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 Post subject: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Wed Dec 09, 2009 12:47 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
I'm moving this discussion from here:

http://twistypuzzles.com/forum/viewtopic.php?f=15&t=15591

bhearn wrote:
wwwmwww wrote:
But combine a Skewb and a 24-Cube and it looks like there IS a new jumbling move half-way though a Skewb turn that just involves the 24-Cube cuts

Yes indeed! But it's hard for me to visualize whether there would be any places to stop in that new rotation that enabled any further moves. Anyway, what a monster puzzle that would be.

I had to go dig my Skewb out again... Yes, there is one. Take a Skewb/24-Cube and make half a Skewb turn. From there you can turn the puzzle 180 degrees along the 24-Cube cuts. And from there you can complete the original Skewb turn. To see this pick up a Skewb and make a half turn. Now look at the plane you just turned along and the plane formed by the 24-Cube cuts you want to turn along. Ignore all other cuts. At this point you can think of the puzzle as a shape mod of a 2x2x1. So you can make many 180 degree turns before finishing the original Skewb turn. I find it interesting that the combination of an edge turn puzzle and a corner turn puzzle contains a 2x2x1 which is a face turn puzzle.

You know what I think this means? I think you could make a skewb+2x2x2+little chop[24-cube] with 6+4+1 or 11 axis of rotation and not 13. Take the Skewb/24-Cube make a half Skewb turn. Mod it back into a cube and add the one additional cut needed to turn a 2x2x1 into a 2x2x2 and you'd have a 2x2x2 plus shape mods of a Skewb and a 24-Cube all in the same puzzle. I can't really picture what it would look like. If I can figure it out I'll try to make a picture of it in POV-Ray and move this over to its own thread. Nope... I'll do that now.

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Thu Dec 10, 2009 3:05 pm

Joined: Sun Oct 08, 2006 1:47 pm
Location: Houston/San Antonio, Texas
Hey Carl, I wish I had time to follow all the fantastic journeys you take. I'm not really sure what you mean by the 11 axis thing, and don't exactly have time to figure it out since I have finals starting tomorrow.

But I can tell where ALL the jumbling moves are, the one you identified I listed as number1:

1)when it is rotated like a skewb 60 degrees, yes the 6 little chop rotations across the rotated plane can all move again, making this rotation jumbleable.

2)when it is rotated like a 2x2 45 degrees, both the 2x2 and little chop rotations across the rotated plane can rotate again, making this move jumbleable as well.

3)when it is rotated like a little chop approximately 71 or 109 degrees, 2 of the closest 4 little chop rotations can move again (same as little chop/helicopter cube/etc) AND

4) at the same angle,2 of the 4 skewb rotations accros the rotation plane can be rotated again making this move jumbleable twice (I bet you didn't know about this one! )

5)when it is rotated like a little chop 90 degrees, the 2 2x2 rotations and the 2 little chop rotations accros the rotation plane can all be rotated again, making this another jumbleable little chop move

No time to explain now but I think 3 and 4 should be counted seperately as they are jumbleable at the same spot for "different" reasons. Anyway this makes a total of 5 unique jumbleable occurences.

(That's some beast of a puzzle. I would be scared to solve that....... )

I'm sure you'll come up with some profound discovery about all this soon, Carl! I'll be sure to check back again soon

Peace,
Matt Galla

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Thu Dec 10, 2009 10:39 pm

Joined: Tue Jun 26, 2007 2:59 pm
Location: Houston, TX, USA
Oh my god... This is actually one puzzle I haven't even solved on computer yet. I don't fully understand jumbleability, I think I would need to physically use a bevel or little chop to understand it, but this puzzle sounds ridiculous.

I will be extremely impressed when we hybridize just the 2x2x2 and Little Chop. I solved that one because it's a uniform dihybrid, it was pretty fun (but of course no jumbling because it was on computer). I think there are definitely people on this forum who can put a 3x3x3 mechanism at the center of a bevel cube (assuming that will work?), but then that makes building a good little chop around it all the harder.

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Fri Dec 11, 2009 6:21 pm

Joined: Sun Mar 11, 2007 3:11 am
Location: Oregon, USA
I explored a mechanism for this a while back, and put it in my notebook under the tentative name "Deep 13". (A cross-reference between the number of axes and Mystery Science Theater 3000.)

None of the pieces would be attached to the core. The core would be a simple sphere, pushing the pieces outward, while the pieces would all interlock around the core to squeeze inward. A huge variety of puzzles could be built on this premise, but it probably would be impractical and unstable. If a single piece popped loose, thereby releasing the tension that held the puzzle together, the whole puzzle would come apart.

Still, here's the part count I came up with:

• 1 = core (sphere)
• 96 = exterior pieces
• 24 = internal pieces to hold the puzzle together on each 2x2x2 axis
• 48 = internal pieces to hold the puzzle together on each Skewb axis
• 72 = internal pieces to hold the puzzle together on each Little Chop axis
• 50 = internal spacers at the intersections between each axis
Total: 291 pieces (14 unique shapes)

It's a novel and totally-unproven mechanism, so I wouldn't even attempt this until I'd tried the concept separately on a each of the individual puzzles: 2x2x2, Skewb, and Little Chop. If it doesn't work for each of those individual puzzles then it certainly wouldn't work as a combination!

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Fri Dec 11, 2009 9:12 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Ok... I've now got a visualization of the puzzle I was thinking of in the first post. Take a Skewb+Little Chop(24-Cube) and make a 60 degree turn on one of the Skewb planes.

The blue line is the intersection of the plane you just turned along and one of the 24-Cube cut planes that is perpendicular to it. The Skewb has 4 axes of rotation. The 24-Cube has 6 axes of rotation. So it you make this new blue axis an axis of rotation too you will now have 11 axes of rotation.

So I add in the blue cut plane. Now mod this shape into a cube and we get.

If you complete the original Skewb turn on this new puzzle we get.

You now have a fully functional 2x2x2, a fully functional Skewb, and a fully functional 24-Cube in one puzzle with only 11 axes of rotation. By the way... this isn't the only puzzle with 11 axes that contains these 3 puzzles. There is atleast 1 other that I'm rendering now.

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Fri Dec 11, 2009 9:49 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Ok... the above puzzle made me think of this post:

http://twistypuzzles.com/forum/viewtopic.php?p=125389#p125389

Can a puzzle have more then one rest-state? I think this one may... it has a state where its a Skewb and a 24-Cube. And one jumbling move away, its in a state where it is a 2x2x2. It makes me wonder just what is considered jumbling on this puzzle and what isn't? In fact... if that extra cut complicates things too much let's ignore it for now.

Let's just look at the Skewb plus 24-Cube. It's not obvious but just putting these 2 puzzles together automatically gives you a 3rd. I'm not sure what the name of it is but its a Rubik's Cheese with a horizontal plane that cuts through the other 3. Where is this other puzze? It's actually two places. You can rotate the puzzle by 60 degrees along one of the Skewb planes or you can rotate the puzzle along one of the 24-Cube planes by acos(1/3). In both of these states you have a Rubik's Cheese with a horizontal cut to play with. These cuts are all complete and I'm wondering if its really fair to call these jumbling moves. If it were really a Cheese they wouldn't be. You can see the first cheese state in the first picture of the post above. Here is the other.

The red planes are Skewb planes and the others are the 24-Cube. The complete red plane is the jumbling type #4 that Allagem pointed out and I made this image just so I could see it. But that red complete arc is the horizontal* cut through a Rubik's Cheese too. So... is it jumbling? I honestly don't know. I tend to think of jumbling moves as bandaged to some degree and the more you make the more they limit future turns. But there is nothing bandaged about these cheese puzzles.

*Well it's not horizontal in this picture. Pretend the intersection of the three 24-Cube planes you see on the left side of the image is the north pole. Those are your three cheese cuts.

Carl

P.S. Just noticed the 24-Cube by itself contains the normal Rubik's cheese puzzle without the horizontal cut plane and there I believe these moves are considered jumbling moves.

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Last edited by wwwmwww on Sat Dec 12, 2009 8:45 am, edited 1 time in total.

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Fri Dec 11, 2009 10:48 pm

Joined: Sat Sep 19, 2009 7:52 pm
Can someone explain this in terms a Pre-algebra student will understand?

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kastellorizo wrote:
Wow... a jumbable puzzle which doesn't jumble?
Is that even possible???

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Fri Dec 11, 2009 10:59 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Skewber wrote:
Can someone explain this in terms a Pre-algebra student will understand?

Pictures are the best tool I have. Here is a picture I just made of the 24-Cheese. Its a 24-Cube in it's Cheese state:

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Sat Dec 12, 2009 10:16 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
wwwmwww wrote:
There is atleast 1 other that I'm rendering now.

Here is the other 11 axes 2x2x2 + Skewb + Little Chop[24-Cube] combo.

In this case all 3 puzzles exist as the same rest state. However the 2x2x2 is a Fisher 2x2x2. If we wanted to we could make this shape mod which makes the 2x2x2 more apparent.

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Sat Dec 12, 2009 11:17 am

Joined: Sun Oct 08, 2006 1:47 pm
Location: Houston/San Antonio, Texas
Oh!
That's what I thought you were doing. It was very difficult to tell what was going on from your first example but I thought I understood and this puzzle confirms it.

You're realigning the axes of the 3 puzzles so that some of them line up. So what you've said is true, each of these puzzles is a 2x2, a skewb, and a little chop all at once..... but that doesn;t make either of these puzzles equivalent to what most people would consider a 2x2+Skewb+Little Chop puzzle - the one with all the standard cuts of these puzzles on a cube shape. The rotation axes have been given a new orientation and thus certain axes have different properties from the original (i.e. the ones that belong to more than one type of puzzle)

So everything you've done is legit (weird, but legit) but not THAT comparable to a normal 2x2+Skewb+Little Chop. Also by doing this you've modified the theoretical shape of the mathematical core (I know, it's just a point, but geometry can work at infinitely small sizes too) which further redefines things such as the "rest state" of the puzzle. In fact, you've changed some of the fundamental symmetries of the puzzle. Welcome to Bram's/Oskar's design world!!!!

Sorry I don't have much else to say, as I'm not really sure what to do about this very odd thing you've done. I just wanted to let you know that at least one person is actually managing to follow you on this twisted journey

Peace,
Matt Galla

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Sat Dec 12, 2009 11:49 am

Joined: Sun Aug 09, 2009 1:46 pm
Location: P.R.China
wwwmwww wrote:
You know what I think this means? I think you could make a skewb+2x2x2+little chop[24-cube] with 6+4+1 or 11 axis of rotation and not 13. Take the Skewb/24-Cube make a half Skewb turn. Mod it back into a cube and add the one additional cut needed to turn a 2x2x1 into a 2x2x2 and you'd have a 2x2x2 plus shape mods of a Skewb and a 24-Cube all in the same puzzle. I can't really picture what it would look like. If I can figure it out I'll try to make a picture of it in POV-Ray and move this over to its own thread. Nope... I'll do that now.

Carl I'm not very sure about your 11 axes claim until your latest update. It seems to me that because there are three pairs of perpendicular cuts in the 24 cube, then adding an extra perpendicular cut(e.g. one from a normal 222 cube) to one of these pair will make a 222 cube(in fisher cube style or whatever), is that right?

Now to reduce cuts as much as possible, rearrange a skewb cut to be the aforementioned extra cut, we have a 'badly' oriented axis system containing the axis sytem of a 222 cube(thus containing a shapemoded 222) without introducing any new cuts other than cuts from skewb and 24 cube. And we have 6+4=10 axes in total.

By thinking in this way, we are actually investigating the most compact combination of some standard axis system which will usually break the standard symmetry.

And I agree to Matt,
[quote="Allagem]
In fact, you've changed some of the fundamental symmetries of the puzzle.
[\quote]

Leslie

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Sat Dec 12, 2009 1:11 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Allagem wrote:
but that doesn;t make either of these puzzles equivalent to what most people would consider a 2x2+Skewb+Little Chop puzzle - the one with all the standard cuts of these puzzles on a cube shape.

True... I didn't mean to imply it was THE 2x2x2 + Skewb + Little Chop[24-Cube] puzzle. Just that there are apparently several 2x2x2 + Skewb + Little Chop[24-Cube] puzzles. These 11 axes ones are new... to me at least... and I find it odd that I didn't even see the second one (which seems much more obvious) until after I had rendered the first.

Allagem wrote:
...which further redefines things such as the "rest state" of the puzzle. In fact, you've changed some of the fundamental symmetries of the puzzle. Welcome to Bram's/Oskar's design world!!!!

Wow!!! What an honor. Pushing the definitions that we come up with here is one of the things I enjoy the most. Take a definition that everyone here seems to agree on and try to break it. Bram and Oskar do it everyday... now if I could just turn this into a something I could hold in my hands as fast as they can.

Allagem wrote:
I just wanted to let you know that at least one person is actually managing to follow you on this twisted journey

A Twisted Jorney with TwistyPuzzles. Sounds like the name of a book about the history of this site. I for one would buy a copy.

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Sat Dec 12, 2009 2:33 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Leslie Le wrote:
Carl I'm not very sure about your 11 axes claim until your latest update. It seems to me that because there are three pairs of perpendicular cuts in the 24 cube, then adding an extra perpendicular cut(e.g. one from a normal 222 cube) to one of these pair will make a 222 cube(in fisher cube style or whatever), is that right?

Correct.
Leslie Le wrote:
Now to reduce cuts as much as possible, rearrange a skewb cut to be the aforementioned extra cut, we have a 'badly' oriented axis system containing the axis sytem of a 222 cube(thus containing a shapemoded 222) without introducing any new cuts other than cuts from skewb and 24 cube. And we have 6+4=10 axes in total.

Very interesting... Yes you could do it with only 10 axes. Let's pick two of the 24-Cube cuts that are perpendicular to the x-z plane. Now put one of the Skewb cuts in the x-z plan and you get a 2x2x2 for free. One problem... that method doesn't fix the other Skewb cuts relative to the 24-Cube. Think of holding a Skewb by two opposite corners and spinning it. The other 3 cuts of the Skewb are free to rotate about the y-axis. Does this results in different 10-axes 2x2x2 + Skewb + Little Chop[24-Cube] combos? Looks like I'll have to render a few and see.
Leslie Le wrote:
By thinking in this way, we are actually investigating the most compact combination of some standard axis system which will usually break the standard symmetry.

And I agree to Matt,
Quote:
In fact, you've changed some of the fundamental symmetries of the puzzle.

That is the point... to look at something new.

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Sun Dec 13, 2009 11:17 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
wwwmwww wrote:
Very interesting... Yes you could do it with only 10 axes. Let's pick two of the 24-Cube cuts that are perpendicular to the x-z plane. Now put one of the Skewb cuts in the x-z plan and you get a 2x2x2 for free. One problem... that method doesn't fix the other Skewb cuts relative to the 24-Cube. Think of holding a Skewb by two opposite corners and spinning it. The other 3 cuts of the Skewb are free to rotate about the y-axis. Does this results in different 10-axes 2x2x2 + Skewb + Little Chop[24-Cube] combos? Looks like I'll have to render a few and see.

Ok... I've played around with this idea a bit and I think I've come up with a better/easier way to view these puzzles. Let me know if these pictures make it any easier to see/understand what is going on.

In the below pics:
The cube with the Red face is the 2x2x2.
The cube with the Blue Face is the 24-Cube aka Little Chop.
The cube with the Green Face is the Skewb.

In this 10-Axis puzzle I put an edge of the Skewb in the same plane as an edge of the 24-Cube.

In this 10-Axis puzzle I put an edge of the Skewb in the same plane as an edge of the 2x2x2.

Since there are many edges there is really only a 15 degree rotation that can be made before you start repeating yourself. Are the above two puzzles effectivly identical or did this rotation create new piece types or chance the type of jumbling moves available? I can't really tell. What if I rotated the Skewb by 7.5 degrees from either position such that none of the edges lined up?

Even if the above two puzzles aren't fundamentally different I know how to make one that is. How is this for a 10-Axis puzzle?

In this state it is a 2x2x2 and a 24-Cube.

In this state it is a Skewb.

For the pictures I made the two states 45 degress apart along a jumbling turn but any N*15degrees rotation could be used as long as N isn't a factor of 4. If N is a factor of 4 you are back to all 3 puzzles having the same rest state again.

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Sun Dec 13, 2009 4:35 pm

Joined: Sat Mar 29, 2008 12:55 am
Location: WA, USA
how many axes of rotation does the normal symmetric 24 cube+skewb+2x2 with the same rest state have?
Also, could you render that original one with its new jumbling moves as well?

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Sun Dec 13, 2009 5:50 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
elijah wrote:
how many axes of rotation does the normal symmetric 24 cube+skewb+2x2 with the same rest state have?

The 24-Cube has 6 axes of rotation. The Skewb has 4 axes of rotation. The 2x2x2 has 3 axes of rotation so the "normal" puzzle has 6+4+3 or 13 axes of rotation

elijah wrote:
Also, could you render that original one with its new jumbling moves as well?

As for what it looks like. It's here:
http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/hexa_fve1.htm

As for the jumpling turns there are 5... from 4 states. The states are easy to render.

(1) Turn the Skewb by 60 degrees.
(2) Turn the 2x2x2 by 45 degrees.
(3) Turn the 24-Cube by 90 degrees.
(4) Turn the 24-Cube by acos(1/3). This is the state with 2 types of jumbling as Matt points out above.

Give me a little while and I can try to make a picture for each of these cases. Personally I want Gelatinbrain to add jumbling to their applet.

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Mon Dec 14, 2009 11:56 am

Joined: Sun Aug 09, 2009 1:46 pm
Location: P.R.China
Vivid illustrations, Carl. You've made it a lot easier to explore these new concept puzzles.

wwwmwww wrote:
Very interesting... Yes you could do it with only 10 axes. Let's pick two of the 24-Cube cuts that are perpendicular to the x-z plane. Now put one of the Skewb cuts in the x-z plan and you get a 2x2x2 for free. One problem... that method doesn't fix the other Skewb cuts relative to the 24-Cube. Think of holding a Skewb by two opposite corners and spinning it. The other 3 cuts of the Skewb are free to rotate about the y-axis. Does this results in different 10-axes 2x2x2 + Skewb + Little Chop[24-Cube] combos? Looks like I'll have to render a few and see.

There are still one degree of freedom for the rest 3 skewb axes, so it is possible to choose a more symmetrical representative(personally don't expect too much coz 360/3 : 360/4=4:3 is a bad ratio but at least a mirror symmetry is likely to be achievable).

Are they different? I'll prefer starting from configurations of axes. Think about putting an equilateral triangle in a square with centers coincide, how many representative configurations(with regard to its symmetries) do we get? I suppose there are 3. Now put a regular tetrahedron inside a cube with one face perpendicular to the main diagonal of the cube, centers coincided. How many different general position do we get? 3 I guess. Note that I havn't taken the Little Chop into consideration.

I'm pretty lost, however, in the way it transforms. Probably a spherical version with cutting planes slightly overtops the sphere(thus indicating the direction of a cut) can tell more about the combinations of facelets. Or could you render a picture for this 10 axis puzzle like.... this?

http://wwwmwww.com/Puzzle/24Skewb.png

Anyway a standard 24+222+Skewb(Deep Cut V-E-F-T cube) cube is already a nightmare, it is for sure that squeezing a DCVEFTC will produce a nightmare in nightmare.

Leslie

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Mon Dec 14, 2009 12:30 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Leslie Le wrote:
There are still one degree of freedom for the rest 3 skewb axes, so it is possible to choose a more symmetrical representative(personally don't expect too much coz 360/3 : 360/4=4:3 is a bad ratio but at least a mirror symmetry is likely to be achievable).

Doesn't this one have mirror symmetry along the plane where the green and blue edges line up? The vertical 2x2x2 turn in the middle of the red face.
http://wwwmwww.com/Puzzle/10Cube.png

And this one has mirror symmetry along the plane where the green and red edges line up? There isn't a cut plane there but its the plane that contains left edge of the red face and the center of the puzzle.
http://wwwmwww.com/Puzzle/10CubeB.png

Leslie Le wrote:
Or could you render a picture for this 10 axis puzzle like.... this?
http://wwwmwww.com/Puzzle/24Skewb.png

Yes... give me a bit.

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Mon Dec 14, 2009 1:06 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Here is something I was working on last night... I wanted to try and make a 3 state puzzle. This isn't possible with 10 axes as I believe the 24-Cube and the 2x2x2 must exist in the same state with that limitation. However if we go back to 11 axes... it becomes possible. This is the most symmetrical representative one I could find.

In this state the 2x2x2 is solved. By the way... there are 5 axes of rotation you can play with in this state as it is a Cheese/2x2x2 Hybrid.
Turn the red face of the 2x2x2 60 degrees clockwise and the Green Skewb is solved.
Turn the left half of the red face of the 2x2x2 acos(1/3) counter-clockwise from the above state and the 24-Cube(Little Chop) is solved.

I named this the JumbleCube (if that's not already taken) as I have no clue what would be considered jumbling and what wouldn't be on this puzzle. I think you could almost consider all moves as jumbling moves. Either that or none as that term may only be well defined for single state puzzles. Then again... you can look at the 24-Cube as a two state puzzle by itself. After an acos(1/3) it's in a Rubik's Cheese state. Interesting questions... but I don't have a clue what the best answers are.

Carl

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Mon Dec 14, 2009 11:20 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
wwwmwww wrote:
This is the most symmetrical representative one I could find.
http://wwwmwww.com/Puzzle/JumbleCube2.png
In this state the 2x2x2 is solved. By the way... there are 5 axes of rotation you can play with in this state as it is a Cheese/2x2x2 Hybrid.
Turn the red face of the 2x2x2 60 degrees clockwise and the Green Skewb is solved.
Turn the left half of the red face of the 2x2x2 acos(1/3) counter-clockwise from the above state and the 24-Cube(Little Chop) is solved.

Wow... by keeping the solved states of each puzzle just one move apart I really limited myself.

Here is a slightly different starting position.

Turn the bottom half of the red face of the 2x2x2 60 degrees clockwise and the Green Skewb is solved.
Turn the left half of the red face of the 2x2x2 acos(1/3) counter-clockwise from the above state and the 24-Cube(Little Chop) is solved.

From here we can see there is a puzzle with the Skewb two moves away. Now rotate the left half of the red face of the 2x2x2 180 degrees before the 60 degree turn of the bottom half of the red face of the 2x2x2.

Puzzle/Top View/Bottom View
Same views just looking at the Skewb.

I love the symmetry of the top half of the puzzle. Now lets see if we can get the bottom half to look as nice. As is all the Skewb surfaces are the same as the 24-Cube surfaces for the bottom half of the puzzle.

We can fix part of that by rotation the bottom half of the 24-Cube by 60 degrees before we murge the puzzles. If we do that we get this:

Now both the Skewb and the 24-Cube states are 2 moves away from the 2x2x2 state.
It's better but there is still some overlap. Can this be fixed? This puzzle is so hard to play with in POV-Ray... I wish I could hold it in my hands to get a better idea of what is going on.

Carl

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Tue Dec 15, 2009 11:08 am

Joined: Sun Aug 09, 2009 1:46 pm
Location: P.R.China
Carl, i have to say sorry that this thread is somewhat lonely though you've apparently done a lot of works... I'm busy with the new cube. In any case, I'd like to contribute as far as i can reach.

wwwmwww wrote:
Doesn't this one have mirror symmetry along the plane where the green and blue edges line up? The vertical 2x2x2 turn in the middle of the red face.
http://wwwmwww.com/Puzzle/10Cube.png

And this one has mirror symmetry along the plane where the green and red edges line up? There isn't a cut plane there but its the plane that contains left edge of the red face and the center of the puzzle.
http://wwwmwww.com/Puzzle/10CubeB.png

You are absolutely right, i see that. These two represenatives has clearly different configurations hence different symmetries.

In my consideration, since we are already on the track of manipulating standard axis systems then what matters is the configuration. There are many independent configurations for L2(2x2x2 axis sys.) & L24(24 cube axis sys.) & Ls(Skewb axis system) with possibly 10,11,12 and 13 axes.

Now the problem seems to find all the representatives(with regard to symmetry) and explore their nature. However, without sexy tools at hand I must say that tracking them with bare axes have to be my first choice. For example, the way i explore jumble states is through vector calculations.

I'm not sure if you'd take them as equivalent approaches. In case you'd like to, shall we set up a few major topics /terms for the class of 'puzzles with reduced symmetry', or PRS. The reason for 'reduced symmetry' is that the axis system for PRS is constructed from standard axis systems(tetrahedron[Lt], hexahedron[Lh], octahedron[Lo], <rhombic> dodecahedron[Ld,Lrd], <rhombic> icosahedron[Li, Lri]) and some of them are either not symmetrically arranged(or arranged out of apparent symmetry) or partly colinear. Let me know if you have a better clarification for this class of puzzles.

Topics around PRS as I can see currently,
1. Representative and
2. Symmetry,
3. Jumble positions,
4. Solutions,
5. Mech.
Suggestions are fully welcome!

I agree that some of them deserves their place in GB. Solving puzzles with broken symmetries seems challenging!

Let's keep it up, Carl!

Leslie

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 Post subject: Re: A 2x2x2 + Skewb + Little Chop[24-Cube]Posted: Tue Dec 29, 2009 1:25 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Here are some more pictures I've been working on.

The cube with the Red face is the 2x2x2.
The cube with the Blue Face is the 24-Cube aka Little Chop.
The cube with the Green Face is the Skewb.

Cyan or Blue+Green is therefore a Little Chop plus a Skewb.
Yellow or Red+Green is therefore a 2x2x2 plus a Skewb.

All these puzzles have the 2x2x2, the Little Chop, and the Skewb sloved at the same time. I got to looking at my post above where I tryed to have 3 different solved states with the same shape as the first 11-axis puzzle pictured above. If I were to find it how would I know it was really a different puzzle? I'm guessing the mech could be the same and its just a shape mode of one of the jumbled states of the puzzle pictured here.

Enjoy,
Carl

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