-prone.

So what is the Clockwork 4x4x4? It's an incredibly difficult 2x2x2 variant.

Edit:

Group size calculations for posterity:

Clockwork 4x4x4:
? (((8! * 24!) / 2) * (24! / (4!^6)) * (3^8 / 3) ) / 24
% = 3700598420782450934937046987249287168000000000

Super Clockwork 4x4x4:
? (((8! * 24!) / 2) * (24! / 2) * (3^8 / 3) ) / 24
% = 353597685596213311120226025957586415841705984000000000

Ratio of Super to Standard:
? (4!^6) / 2
% = 95551488


Also, this puzzle is what I'd call "deep-cut" and I think Carl would call Order-1.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Wed Nov 14, 2012 5:04 pm 
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Jorbs3210 wrote:
I love this thing, and I'd also like to see a "real" version of this as well.
You mean a mass produced version? The one presented here certainly isn't "virtual".
bmenrigh wrote:
Here is a pure 3-cycle of the X-centers:
[[R, R'2&2, R'&4, U, U'2&2, U'&4]x15, B'2, F'2&2, [U', U2&2, U&4, R', R2&2, R&4]x15, F2&2, B2]x3, R2, R2&4, [[R, R'2&2, R'&4, U, U'2&2, U'&4]x15, B'2, F'2&2, [U', U2&2, U&4, R', R2&2, R&4]x15, F2&2, B2]x3, R'2, R'2&4
(1108 moves on a regular 4x4x4, 374 on the Clockwork 4x4x4)
That is just crazy!!! And I have NO idea how you generate these. Is there a tool I can plug these sequences into so I can see them in action? I'd go crazy trying this on a real 4x4x4 and trying to keep everything strait.
bmenrigh wrote:
I think I'm getting close to a 3-cycle for the edge-wings too.
I was about to ask you to PLEASE post when you find one... but I see you already have.
bmenrigh wrote:
Here is a 3-cycle for the edge-wings:
[F'2, F'2&4, U, U'2&2, U'&4, F'2, F'2&4, U', U'2&2, U&4]x10
(100 moves on a regular 4x4x4, 40 moves on the Clockwork 4x4x4)
So how is it that it was easier to find a 1108 move sequence over a 100 move sequence?
Timur wrote:
No problem, I will, but the model has about 2.8mln polygons, so it can be ordered only through a Shapeways backdoor for high poly models. Else I have to divide it into 2 or 3 parts.
Backdoor is fine by me... I'll send you a PM later today...
Timur wrote:
Well I don't see a straightforward way of doing this, but it should be possible with a little more complicated mech :)
Only a little... LOL!!! Before you try though I really want to know if it would even add anything to the puzzle. I suspect the inner 2x2x2's state is directly determined by the outer 4x4x4's corners state. Is so even if you make these pieces visable it wouldn't add any further states to the puzzle. In other words if one solved the puzzle as you have presented it then the inner 2x2x2 would automatically be solved even without being able to see it.
Andreas Nortmann wrote:
@ Carl: You are right. This is the number for the Super-variant.
The number for the non-super variant can be calculated by dividing by 95551488=24^5*12
How hard would it be to check the Multi- (or Circle) variant where the inner 2x2x2 was visable from the surface? I strongly suspect that you'd get the same numbers.
bmenrigh wrote:
So what is the Clockwork 4x4x4? It's an incredibly difficult 2x2x2 variant.
Curious... may I ask how you'd compare it (difficulty wise) with the Complex 3x3x3? Granted I'm sure that is a very subjective question but I suspect you are the only one that could say much on that topic at the moment.
bmenrigh wrote:
Also, this puzzle is what I'd call "deep-cut" and I think Carl would call Order-1.
Yes... and I've already called it Order=1 one above. LOL!!! As for deep cut... I could see that. The central cut plane divides the puzzle into two isomorphic halves. One can view this cut plane as the independant one and the other two as dependant. But I don't think I've seen a definition of deep cut that would "naturally" cover this puzzle. Though I don't think ANY definition I've seen even address the notion of gears and how they could be used in a deep cut puzzle.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Wed Nov 14, 2012 6:27 pm 
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wwwmwww wrote:
bmenrigh wrote:
Here is a pure 3-cycle of the X-centers:
[[R, R'2&2, R'&4, U, U'2&2, U'&4]x15, B'2, F'2&2, [U', U2&2, U&4, R', R2&2, R&4]x15, F2&2, B2]x3, R2, R2&4, [[R, R'2&2, R'&4, U, U'2&2, U'&4]x15, B'2, F'2&2, [U', U2&2, U&4, R', R2&2, R&4]x15, F2&2, B2]x3, R'2, R'2&4
(1108 moves on a regular 4x4x4, 374 on the Clockwork 4x4x4)
That is just crazy!!! And I have NO idea how you generate these. Is there a tool I can plug these sequences into so I can see them in action? I'd go crazy trying this on a real 4x4x4 and trying to keep everything strait.

You're opening a huge can of worms with this question :lol: First, you can directly copy and paste my sequence with the nested [...]xN stuff and everything directly into Gelatinbrain's program to see the effect:
Attachment:
clockwork_4x4x4_center_3-cycle.png
clockwork_4x4x4_center_3-cycle.png [ 19.82 KiB | Viewed 8022 times ]


Second, how I found this sequence is shamefully mindless.

First, I found a very simple sequence that is equivalent to the identity on the 2x2x2: [R, U]x15. The reason is that it is a 5-cycle + it twists corners. every []x5 the corners are back in place twisted. After [[]x5]x3 they corners have not changed at all. I then manually performed this sequence (15 times) on the Clockwork 4x4x4 to see the effect:
Attachment:
clockwork_4x4x4_center_3-cycle_90_in.png
clockwork_4x4x4_center_3-cycle_90_in.png [ 23.47 KiB | Viewed 8022 times ]


So that makes a big mess. Time to look for a way to turn it into a less-messy commutator. Notice that with the move I have highlighted we can make it a lot cleaner:
Attachment:
clockwork_4x4x4_center_3-cycle_90_in_commute.png
clockwork_4x4x4_center_3-cycle_90_in_commute.png [ 32.8 KiB | Viewed 8022 times ]

The reason only the outer face and far-slice are moving in this Y portion of the commutator is that it is a half-turn and the close-slice doesn't move because of the way Timur has geared the puzzle.

Looking only at the edge-wings, this commutator will create two disjoint 3-cycles and some super-complex set of cycles in the X-centers. We'll worry about that later. Here is the result of that commutator:
Attachment:
clockwork_4x4x4_center_3-cycle_184_in.png
clockwork_4x4x4_center_3-cycle_184_in.png [ 23.31 KiB | Viewed 8022 times ]


To get rid of the the two 3-cycles in the edge-wings I apply the whole commutator three times:
Attachment:
clockwork_4x4x4_center_3-cycle_2-2-swap.png
clockwork_4x4x4_center_3-cycle_2-2-swap.png [ 22.62 KiB | Viewed 8022 times ]

Lucky for us, all that is left is a 2-2 swap. Using this as the X part of a commutator to turn it into a 3-cycle is trivial (which is what I did).

As you can see, almost no thought was involved in this process. The advantage this technique has by being trivial is made up for by the extremely long move-sequences it creates. For what it's worth, this "do random stuff" commutator construction works for basically any piece on basically any twisty puzzle. The deeper you nest the commutators or cancel some cycles by applying a sequence over and over the longer the overall sequence gets. >10,000 moves for a single 3-cycle sequence is not unreasonable.

wwwmwww wrote:
bmenrigh wrote:
Here is a 3-cycle for the edge-wings:
[F'2, F'2&4, U, U'2&2, U'&4, F'2, F'2&4, U', U'2&2, U&4]x10
(100 moves on a regular 4x4x4, 40 moves on the Clockwork 4x4x4)
So how is it that it was easier to find a 1108 move sequence over a 100 move sequence?
It's deceiving. It wasn't easier. It was luckier. Since I already had a pure 3-cycle for the centers I didn't have to worry about a pure 3-cycle for the edge-wings. I just tried random 2x2x2 identity-sequences until I found one that had cycles that could be canceled easily.

wwwmwww wrote:
How hard would it be to check the Multi- (or Circle) variant where the inner 2x2x2 was visable from the surface? I strongly suspect that you'd get the same numbers.
The corners of the Clockwork 4x4x4 are basically 2x2x2 corners. Unless you have something else in mind, the Circle 4x4x4 with the inner 2x2x2 showing would not add any new states (and the outer 2x2x2 corners would always stay lock-step in sync with the inner 2x2x2).

wwwmwww wrote:
bmenrigh wrote:
So what is the Clockwork 4x4x4? It's an incredibly difficult 2x2x2 variant.
Curious... may I ask how you'd compare it (difficulty wise) with the Complex 3x3x3? Granted I'm sure that is a very subjective question but I suspect you are the only one that could say much on that topic at the moment.
The X-center 3-cycle I've listed here is not usable for a physical solve. I'm sure a much shorter 3-cycle exists but it's hard to find it because manual simulating the movement of the Clockwork 4x4x4 is tedious and error-prone. If Gelatinbrain added a Clockwork 4x4x4 option then I'm sure I could find much shorter sequences.

Stepping back and just comparing the two puzzles without focusing on sequence length, the Complex 3x3x3 is so much harder you can barely compare them. The Clockwork 4x4x4 has 3 piece types. The Complex 3x3x3 has 10. Another measure of difficulty is how "interconnected" the pieces are. Without explaining what I mean by "interconnected", the Complex 3x3x3 is maximally interconnected. The corners and edge-wings on the Clockwork 4x4x4 are partially interconnected because of their linked parities. Because it's natural to solve the corners first this actually doesn't increase the difficulty in any way.

My apologies for waving my hands about what I mean by "interconnected" but I've been working on a complicated idea for the measure of difficulty of a puzzle. For moderately sized puzzles a computer is needed to measure the interconnectedness of its pieces. I have agreed to too many twisty tasks / duties to have made much progress on this. It's coming but I have to fulfill other twisty obligations first.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Thu Nov 15, 2012 6:11 pm 
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I'm not sure what notation everyone is using, but if we pretend the thing is a 2x2x2 and that the whole thing is a 2x2x2 with some stuff added to the middle, then the sequence (R6F6)3 might be useful, and also some things like (R6F6U6F6)2 and generally things involving R6, U6, and F6. I don't have a 4x4x4 handy to try them out with though.


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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Thu Nov 15, 2012 6:36 pm 
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Bram wrote:
I'm not sure what notation everyone is using, but if we pretend the thing is a 2x2x2 and that the whole thing is a 2x2x2 with some stuff added to the middle, then the sequence (R6F6)3 might be useful, and also some things like (R6F6U6F6)2 and generally things involving R6, U6, and F6. I don't have a 4x4x4 handy to try them out with though.

Hey Bram, the notation I'm using is just Gelatinbrain's notation since that's easiest for sharing long sequences (copy and paste into the program).

I'm not sure what you mean by R6, U6, etc. What does the 6 signify? It is a "half-turn"?

If by (R6F6)3 you mean R2, F2, R2, F2, R2, F2 which has this effect on the 4x4x4:
Attachment:
4x4x4_2-2-swap.png
4x4x4_2-2-swap.png [ 11.53 KiB | Viewed 7899 times ]


On the Clockwork 4x4x4 it has no effect (identity sequence).

Assuming the same for (R6F6U6F6)2 then it too has no effect on the Clockwork 4x4x4.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Thu Nov 15, 2012 8:31 pm 
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By R6 I'm referring to what it actually feels like when you hold it - keep the left face in place, and rotate the right face clockwise 540 degrees. This will result in one of the middle slices returning to exactly where it started and another one rotating 180 degrees.


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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Thu Nov 15, 2012 9:04 pm 
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Bram wrote:
By R6 I'm referring to what it actually feels like when you hold it - keep the left face in place, and rotate the right face clockwise 540 degrees. This will result in one of the middle slices returning to exactly where it started and another one rotating 180 degrees.
Gotcha. That makes a lot of sense. When I was saying "R" what I really meant is R'3 and the associated slice movements.

A neat property of the Clockwork 4x4x4 is if you restrict yourself to only {U,D,F,B,R,L}6 moves (half-turn-only) then the puzzle only has 8 pieces which correspond to the corners of a half-turn-only 2x2x2.

Here is a single corner highlighted:
Attachment:
clockwork_4x4x4_half_only_corner.png
clockwork_4x4x4_half_only_corner.png [ 8.79 KiB | Viewed 7846 times ]


So any sequence of moves to preserve the corners of the puzzle must make use of non-6 / non-half-turn moves if it is to have any hope of moving the other pieces in the puzzle.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Mon Dec 31, 2012 11:07 pm 
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Bump...

I changed my RefleCube simulator a little bit and made a simulator of Clockwork 4x4x4, together with the 3x3x3 and 5x5x5 versions. I haven't studied 4x4x4 yet and I don't know if it's possible to solve it without macros (which my simulator doesn't support). But I enjoyed solving the 3x3x3 and 5x5x5. They are pretty easy to solve.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Jan 01, 2013 1:53 pm 
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schuma wrote:
I changed my RefleCube simulator a little bit and made a simulator of Clockwork 4x4x4, together with the 3x3x3 and 5x5x5 versions. I haven't studied 4x4x4 yet and I don't know if it's possible to solve it without macros (which my simulator doesn't support). But I enjoyed solving the 3x3x3 and 5x5x5. They are pretty easy to solve.
NICE!!!! Timur and I have already talked a bit about the Clockwork 5x5x5. I must confess I'm a little disappointed to hear that its easy to solve. I was hoping it would create a several new piece types and be on par with the Clockwork 4x4x4 or even harder. As it is not I'm now curious, can the Clockwork 5x5x5 be shown to be equivalent to the Clockwork 3x3x3? I know pieces like the corners of a Clockwork 5x5x5 can't be scrampled, they all act as one piece though that piece can be flipped 180 degree relative to the core. So how many permutations does the Clockwork 5x5x5 have in relation to the Clockwork 3x3x3?

Carl

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Jan 01, 2013 3:28 pm 
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wwwmwww wrote:
NICE!!!! Timur and I have already talked a bit about the Clockwork 5x5x5. I must confess I'm a little disappointed to hear that its easy to solve. I was hoping it would create a several new piece types and be on par with the Clockwork 4x4x4 or even harder. As it is not I'm now curious, can the Clockwork 5x5x5 be shown to be equivalent to the Clockwork 3x3x3? I know pieces like the corners of a Clockwork 5x5x5 can't be scrampled, they all act as one piece though that piece can be flipped 180 degree relative to the core. So how many permutations does the Clockwork 5x5x5 have in relation to the Clockwork 3x3x3?


Thanks. I don't know much about what the pieces types are in this kind of puzzles, but I guess we do have new things here in Clockwork 5x5x5. But it is as easy as, or maybe easier than the clockwork 3x3x3.

Have you tried it? From a scrambled Clockwork 5x5x5, I always very quickly end up with some pretty patterns. Then I only need to solve the patterns....

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Jan 01, 2013 4:59 pm 
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schuma wrote:
Have you tried it? From a scrambled Clockwork 5x5x5, I always very quickly end up with some pretty patterns. Then I only need to solve the patterns....
Yes, I have played with it a bit. I'm not much of a solver but I can get very close to solved in as little as 14 moves. First time only swapping the t-centers on two opposite faces. And I've now also nearly solved it except for the x-centers a few times. Unlike the Clockwork 4x4x4 which has nearly as many states as a normal 4x4x4 its clear this doesn't have any where close to the number of states as a normal 5x5x5. I think it may reduce to two Clockwork 3x3x3s which always move opposite each other so once one is solved so is the other. In other words I think it may have the same number of states as a Clockwork 3x3x3. I'll have to think about this one more. As poor as a solver as I am I also must confess I'm better at solving (and understanding the correct picture) when I can actually hold the 3D puzzle in my hands. I've never been very good with puzzle simulators. Not trying to knock the great work that you've done.

Carl

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Tue Jan 01, 2013 10:14 pm 
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Carl, I computed the number of permutations using gap. I assume the middle layer doesn't move to have a static reference so that I don't need to divide the number by 24... Anyway:

Clockwork 3x3x3: 6144
Clockwork 5x5x5: 98304 (= 6144 * 16)


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gap_clockwork.txt [3.68 KiB]
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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Wed Jan 02, 2013 3:18 am 
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schuma wrote:
Clockwork 5x5x5: 98304 (= 6144 * 16)

That's an interesting case - a higher order puzzle has less possible states! I wonder whether there are other examples of that.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Wed Jan 02, 2013 10:48 pm 
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Timur wrote:
schuma wrote:
Clockwork 5x5x5: 98304 (= 6144 * 16)
That's an interesting case - a higher order puzzle has less possible states! I wonder whether there are other examples of that.
I read this in the morning and it JUST hit me 5 minutes ago what you meant. Boy am I slow. All day I've been thinking... How is 16 times as many states... less states? Only now did it hit me you are comparing the Clockwork 4x4x4 to the Clockwork 5x5x5. As for the use of the word "order"... I'd consider all 3 of these puzzles Order=1... in that they each only have one independent layer per axis of rotation.

I still think the Clockwork 3x3x3 and the Clockwork 5x5x5 are very closely related but I'll have to think a bit more before I understand where that factor of 16 is coming from.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Wed Jan 02, 2013 11:07 pm 
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I'm not sure I understand the Clockwork 3x3x3 or 5x5x5 yet. Although I can solve them, I don't have a strategy / algorithm for doing so. Stochastic hill climbing towards a solution works every time. The Clockwork 3x3x3 seems very similar to the Gear Cube (same puzzle minus some orientation?) and I don't really have a strategy for the Gear Cube either.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Wed Jan 02, 2013 11:22 pm 
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bmenrigh wrote:
The Clockwork 3x3x3 seems very similar to the Gear Cube (same puzzle minus some orientation?)
Yes, its exactly the same puzzle. Just take the stickers off the gear parts (which can be solved trivially after the other parts are solved). And maybe I should point out the Clockwork 3x3x3 can also be made from the Gear Cube kit. Oskar first called it the Geary Cube.

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 Post subject: Re: Clockwork 4x4x4 Cube by Timur
PostPosted: Thu Jan 03, 2013 2:39 am 
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I just solved the clockwise 4x4x4 on gelatinbrain, using macros of multi-layer turns to simulate a clockwork turn.
All my algorithms is based on a simple algorithm:

(clockwork turn notation) [U2, F, U2, R]x3
or (gelatinbrain notation) [U2,U2&4, F,F2&2,F'&4, U2,U2&4, R,R2&2,R'&4]x3

It swaps two pairs of edges and two pairs of centers. I use variations of this algorithms as atoms to construct three cycles. Since this algorithm is not long and kind of easy to remember, I suspect solving this puzzle without macro may actually be possible.

wwwmwww wrote:
Yes, its exactly the same puzzle. Just take the stickers off the gear parts (which can be solved trivially after the other parts are solved).

Well, on the standard gear cube the edge stickers are only on the gear parts. If you peel them off you lose all the edges. I think you mean that one should start with this sticker variation where there are stickers on the non-gear parts of edges, then remove the stickers off the gear parts.

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