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 Post subject: Incomprehensible Cube by OSKAR
PostPosted: Sat Sep 21, 2013 6:46 am 
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Hi Twisty Puzzles fans,

Incomprehensible Cube was inspired by an idea from Bram Cohen. It looks like a regular Rubik's Cube, but it has two edges with two peek holes each. Those are the "special edges". The special-edges layer and the two layers behind it can all turn independently. Like a regular Rubik's Cube. The slice layer between the special edges turns like a slice layer, while the two special edges remain coupled to each other. The opposite faces at the top and bottom are connected with a rod, which can be observed through the peek holes.

This puzzle is another demonstration that a slice move is different from an opposite faces move. This was also what distinguished Double Trouble A and B.

The operation of the puzzle as a whole is quite incomprehensible, hence the puzzle's name. The peek holes help understand the operation and orientation of the mechanism. The mechanism is deeper-than-origin cut.

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

Enjoy!

Oskar
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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Sat Sep 21, 2013 7:47 am 
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Incomprehensible - how you can invent such things :)

I have no idea how hard it will be to solve. :roll:

I like such variations that look like a Rubik's Cube at the first glance, but are quite different, indeed.

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Sat Sep 21, 2013 7:53 am 
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A slice only + opposite faces only + deep-cut Rubik's cube. It's obvious really, I'm surprised someone has not done this before :lol: How many more devilish variations on the 3x3x3 have you waiting in the wings?

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Sat Sep 21, 2013 9:38 am 
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I would call this puzzle
(UD, L, Ls, Fs)
This group has 231158159769600 permutations

Without the linked opposite layers it would be
(U, D, L, Ls, Fs) which I implemented here
and has 1966000148840448000 permutations

There is a factor of 8505 between both numbers wherever that comes from ...

EDIT: All numbers include the face pieces (without orientation) as well which is necessary for all puzzles where the core is not fixed in space.
EDIT2: I found a huge mistake. See below.


Last edited by Andreas Nortmann on Sun Sep 22, 2013 4:28 am, edited 2 times in total.

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Sat Sep 21, 2013 11:22 am 
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At TwistyPuzzles.com our Museum moderator comprehends the incomprehensible.

Dave

Andreas Nortmann wrote:
I would call this puzzle
(UD, L, Ls, Fs)
This group has 231158159769600 permutations

Without the linked opposite layers it would be
(U, D, L, Ls, Fs) which I implemented here
and has 1966000148840448000 permutations

There is a factor of 8505 between both numbers wherever that comes from ...

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Sat Sep 21, 2013 3:02 pm 
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I think the one thing that strikes me as amazing (other than your prolific puzzle creation) is the number of puzzles that have been based of of Erno Rubik's original 3x3x3 mechanism.

Without that invention, I wonder if any of the puzzles we enjoy now would have ever been created.

That elegant 3x3x3 x-y-z plane core that Rubik came up with is like the Rosetta Stone of twisty puzzle creations.

Nice puzzle, Oskar. As always!


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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Sat Sep 21, 2013 3:04 pm 
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Andreas Nortmann wrote:
I would call this puzzle
(UD, L, Ls, Fs)
This group has 231158159769600 permutations

Without the linked opposite layers it would be
(U, D, L, Ls, Fs) which I implemented here
and has 1966000148840448000 permutations

There is a factor of 8505 between both numbers wherever that comes from ...


That is... mysterious.

How do those numbers break down if you include only edges or only corners?


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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Sat Sep 21, 2013 3:54 pm 
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8505 = 3^5 * 5 * 7

Not sure where the prime factors 5 and 7 come from, maybe a factorial expression somewhere? :?

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Sun Sep 22, 2013 4:35 am 
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Bram wrote:
That is... mysterious.
How do those numbers break down if you include only edges or only corners?

I worked half an hour to find answer and then found the mistake in my original calculation.

231158159769600 is the number of permutations for (UD, L, Ls, Fs)
This means U and D can only be turned clockwise by the same time => a move like in the Gear Cube.

The "Incomprehensible Cube" is (UD', L, Ls, Fs) and has 1966000148840448000 permutations like (U, D, L, Ls, Fs).


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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Sun Sep 22, 2013 5:56 am 
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DLitwin wrote:
At TwistyPuzzles.com our Museum moderator comprehends the incomprehensible.

Dave
...
Our great Museum moderator comprehends things that remain incomprehensible to me.

I cannot understand the following:
Andreas Nortmann wrote:
... The "Incomprehensible Cube" is (UD', L, Ls, Fs) and has 1966000148840448000 permutations like (U, D, L, Ls, Fs).
I didn't search for this notation, I just try to interpret what I'm reading here.
The subgroup of the Incomprehensible Cube can be generated if I restrict myself to use the set of quarter turns {UD', U'D, L, L',Ls, Ls', Fs, Fs'} where UD' is a combined move of U and D' and U and D' can never be done independently. Ls is the inner slice between L and R and Fs is the inner layer between F and B.

If this interpretation is correct, how can (U, D, L, Ls, Fs) generate the same subgroup? In the latter case U and D can be turned independently. So I could do [U D] on a solved cube and reach a state that cannot be reached by combined UD' or U'D turns. At least I do not comprehend how such a state would be possible.
What am I missing here?

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Sun Sep 22, 2013 9:42 am 
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Andreas Nortmann wrote:
231158159769600 is the number of permutations for (UD, L, Ls, Fs)
This means U and D can only be turned clockwise by the same time => a move like in the Gear Cube.

The "Incomprehensible Cube" is (UD', L, Ls, Fs) and has 1966000148840448000 permutations like (U, D, L, Ls, Fs).


Well that means that solving the incomprehensible cube probably isn't all that weird, but there's still something very strange and interesting and worth investigating further with that other one. What are the number of possible moves of (UD, Ls, Fs) and (U, D, Ls, Fs)? How about (UD, L, Fs) and (U, D, L, Fs)? Or ever the seemingly much less likely to be strange (UD, F, B, R, L) and (U, D, F, B, R, L)?


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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Mon Sep 23, 2013 8:36 am 
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I am not that great as my mistake prooves. Anyway:
Konrad wrote:
I didn't search for this notation, I just try to interpret what I'm reading here.
The subgroup of the Incomprehensible Cube can be generated if I restrict myself to use the set of quarter turns {UD', U'D, L, L',Ls, Ls', Fs, Fs'} where UD' is a combined move of U and D' and U and D' can never be done independently. Ls is the inner slice between L and R and Fs is the inner layer between F and B.
Correct so far.
As a side note: That means (U, D, Fs) is just a different notation of (U, D, Bs)
Konrad wrote:
If this interpretation is correct, how can (U, D, L, Ls, Fs) generate the same subgroup? In the latter case U and D can be turned independently. So I could do [U D] on a solved cube and reach a state that cannot be reached by combined UD' or U'D turns. At least I do not comprehend how such a state would be possible.
What am I missing here?
The state you reach by [U D] on the more allowing cube can be reached by the more restricted one. You just have to go through a detour.
You might want to take a look into this often ignored site of Jaap:
http://www.jaapsch.net/puzzles/subgroup.htm
Below the first table you can read this equivalence:
D = F2R2D2F2U2R2F2 U F2R2U2F2D2R2F2
Thereby Jaap prooves that the subgroup (U, D, F, B, R, L) is identical to (D, F, B, R, L)
EDIT: Jaaps equivalence prooves something else but U can still be emulated with (D, F, B, R, L)

All this holds true only if you ignore orientation of faces.
If you include orientation of faces then Jaaps example does not work.
If you include orientation of faces then GAP says (UD', L, Ls, Fs) and (U, D, L, Ls, Fs) have still identical number of permutations => There must be a sequence which emulates [U] just with (UD', L, Ls, Fs).
Bram wrote:
What are the number of possible moves of (UD, Ls, Fs) and (U, D, Ls, Fs)? How about (UD, L, Fs) and (U, D, L, Fs)? Or ever the seemingly much less likely to be strange (UD, F, B, R, L) and (U, D, F, B, R, L)?
(UD, Ls, Fs) => 27648 permutations
(U, D, Ls, Fs) => 990904320 permutations
(UD, L, Fs) => 38526359961600 permutations
(U, D, L, Fs) => 327666691473408000 permutations
(UD, F, B, R, L) is identical to (U, D, F, B, R, L) which should not surprise after Jaaps equivalence.

Bram: Do you want to have the GAP-code? For calculating the group sizes it is fairly simple.


Last edited by Andreas Nortmann on Tue Sep 24, 2013 9:30 am, edited 1 time in total.

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Mon Sep 23, 2013 11:07 am 
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Andreas Nortmann wrote:
(UD, Ls, Fs) => 27648 permutations
(U, D, Ls, Fs) => 990904320 permutations
(UD, L, Fs) => 38526359961600 permutations
(U, D, L, Fs) => 327666691473408000 permutations


Well that's very interesting.

327666691473408000 / 38526359961600 = 8505 = 3^5 * 5 * 7, so the second case preserves the extremely weird factor we observed earlier. Am I correct in assuming that the size of (UD, R, L, Fs) is the same? How does (UD, R, L) compare? A funny note about that second one - that subgroup is used in the supergroup sequence (RLU2R-L-U)2 which rotates one face center 180 degrees.

I still have no idea where 8505 comes from. Why 3^5 when there are 8 corners? Where does the 5*7 come from?

There's a little bit of a hint in the first case. 38526359961600 / 990904320 = 35840 = 2^10 * 5 * 7, so we still have the 5*7 but the lower number is different, so apparently they're separate things. The number 990904320 is also very strange. It's 2^20*3^3*5*7. There are 12 edges but the number 11 somehow doesn't show up in the number of positions. Does (UD Fs) have nontrivial structure? I'm guessing that one is similar and it's easy to build, so it would be interesting to experiment with.

Then of course there are the flipping axes variants. For example the other variants of (UD Fs) are (UsU2 Fs) (UD FB-) and (UsU2 FB-).

Andreas Nortmann wrote:
Bram: Do you want to have the GAP-code? For calculating the group sizes it is fairly simple.

What's required to run it?


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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Mon Sep 23, 2013 12:31 pm 
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Bram wrote:
I still have no idea where 8505 comes from. Why 3^5 when there are 8 corners? Where does the 5*7 come from?

I'm not sure either but it seems you're assuming the factors of 3 came from corner twists. Another possibility is that the 5, 7, and some of the 3s came from a factor of 7! or 8!. Perhaps in the smaller group the corner permutation is fixed (or maybe some of the edge permutation?) relative to the other pieces but in the larger group the corners (or some edges) are freed up.

8505 could be (3^3 * 8!) / 2^7 but even if it is, there is still some mystery lurking.

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Mon Sep 23, 2013 8:00 pm 
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That's really amazing to be able to even see the central rod in there. Crazy design, and well named!

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Tue Sep 24, 2013 12:47 am 
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For solving, you can do this one by first solving the corners using things like UD-LU-D to basically do F for the corners, which basically makes it like solving the corners using the (R L F B) subgroup which is fairly normal, then solve the edges using the (L Fs) subgroup. The edges on the left-right slice are slightly annoying to deal with, but that covers the essential problems.


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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Tue Sep 24, 2013 5:56 am 
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I don't get it, so I must be doing something right.

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Tue Sep 24, 2013 7:50 am 
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Andreas Nortmann wrote:
... The state you reach by [U D] on the more allowing cube can be reached by the more restricted one. You just have to go through a detour.
You might want to take a look into this often ignored site of Jaap:
http://www.jaapsch.net/puzzles/subgroup.htm
Below the first table you can read this equivalence:
D = F2R2D2F2U2R2F2 U F2R2U2F2D2R2F2
Thereby Jaap prooves that the subgroup (U, D, F, B, R, L) is identical to (D, F, B, R, L)

....
Bram: Do you want to have the GAP-code? For calculating the group sizes it is fairly simple.
Thank you Andreas. :D I find it still surprising that the two subgroups can reach the identical number of permutations (states).

What would I need to run GAP? (I run Windows 7 or 8 on my computers, but I have not got any compiler environments.)

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 Post subject: Re: Incomprehensible Cube by OSKAR
PostPosted: Tue Sep 24, 2013 9:38 am 
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Continuation here:
viewtopic.php?f=1&t=26081


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