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 Post subject: V-Cube 6 Pattern Request
PostPosted: Sat Jan 17, 2009 12:47 pm 
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Hey all. I know a few of you have the knack for solving cubes into patterns, so I would like to request one. Has anyone got a V-6 into a position where each side is a Latin square with all six colors? (Of course Latin squares could be done on any cube but this is the only one where each side contains one of every color on the cube.)


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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Sun Jan 18, 2009 3:59 am 
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Sorry, but what is a Latin square again?? :oops:

I don't know if this helps but it is actually very easy to determine if a pattern is possible or not on a 6x6x6 in particular: The only thing you have to check is the corners: If the corners are in a reachable position (and ofcourse all other pieces are there and accounted for) then the rest of the cube is guaranteed to be reachable.

(Also I'm trying to bump this up to the top of the list because I hate seeing a topic get shoved down so quickly with 0 responses :wink: )

Peace


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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Sun Jan 18, 2009 4:16 am 
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Allagem wrote:
Sorry, but what is a Latin square again?? :oops:


What is a latin square? ;-)

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Sun Jan 18, 2009 4:30 am 
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TomZ, that is the most amazing thing I have seen in a while.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Sun Jan 18, 2009 8:05 am 
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I was able to find a valid corner position and I'm sure there are more so I don't see why this position can't be achieved, it's more a matter of putting in the time.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Sun Jan 18, 2009 12:36 pm 
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Tomz thats amazing. Do you want one side or the whole cube.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Sun Jan 18, 2009 12:50 pm 
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How did you do that Tomz?

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Sun Jan 18, 2009 12:55 pm 
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pierce wrote:
How did you do that Tomz?

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Sun Jan 18, 2009 12:56 pm 
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Bigandrewgold wrote:
pierce wrote:
How did you do that Tomz?

http://letmegooglethatforyou.com/
http://lmgtfy.com/


oh ok thanks Bigandrewgold
:)

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Wed Mar 04, 2009 3:29 am 
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Allagem wrote:
I don't know if this helps but it is actually very easy to determine if a pattern is possible or not on a 6x6x6 in particular: The only thing you have to check is the corners: If the corners are in a reachable position (and ofcourse all other pieces are there and accounted for) then the rest of the cube is guaranteed to be reachable.


I think I know what you meant, but you've accidentally oversimplified the situation. Once you've determined exactly where you want to group the parts on a 6x6x6 your method will determine if that grouping is possible. But for most cube patterns (on all cube sizes) the hardest part is establishing whether a valid group exists for the desired pattern. (The cube is a closed system.) I think this fits into the topological groups section in Wikipedia's Group Theory article.

Here's an oversimplified example of what I mean. The pattern illustrated below is actually impossible on a 6x6x6 even though the corners are perfectly feasible and it has exactly the right combination of cubie-faces of each color and type. (The picture is fake.) If you study it, you'll quickly see several reasons it's impossible.
Image
Attachment:
File comment: Important! This is a FAKE pattern.
6x Impossible.JPG
6x Impossible.JPG [ 77.23 KiB | Viewed 3991 times ]


Offhand I'm undecided whether the Latin Square pattern is possible, but I'm leaning toward "impossible." But I can't decide how to express it rigorously enough to test whether it's possible or not. (IE: how to formulate a grouping formula that can be tested)

This could be important because it underlies the basic challenge for all pattern making. Is there an application of Group Theory that would help express this? Should it be algebraic or topological in nature? Does anyone know if this has already been resolved for cubes??

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Wed Mar 04, 2009 4:53 am 
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I lean towards the possibility of such (a) latin squares pattern(s). However i strongly doubt that any such pattern would show any kind of beauty or striking features. So for me - i simply dont see it worthwhile to look for them, far less try to sequence them :|

Per

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Wed Mar 04, 2009 6:16 am 
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this pattern is actually kinda cool on a 3x3 (I've managed 3 sides - a corner)


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File comment: 3 sides with pattern on 3x3
latinsqcube.jpg
latinsqcube.jpg [ 164.19 KiB | Viewed 3434 times ]
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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Wed Mar 04, 2009 4:35 pm 
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A Latin square pattern ought to be possible simply because there are so many possible 6x6 latin squares. However, it probably wouldn't look very pretty. So I'm not gonna try to find one :P

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Thu Mar 05, 2009 12:10 pm 
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As long as we're talking about this again, it might be useful to know that any position is reachable on a 6x6x6 provided the corners alone are reachable. So, find a Latin square pattern where just the corners check out, and the rest is possible guaranteed (if of course all the pieces used are valid pieces, no Y/Y edges!!! :wink: )

Peace


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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Thu Mar 05, 2009 8:51 pm 
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Allagem wrote:
As long as we're talking about this again, it might be useful to know that any position is reachable on a 6x6x6 provided the corners alone are reachable. So, find a Latin square pattern where just the corners check out, and the rest is possible guaranteed (if of course all the pieces used are valid pieces, no Y/Y edges!!! :wink: )

Yeah, my photo was an bad example because it used fictitious pieces. (Y/Y among them.)

But I still suggest you're grossly understating the scale of the challenge. Any proposed pattern must simultaneously accommodate all the constraints imposed on every single part, not just the corners. I admit the corners impose the toughest restrictions, but all the parts have some constraints. The Latin Square premise doesn't allow any slack, so constraints that would ordinarily seem trivial become suddenly restrictive.

Here's what I mean by that. If you wrote letters on each fact of a 6x6 according to the diagram below, the cube would have exactly 24 instances of each letter. (Four on each face, as shown.) No matter how you turn the cube, two different letters cannot exchange places. You cannot move a square labeled F into a square labeled H even though they're both inner-face pieces. After every move the letters on each face would remain in exactly the same positions.
Attachment:
6x6 letter labels.GIF
6x6 letter labels.GIF [ 3.09 KiB | Viewed 4251 times ]

If you aren't convinced, think about disassembling a 6x6 and trying to swap F and H by hand. The pieces won't even fit into each other's positions...they're physically incompatible!

If the Latin Square is possible, I doubt it could be a random or messy pattern like qqref's comment seems to imply. If the pieces aren't meticulously organized, one or more faces would undoubtedly break the rules of a Latin Square. I think the Latin Square could only be achieved with a very carefully ordered arrangement, like the one Disjunto showed, to carefully satisfy all the requirements.

I think it's still worth looking, though.

To make up for the fake photo I posted earlier, here's a sketch of an impossible pattern using real pieces from the red, green, and yellow sides of a 6x6. The corners and edges are solvable, the total number of pieces is strictly correct, but it doesn't work because the middle-face cubies can't move between these positions.


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6x6 different impossible.GIF
6x6 different impossible.GIF [ 6.86 KiB | Viewed 4243 times ]
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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Thu Mar 05, 2009 9:52 pm 
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Still not so much real pieces. You have 5 green "I" pieces (in accordance with your letter diagram), 6 yellow, and 1 red. And then you have 5 red "E" pieces and 3 green. But if all you were trying to show is that there are 4 separate types of center pieces, you have done that.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Thu Mar 05, 2009 10:39 pm 
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Could you show a picture of a Latin Square on a 5x5x5?

I'm just not convinced its possible to do passed 4. Plus, if you can do it with 6, you should be able to do it with 5.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Thu Mar 05, 2009 10:45 pm 
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That's kinda what I meant by VALID pieces.... :?

But my point was that if you have the right number of the right types of pieces in valid locations, there is no position (excluding corners) that a 6x6x6 can be physically assembled into that it can't be solved into. There is that better?? :P

If you want an example of what I mean, take apart your 6x6x6 and reassemble it scrambled. As long as the corners are in a legal state, the rest of the puzzle is guaranteed to be in a legal state.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Thu Mar 05, 2009 11:15 pm 
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I found something that I think could be used to solve the problem.
Suppose you have the 6x6 so that on each side the four 3x3 blocks at the corners are each Latin squares of one of two sets of three colors (say, URF and BLD). Each face should only have three colors... so any block from U, R, and F will contain a Latin square of the U, R, and F colors (and similar for B, L, and D).

Then do this 2x2 maneuver:
y2 F R U2 F2 R' F U R' U2 R' F'

This 2x2 maneuver puts each face into a 'checkerboard' pattern where each face has a 2x2 grid of alternating 3x3 blocks from U, R, or F and B, L, or D. So if you can somehow get the 6x6 properly 'blocked out', you could do the 2x2 pattern and get a 6-color latin square on each side (instead of four separate 3-color ones all on the same 3 colors). Hope that made sense.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Fri Mar 06, 2009 3:14 am 
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Allagem wrote:
That's kinda what I meant by VALID pieces.... :?

But my point was that if you have the right number of the right types of pieces in valid locations, there is no position (excluding corners) that a 6x6x6 can be physically assembled into that it can't be solved into. There is that better?? :P
You know, I think we're just arguing the same side of the same issue for different (or irrelevant) reasons.

What you'd said was certainly true, but I just couldn't see how it would help me work toward a solution. But in hindsight that's probably because I approach pattern making (and solving) in an unconventional way. I could only try your test after I'd already solved enough of the pattern to know the corners were valid, so the test seemed redundant. If I tried to use it before solving the rest of the cube, the test seemed to risk confirming 'impossible' patterns as valid. Either way it didn't help me.

I'll bet your suggestion would've been more readily accepted by a typical cuber. I never learned any conventional algorithms.

qqwref wrote:
I found something that I think could be used to solve the problem.
Suppose you have the 6x6 so that on each side the four 3x3 blocks at the corners are each Latin squares of one of two sets of three colors (say, URF and BLD). Each face should only have three colors... so any block from U, R, and F will contain a Latin square of the U, R, and F colors (and similar for B, L, and D).

Then do this 2x2 maneuver:
y2 F R U2 F2 R' F U R' U2 R' F'

This 2x2 maneuver puts each face into a 'checkerboard' pattern where each face has a 2x2 grid of alternating 3x3 blocks from U, R, or F and B, L, or D. So if you can somehow get the 6x6 properly 'blocked out', you could do the 2x2 pattern and get a 6-color latin square on each side (instead of four separate 3-color ones all on the same 3 colors). Hope that made sense.

It took a little re-reading to understand what you meant with the 'checkerboard' thing. That's a very clever idea, checkerboarding between two different tri-color groups instead of checkerboarding two distinct colors. It effectively relaxes the ordinary corner restrictions because you can freely rotate two of the corners in order to get the ideal orientation of the other six. Clever.

There's still a challenge of getting coherent tri-color Latin Squares on four quadrants of each face, but that's a hugely simpler problem.

I don't recognize the 'y2' move though. Is that some non-2x2 move or a typo?


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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Fri Mar 06, 2009 3:34 am 
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VeryWetPaint wrote:
I don't recognize the 'y2' move though. Is that some non-2x2 move or a typo?
wca notation (specifically, rule 12b3)

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Fri Mar 06, 2009 5:32 am 
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Someone could probably write a simple (?) program to try to search for the possibility of a latin square 6x6x6 "pretty" pattern. No need to sequence it. Just show that it is a possible state (quite trivial for even sized cubes).

Again - i think they should exist. But the most striking pattern feature - same color lines - is out of the question. I would call it a pattern only if the 6 faces show some kind of similarity ...

Per

PS! I would call this a soduko pattern ;-)

PPS! It would probably suffice to prove the possibility for setting up the edges and corners. It's trivially possible (?) to set up the centers i guess.

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Last edited by perfredlund on Fri Mar 06, 2009 6:24 am, edited 1 time in total.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Fri Mar 06, 2009 6:16 am 
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perfredlund wrote:
Someone could probably write a simple (?) program to try to search for the possibility of a latin square 6x6x6 "pretty" pattern. No need to sequence it. Just show that it is a possible state (quite trivial for even sized cubes).
Maybe I'm the only one... but I think it's no fun to have a pattern and no sequence :P

Well! I think I found it. Here's the 4-3x3-latin-squares pattern (you can find a sequence by basically treating it as a 5x5):
Image
And here - drum roll please! - is the finished latin square pattern:
Image
I think it's kind of pretty, actually. I assure you this is possible, because I have it on my 6x6 right now :)

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Fri Mar 06, 2009 6:30 am 
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Hi :-)

I can see that you first do a "5x5x5" pattern to achieve the first diagram. Then you treat it as a 2x2x2 cube to go to the final latin squares pattern, right?

Can you show exactly the "pattern" you do to go from 1st to 2nd diagram?? Yes i could find it out, but im lazy today :twisted:

Per

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Fri Mar 06, 2009 5:48 pm 
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It's just the 2x2 maneuver I mentioned above; perform the moves with three-layer turns.

The whole move sequence (unoptimized) is
F R' B' D2 B2 F' D2 L D' F R' F' U' F' R' B R U2
TF TR' TB' TD2 TB2 TF' TD2 TL TD' TF TR' TF' TU' TF' TR' TB TR TU2
T3B T3L T3U2 T3B2 T3L' T3B T3U T3L' T3U2 T3L' T3B' (47 btm)
And here's the whole thing in an applet.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Mon Mar 09, 2009 5:00 am 
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Gotcha!!!

Only the 3 visible faces have the latin square feature. The other 3 faces DO NOT!!
Nice try though :)

Please keep looking :roll:

Per

PS!! Hmm. Now i see that ur sequencing is indeed correct. Why it came up incorrect first time?? :oops: Nice work, how you came up with the idea ??

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Mon Mar 09, 2009 5:06 am 
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Wait, can you show me where the other 3 faces don't have the latin square feature? On the applet it looks to me like all faces are good.

EDIT: Hah, never mind. Yeah, so, I was thinking that it's hard to construct a 6x6 latin square from scratch (using numbers), but that you can make one by splitting 1,2,3 and 4,5,6 into two groups like that (you can do any even latin square like this, it's a nice trick), and making 2x2 and 3x3 latin squares is extremely simple. So then I was wondering if there was a nice 2x2 pattern that would make a 6x6 latin square out of only 3x3 ones (that is, a 2x2 pattern that had a checkerboard on each face if you consider U,R,F and D,L,B the same color), and after a bit of fiddling I found it (yay!) so I knew I could make a latin square this way. The last part was to find the 5x5 pattern that made the 3x3 latin squares, but after making a picture I realized it was basically trivial, just the same 3x3 pattern twice, so there you go :)

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Last edited by qqwref on Mon Mar 09, 2009 5:09 am, edited 1 time in total.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Mon Mar 09, 2009 5:08 am 
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I corrected my post - weird mistake. When i first opened the applet the 3 hidden faces were not latin squares :mrgreen:

Per

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Mon Mar 09, 2009 5:15 am 
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VeryWetPaint wrote:
Allagem wrote:
As long as we're talking about this again, it might be useful to know that any position is reachable on a 6x6x6 provided the corners alone are reachable. So, find a Latin square pattern where just the corners check out, and the rest is possible guaranteed (if of course all the pieces used are valid pieces, no Y/Y edges!!! :wink: )

Yeah, my photo was an bad example because it used fictitious pieces. (Y/Y among them.)

But I still suggest you're grossly understating the scale of the challenge. Any proposed pattern must simultaneously accommodate all the constraints imposed on every single part, not just the corners. I admit the corners impose the toughest restrictions, but all the parts have some constraints. The Latin Square premise doesn't allow any slack, so constraints that would ordinarily seem trivial become suddenly restrictive.

Here's what I mean by that. If you wrote letters on each fact of a 6x6 according to the diagram below, the cube would have exactly 24 instances of each letter. (Four on each face, as shown.) No matter how you turn the cube, two different letters cannot exchange places. You cannot move a square labeled F into a square labeled H even though they're both inner-face pieces. After every move the letters on each face would remain in exactly the same positions.
Attachment:
6x6 letter labels.GIF

If you aren't convinced, think about disassembling a 6x6 and trying to swap F and H by hand. The pieces won't even fit into each other's positions...they're physically incompatible!

If the Latin Square is possible, I doubt it could be a random or messy pattern like qqref's comment seems to imply. If the pieces aren't meticulously organized, one or more faces would undoubtedly break the rules of a Latin Square. I think the Latin Square could only be achieved with a very carefully ordered arrangement, like the one Disjunto showed, to carefully satisfy all the requirements.

I think it's still worth looking, though.

To make up for the fake photo I posted earlier, here's a sketch of an impossible pattern using real pieces from the red, green, and yellow sides of a 6x6. The corners and edges are solvable, the total number of pieces is strictly correct, but it doesn't work because the middle-face cubies can't move between these positions.


No cubie can cross its orbital!! :o

Per

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Mon Mar 09, 2009 6:38 am 
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You're saying you could make a 8x8x8 pattern with latin squares also ? Hmmm :twisted: You could of course just simulate a 6x6x6 cube :wink:

Per

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Mon Mar 09, 2009 1:18 pm 
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Well, no, unless you can show me an 8x8 with 8 colors :) (Actually you should probably show me an 8x8 first :D) But if I wanted to construct an 8x8 Latin Square that's pretty much how I'd do it.

P.S. You know the stickering style where every octant has one color? I think it might be possible to do an 8x8 latin square pattern with that. It seems like there are the right number of edges/corners for it to work out, at least.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Mon Mar 09, 2009 1:21 pm 
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Hmm... would an 8x8x8x8 hypercube be able to twist into a pattern with a latin cube on each face? :P


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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Tue Mar 10, 2009 7:54 am 
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Jared, with your obsession for latin squares, go play soduko instead :-P
I also always found soduko very dull. I acknowledge the difficulties in solving soduko - but it doesn't tickle any urge in me ... I always passed when i saw them in magazines and newspapers. I dont understand the craze about it :|

Per

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Tue Mar 10, 2009 12:33 pm 
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First of all, it's "sudoku". :P

Second, incidentally, I do do those every so often, but lately I've been getting into other pencil puzzles too. I have a few books that I ordered from Nikoli's Japanese website that I'm working on, none of which include sudoku but all of which are full of brilliant and challenging puzzle designs. And I like word puzzle magazines too! I have almost as many Penny Press magazines as I do twisty puzzles, I think. But this is getting off-topic.


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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Tue Mar 10, 2009 2:12 pm 
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Hah, I used to do sudoku too, but I got bored of the standard 'nine 3x3 boxes' one pretty quickly (I also don't understand why people spend months playing with those, unless they are practicing to complete it as fast as possible). What I was interested in was the variations, the ones with weird-shaped or extra boxes, or new rules, or marked subsets that had to add or multiply to a certain number, or whatever it was (I guess you could call those the custom mods of sudoku :P). I even ended up making a few puzzles and a rudimentary solver program.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Tue Mar 10, 2009 9:56 pm 
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qqwref wrote:
Hah, I used to do sudoku too, but I got bored of the standard 'nine 3x3 boxes' one pretty quickly (I also don't understand why people spend months playing with those, unless they are practicing to complete it as fast as possible). What I was interested in was the variations, the ones with weird-shaped or extra boxes, or new rules, or marked subsets that had to add or multiply to a certain number, or whatever it was (I guess you could call those the custom mods of sudoku :P). I even ended up making a few puzzles and a rudimentary solver program.

Once the 9x9x9 comes out, you can make a sliding puzzle alike the 15 puzzle, only the solution is Sudoku. I always wanted to do that. It'd be so hard, though there wouldn't be one fixed solution. But that makes it even harder. Try a normal 80-puzzle, then imagine 6 intermixable ones.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Wed Mar 11, 2009 3:36 am 
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I don't think it'd be THAT hard actually, there are so many equivalent pieces that setting it up would be a breeze.

Unfortunately you can't make a perfect cube 9x9, so it's not really feasable to put a sliding puzzle on the side of it. Not only will all the pieces not be the same size, but the face itself will be curved, so you'd have a really hard time getting the pieces to slide properly.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Wed Mar 11, 2009 9:16 pm 
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The challenge wouldn't be putting the right pieces on the right faces in the correct orientation, but there are millions of positions on a 80-puzzle that are not possible. Extra parity cases! But that's not all. There are 6 faces, each with 80 pieces. That means you must order them correctly facing the correct direction and maybe they are in an impossible order. Very time consuming. If you have not made a 4x4x4 with a 15 puzzle on every face, or have never tried one, don't question the might of the 80 puzzle.

I know the pieces are not the same, so I guess it would be far easier to make a program that would make this kind of puzzle. But it is possible to make it from a pillowed 9x9x9.

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Thu Mar 12, 2009 5:15 am 
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Rentlix wrote:
Very time consuming. If you have not made a 4x4x4 with a 15 puzzle on every face, or have never tried one, don't question the might of the 80 puzzle.

I know the pieces are not the same, so I guess it would be far easier to make a program that would make this kind of puzzle. But it is possible to make it from a pillowed 9x9x9.


Heh, has anyone actually made a 4x4 with a 15 puzzle on each face? I thought that was just an idea that nobody put into practice yet. I think the 80 twisty puzzle would definitely be VERY time consuming (especially if each face had numbers of a different color) but solving it really shouldn't be out of reach if you can already do a 9x9 supercube and an 80 puzzle. Seems like it would just take forever :roll:

Are you sure it's possible to make it from a pillowed 9x9, though? I've never even seen one...

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 Post subject: Re: V-Cube 6 Pattern Request
PostPosted: Fri Mar 13, 2009 6:40 pm 
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qqwref wrote:
Rentlix wrote:
Very time consuming. If you have not made a 4x4x4 with a 15 puzzle on every face, or have never tried one, don't question the might of the 80 puzzle.

I know the pieces are not the same, so I guess it would be far easier to make a program that would make this kind of puzzle. But it is possible to make it from a pillowed 9x9x9.


Heh, has anyone actually made a 4x4 with a 15 puzzle on each face? I thought that was just an idea that nobody put into practice yet. I think the 80 twisty puzzle would definitely be VERY time consuming (especially if each face had numbers of a different color) but solving it really shouldn't be out of reach if you can already do a 9x9 supercube and an 80 puzzle. Seems like it would just take forever :roll:

Are you sure it's possible to make it from a pillowed 9x9, though? I've never even seen one...

I have been making a 15 puzzle on every face of a 4x4x4. The process has been going slowly since I rarely put the time into it. I have made the grooves, but not the attachment number panels. As far as the pillowing, it could prove difficult to build. I'm sure someone who really tried could make it from a pillowed cube, but that would take serious modding skills.

Yes, the point is the extensive time to solve. There isn't much of a challenge unless you are trying to PR.

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