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 Post subject: Vladimir's Cheese by OSKARPosted: Sat Feb 09, 2013 6:49 am

Joined: Mon Nov 30, 2009 1:03 pm
Hi Twisty Puzzles fans,

Vladimir's Cheese is a design that was requested by Vladimir Yaroslavskiy. He wanted a "deep cut" version of The Heptagon. This was achieved by shape-modding a 3-layer 14-section Cheese Block. This is a puzzle with a quite low "justice factor".

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: Vladimir's Cheese by OSKARPosted: Sat Feb 09, 2013 7:43 am

Joined: Sun Mar 15, 2009 12:00 am
Location: Jarrow, England
I really like the shape, another great puzzle. And I'm sure that the idea of "justice factor" (JF?) is going to spark plenty of discussions on these forums. I suppose a 1x3x3 Floppy cube has a JF of 100%?

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sat Feb 09, 2013 8:52 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Oskar wrote:
This is a puzzle with a quite low "justice factor".
What a MEAN prank.
Usually I hate it when people generate new terms without making sure everybody can understand them easily. And now you accused (?) me for creating such a term. *grumbl*

The 2x2x2 by Larry Nichols (implemented with magnets) has 100%
My first guess for the puzzle with the lowest "visibility factor" is the Pentultimate with (12+20)/(12+20+12+20+30+30+60+1) = 17.3 %
The "traditionally implemented" Big Chop would have 120/2845 = 4.2%
Please correct my somebody when I am wrong.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sat Feb 09, 2013 9:38 am

Joined: Mon Mar 30, 2009 5:13 pm
A Rubik's Cube has a 0% justice factor - if you solve it blind.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sat Feb 09, 2013 2:06 pm

Joined: Sun Mar 06, 2011 12:46 pm
Location: Wichita
how about the 10.01% justice factor of the chopasaurus? (92/919) Excellent puzzle Oskar!

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sat Feb 09, 2013 2:32 pm

Joined: Fri Dec 28, 2012 1:50 pm
Location: Near Las Vegas, NV
Andreas Nortmann wrote:
The 2x2x2 by Larry Nichols (implemented with magnets) has 100%
My first guess for the puzzle with the lowest "visibility factor" is the Pentultimate with (12+20)/(12+20+12+20+30+30+60+1) = 17.3 %
The "traditionally implemented" Big Chop would have 120/2845 = 4.2%
Please correct my somebody when I am wrong.

For deepcut puzzles this isn't entirely correct, as some pieces need to be bandaged in the internal mechanism to prevent internal misalighments. This reduces the number of total pieces quite a bit. If you take this into account, and count all of the bandaged pieces as "one piece", then the "justice factor" would increase: For the Pentultimate this means it would not be 17.3%, but rather 20.9%. Therefore I think Andrew Cormier's Chopasaurus is the puzzle with the lowest justice factor at 17.6% taking the bandaging into account. I don't even want to attempt to calculate the justice factor for the Big Chop...
cubedude76 wrote:
how about the 10.01% justice factor of the chopasaurus? (92/919)

This isn't completely right because some pieces in the Chopasaurus screw together to make one piece. In my calculation I've counted these pieces that screw together as "one" piece, and there are actually only 521 total parts (I hope I did that right!)

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sat Feb 09, 2013 7:59 pm

Joined: Thu Dec 21, 2006 5:32 pm
Location: Bay Area, CA
Putting aside the matter of justice (does this make me an outlaw?) I have to challenge this being called a deep cut puzzle. While its cuts do indeed intersect in a single central point, only the groups of pieces divided by horizontal plane are isomorphic. The more interesting cuts yield a 6/8 distribution.

By my proposed nomenclature here this would be a:
(6/8x7, 7/7x1)

Certainly the cuts are perhaps "central cuts" but only one (horizontal) meets this definition of "deep cut".

Dave

P.S. Wonderful puzzle!

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Feb 10, 2013 5:20 am

Joined: Mon Nov 30, 2009 1:03 pm
DLitwin wrote:
I have to challenge this being called a deep cut puzzle.
Dave,

Vladimir's Cheese is a shapemod of a two-layer Cheese Block. So what are you saying?
1) A two-layer Cheese Block is not a deep-cut puzzle?
2) Shapemodding can change a deep-cut puzzle into a non-deep-cut one?

Oskar

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Feb 10, 2013 7:35 am

Joined: Fri Nov 05, 2010 2:20 am
Location: Wherever
Speaking of Justice Factor, what is the justice factor of the Novaminx, or Curvy copter skewb?

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Feb 10, 2013 10:12 am

Joined: Thu Sep 24, 2009 12:21 pm
Location: Chichester, England
rubikcollector123 wrote:
Speaking of Justice Factor, what is the justice factor of the Novaminx, or Curvy copter skewb?

Novaminx:

Visible parts -

12 face centres,
20 corners,
30 edges,
60 X-centres,
120 chiral pieces.

Total: 242

Internal parts -

1 core,
20 axis pieces,
30 Dino Dodecahedron edges,
60 FTI centres,
60 FTI tips.

Total: 171

242 / (242+ 171) * 100 = 58.6%

Curvy Copter Skewb:

Visible parts -

8 corners,
24 small edge parts,
24 large edge parts,
24 shields,
24 slivers.

Total: 104

Internal parts -

1 core,
4 axis pieces,
4 Skewb corners,
6 Skewb centres.

Total: 15

104 / (104 + 15) * 100 = 87.4%

Back on topic:

Very nice puzzle Oskar! Although I too am a little confused by how the Cheese Block (a deep cut puzzle) has been bandaged and shape modded into what I would consider a non-deep cut puzzle. What would the regular Cheese Block look like if it were transformed into a heptagonal prism? My mind can't visualise how you go from the Cheese Block to this puzzle.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Feb 10, 2013 10:43 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
Dave,

Vladimir's Cheese is a shapemod of a two-layer Cheese Block. So what are you saying?
1) A two-layer Cheese Block is not a deep-cut puzzle?
2) Shapemodding can change a deep-cut puzzle into a non-deep-cut one?

Oskar
I'm not Dave but I do agree with him. Here are my answers...
1) No. The two-layer Cheese Block is clearly a deep-cut puzzle.
2) Yes. That is exactly what he is saying. Note most 2x2x2 puzzles could be considered shapemoded 3x3x3 puzzles where the slice layer has just been hidden inside the puzzle. Isn't this exactly what you've none, just hidden pieces of the two-layer Cheese Block inside the Vladimir's Cheese?

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Feb 10, 2013 11:16 am

Joined: Sun Dec 13, 2009 5:48 pm
To answer the question posed on youtube, Jason Smith's first Pentultimate has a visibility factor on 8.7804878048780487804878048780­488%

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Feb 10, 2013 11:19 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Andreas Nortmann wrote:
Oskar wrote:
This is a puzzle with a quite low "justice factor".
What a MEAN prank.
Usually I hate it when people generate new terms without making sure everybody can understand them easily. And now you accused (?) me for creating such a term. *grumbl*
Actually I quite like the term.
Andreas Nortmann wrote:
The 2x2x2 by Larry Nichols (implemented with magnets) has 100%
Actually there are quite a few twisty puzzles out there with a justice factor of 100%. They don't have to use magnets.

1) Scott Elliott's 1x2x2
2) Fused Cube
3) Doctor Cube
4) Doctor Skewb
5) Circle 3x3x3
And I'm sure there are many others...
Andreas Nortmann wrote:
My first guess for the puzzle with the lowest "visibility factor" is the Pentultimate with (12+20)/(12+20+12+20+30+30+60+1) = 17.3 %
The "traditionally implemented" Big Chop would have 120/2845 = 4.2%
Some issues with the definition of Justice factor here. As benpuzzles points out many of these puzzles would use internal bandaging to reduce the total piece count and prevent misalignment of hidden layers... thought that isn't always the case. Just look at the V-Cube 6x6x6. However there is another issue. What about puzzles which jumble... like the Big Chop. The surface of the Big Chop only has 120 surface pieces but once its jumbled I suspect there are other pieces that are given some surface area. How are pieces which get seen on the surface but which are hidden in the solved state counted? Here is the simplest example I can find.

Note 6 of the 12 hidden edge centers are seen in this state. Do they get counted in the numerator of the justice factor calculation?

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Feb 10, 2013 11:35 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
And I guess this should have been in my first post... GREAT PUZZLE!!! Sorry... I just read the entire thread and I've sort of been replying in reverse order to the posts which have been made. I also really like the idea of the justice factor. Anyways on YouTube you posted this question:

Why is the shape not a cylinder, like the﻿ original Rubik's Cheese?

I believe the answer to that question is that if you had it would have had open areas on the surface... correct? Either that or you'd expose some of your hidden pieces.

I believe the arcs formed where the cuts intersect the curved surface all lie on the surface of a sphere centered on the center of the puzzle. If that is the case then the curved edges don't lie on the surface of a sphere but probably something very closely related to an ellipsoid.

Question: Is it possible to get this in a state where it looks like a single edge pair has been flipped?

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Feb 10, 2013 1:15 pm

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
Why is the shape not a cylinder, like the﻿ original Rubik's Cheese? ... I believe the answer to that question is that if you had it would have had open areas on the surface... correct? Either that or you'd expose some of your hidden pieces.
Correct.
wwwmwww wrote:
I believe the arcs formed where the cuts intersect the curved surface all lie on the surface of a sphere centered on the center of the puzzle. If that is the case then the curved edges don't lie on the surface of a sphere but probably something very closely related to an ellipsoid.
Incorrect. Those shapes are circular cylinders, and the radius is smaller than the radius of the puzzle.

Oskar

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Feb 10, 2013 7:28 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
wwwmwww wrote:
I believe the arcs formed where the cuts intersect the curved surface all lie on the surface of a sphere centered on the center of the puzzle. If that is the case then the curved edges don't lie on the surface of a sphere but probably something very closely related to an ellipsoid.
Incorrect. Those shapes are circular cylinders, and the radius is smaller than the radius of the puzzle.
So you are telling me that the 7 circular cylinders that define the curved surfaces of this puzzle do NOT intersect at a point? Because if they do than that point must be the center of the puzzle and the arcs that cut those cylinders must all lay on the surface of a sphere centered on that point. And if so then the curved edges are indeed ellipses. So the shape defined by the curved edges IS exactly an ellipsoid. If your circular cylinders actually don't all cross at the center of the puzzle I'd love to understand why.

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Feb 11, 2013 1:02 am

Joined: Sun Nov 23, 2008 2:18 am
The puzzle sounds cool, though it is hard to appreciate it when I cannot see the video or images.

As for the Justice factor, it sounds more like a measure of mechanical elegance rather than something inherent to a puzzle.

So, if we count pieces that are in contact and do not move relative to each other(in other words, ease of assembly is the only practical reason for them to be separate parts) as a single part, does the knucklehead mechanism always have a justice factor of 100%? Also, would puzzles like the Void Cube, Holey Megaminx, and Minimal Twist have 100% Justice factor due to how their mechanism are completely exposed, even if the exposure is not obvious?

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Feb 11, 2013 6:08 am

Joined: Tue Oct 13, 2009 1:13 am
Location: Saint-Petersburg, Russia
I'm the requester for this puzzle: I suggested idea of 7-sector cheese
and asked Oskar if this is possible to implement this puzzle.
And now you can see ready and assembled puzzle.

I received the cheese (#001), this is extremely brilliant puzzle!
I don't know how it works inside (there should be complex
mechanism), but from my, player, point of view - the puzzle
turns absolutely smooth! Great job!

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Feb 11, 2013 9:03 am

Joined: Sun May 29, 2011 2:56 pm
Location: New York
A deep cut version of the Heptagon would jumble, no?

-Doug

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Tue Feb 12, 2013 12:40 am

Joined: Thu Dec 21, 2006 5:32 pm
Location: Bay Area, CA
Oskar wrote:
Vladimir's Cheese is a shapemod of a two-layer Cheese Block. So what are you saying?
1) A two-layer Cheese Block is not a deep-cut puzzle?
2) Shapemodding can change a deep-cut puzzle into a non-deep-cut one?
1.) I agree with Carl, a two layer cheese block would be deep cut (14/14x8).

2.) Yes, if "shape mod" is stretched to be an imprecise term. At what point does a shape mod represent a new puzzle? When significant pieces are hidden, or hidden pieces begin to be revealed, we start to get into the realm of changing the puzzle. From the perspective of my deep cut definition, the shape mods I spoke of here that had identical signatures (skewb = 7/7x4, Halpern Meier = 7/7x4) didn't reduce or add visible pieces, just their shape.
Once visible pieces start to change (the Kilominx as a "shape mod" of the Megaminx wherein edges are lost is a great example of this, and the 2x2x2/3x3x3 is the cubical analog), can we really say we are speaking of the same puzzle? Mechanically perhaps, and in some cases only with bandaging (Kilominx no, 2x2x2 yes).

The innards of how a puzzle is constructed should be opaque to how we see the resulting form. As Carl points out a deep cut (2x2x2) may be based on a non-deep cut (3x3x3), but it could also be implemented "pure" with magnets. The abstract 2x2x2 puzzle is conceptually pure and functions the same, regardless of is internals.

In your case here you have the potential for the reverse: A non-deep cut (Vladimir's Cheese) could be in theory based on a deep cut mechanism (two layer cheese block). But of course that would likely have been based on a three layer with bandaging, so perhaps we double switch from non-deep-cut to deep-cut, to non-deep-cut again!

Thoughts? I am tempted to find some way to merge this into the deep cut thread, no discussion of deep cut would be complete without these considerations.

Dave

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Tue Feb 12, 2013 2:20 pm

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
So you are telling me that the 7 circular cylinders that define the curved surfaces of this puzzle do NOT intersect at a point?
Carl, indeed that is what I am saying. Any smooth curve would suffice, as long as the equator is further away from the origin than the middle of an edge.
Doug Roth wrote:
A deep cut version of the Heptagon would jumble, no?
Doug, what would that look like from the outside? Can you sketch it?
DLitwin wrote:
In your case here you have the potential for the reverse: A non-deep cut (Vladimir's Cheese) could be in theory based on a deep cut mechanism (two layer cheese block). But of course that would likely have been based on a three layer with bandaging, so perhaps we double switch from non-deep-cut to deep-cut, to non-deep-cut again!
Dave, whereas I agree with the points by Carl and you, I am going to argue that that Vladimir's Cheese is deep-cut, just for argument's sake. The argument would be that the exterior of the puzzle requires a mechanism that is unarguably deep-cut at some shell level.

By the way, this whole yes/no deep-cut discussion inspires a challenge to Twisty Puzzles fans. Can you shapemod another deep-cut puzzle (e.g. 2x2x2, Skewb, Pentultimate) into a non-deep-cut puzzle? Note that you are only allowed to add material to the basic deep-cut sphere design (like I did with 2-layer cheese block --> Vladimir's Cheese), but you cannot add or remove pieces themselves.

Oskar

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Tue Feb 12, 2013 4:29 pm

Joined: Fri Dec 28, 2012 1:50 pm
Location: Near Las Vegas, NV
Oskar wrote:
By the way, this whole yes/no deep-cut discussion inspires a challenge to Twisty Puzzles fans. Can you shapemod another deep-cut puzzle (e.g. 2x2x2, Skewb, Pentultimate) into a non-deep-cut puzzle? Note that you are only allowed to add material to the basic deep-cut sphere design (like I did with 2-layer cheese block --> Vladimir's Cheese), but you cannot add or remove pieces themselves.

Oskar

Here's my approach: start with a Skewb, which is a deep-cut corner-turning cube, and shape-mod it into an offset skewb. The tetrahedral symmetry is self-dual, though. I don't think the offset skewb is deep-cut, but does this count?

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Tue Feb 12, 2013 4:32 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Is the axis which the pieces turn around slightly off of the apparent center of what's getting rotated?

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Tue Feb 12, 2013 4:54 pm

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland

This is what I'd guess you've done: hidden the middle layer and half the pieces on the top & bottom layers. Correct?

Meaning the concept could be (in theory) extended to cheeses with 2(2n+1) = 4n + 2 wedges.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Tue Feb 12, 2013 5:22 pm

Joined: Sun May 29, 2011 2:56 pm
Location: New York
Oskar wrote:
Doug Roth wrote:
A deep cut version of the Heptagon would jumble, no?
Doug, what would that look like from the outside? Can you sketch it?

Well, it's just that the Heptagon jumbled, so logically, a deep cut version should jumble to. I think it would look the same as it does now...

-Doug

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Wed Feb 13, 2013 3:55 am

Joined: Mon Nov 30, 2009 1:03 pm
Coaster1235 wrote:
This is what I'd guess you've done: hidden the middle layer and half the pieces on the top & bottom layers. Correct?
Almost. I also had to bandage one of the visible pieces to a hidden equatorial piece.

Oskar

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Thu Feb 14, 2013 1:20 am

Joined: Thu Dec 21, 2006 5:32 pm
Location: Bay Area, CA
Oskar wrote:
Dave, whereas I agree with the points by Carl and you, I am going to argue that that Vladimir's Cheese is deep-cut, just for argument's sake. The argument would be that the exterior of the puzzle requires a mechanism that is unarguably deep-cut at some shell level.
The underlined portion is an interesting postulate, assuming a shells based mechanism. But how do we support the requirement of a shells mechanism?
A magnetic mechanism (say, all pieces grip a metal sphere in the center) has no shells and is by the isomorphic argument not deep cut. Or does one consider that inner metal sphere a single shell? Either way, I think at this basic level not deep cut. It may meet the "cut surfaces intersect in a single point definition, but this highlights why isomorphism is an important part of the definition.
Oskar wrote:
By the way, this whole yes/no deep-cut discussion inspires a challenge to Twisty Puzzles fans. Can you shapemod another deep-cut puzzle (e.g. 2x2x2, Skewb, Pentultimate) into a non-deep-cut puzzle? Note that you are only allowed to add material to the basic deep-cut sphere design (like I did with 2-layer cheese block --> Vladimir's Cheese), but you cannot add or remove pieces themselves.
I suppose this might be possible if by adding material you "hide" another piece. Is this considered removing it? I suppose I consider this removed in the sense it moves from puzzle piece to mechanism piece.
Without hiding pieces, any shape modification will not alter the isomorphic nature of the starting deep cut puzzle and therefore could not create a non-deep cut.

So what can we say about the interaction of deep and shallow cut puzzles?

Clearly one can make a deep cut puzzle from a non-deep cut puzzle by hiding (and possibly bandaging). The 3x3x3 -> 2x2x2 is the most obvious example. Note bandaging in this case solves an internal alignment issue but I'll not consider that required, just one implementation.

How about building a non-deep cut on a deep cut? By adding pieces (say, edges that hook under the corners and centers under the edges) we can accomplish this. We have more alignment issues but again solving them is not our focus in this exercise.

Dave

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Feb 18, 2013 2:25 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
wwwmwww wrote:
So you are telling me that the 7 circular cylinders that define the curved surfaces of this puzzle do NOT intersect at a point?
Carl, indeed that is what I am saying. Any smooth curve would suffice, as long as the equator is further away from the origin than the middle of an edge.
First to make sure we are on the same page, have I edited this image correctly?
Attachment:

VladimirEdit.jpg [ 57.99 KiB | Viewed 2730 times ]

And if so, may I ask why the point on the equator must be further away? This picture makes me think that maybe the point on the edge needs to go under the point on the equator as the axis of rotation isn't perpendicular to the apparent plane that cuts the cylindrical surface.
Attachment:

VladimirUnder.jpg [ 38.9 KiB | Viewed 2730 times ]

However this image appears to show that they pass right next to each other.
Attachment:

VladimirUnder2.jpg [ 36.04 KiB | Viewed 2730 times ]

So its really hard to tell what is going on here. Maybe the question to ask is what would this look like if the axis of the cylindrical surface did pass through the origin?

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Feb 18, 2013 2:40 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
Doug Roth wrote:
A deep cut version of the Heptagon would jumble, no?
Doug, what would that look like from the outside? Can you sketch it?
To me a deep cut version of the Heptagon would look just like this however each cut that went through an edge would have a cut going though the opposite corner. So the top and bottom faces would again have 14 pieces meeting at the center. If 7 of these pieces were to turn together no I don't think it would jumble. However if it were to turn just like this puzzle, i.e. 6 of these new pieces were to rotate together then yes, you could make a 90 degree turn of an edge and the cut that now runs through the corner that is on the axis of rotation will line up with the equator cut after 90 degrees.

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Feb 18, 2013 3:02 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
DLitwin wrote:
Oskar wrote:
Dave, whereas I agree with the points by Carl and you, I am going to argue that that Vladimir's Cheese is deep-cut, just for argument's sake. The argument would be that the exterior of the puzzle requires a mechanism that is unarguably deep-cut at some shell level.
The underlined portion is an interesting postulate, assuming a shells based mechanism. But how do we support the requirement of a shells mechanism?
And even if you did how do you prove there isn't a shells mechanism out there that could support this puzzle that isn't deep-cut? Granted I haven't tried to design one but saying "requires a mechanism that is unarguably deep-cut at some shell level" implies you don't expect anyone to argue this point and that is exactly what I seem to be doing. It just appears to be stated without proof and granted I haven't tried to disprove it but it seems a bit much to accept as fact.
DLitwin wrote:
It may meet the "cut surfaces intersect in a single point definition, but this highlights why isomorphism is an important part of the definition.
Yes, isomorphism I like to think is the key as well. Its also why I like to think of the Slice Kilominx as deep cut but that one also leads to arguments.
Oskar wrote:
By the way, this whole yes/no deep-cut discussion inspires a challenge to Twisty Puzzles fans. Can you shapemod another deep-cut puzzle (e.g. 2x2x2, Skewb, Pentultimate) into a non-deep-cut puzzle? Note that you are only allowed to add material to the basic deep-cut sphere design (like I did with 2-layer cheese block --> Vladimir's Cheese), but you cannot add or remove pieces themselves.
Does this puzzle count?

Bandaged WitTwo I Magic

It is a deep cut 2x2x2. No material has been added or removed. However the 2x2x2 parts are now clear and they allow you to see the Fused Cube inside. So the puzzle which you are actually solving is a Fused Cube which isn't deep cut.

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Feb 18, 2013 1:40 pm

Joined: Mon Nov 30, 2009 1:03 pm
Carl,
wwwmwww wrote:
First to make sure we are on the same page, have I edited this image correctly?
Yes. Indeed, the center of the edge goes under the equator. I suggest that you try and draw the outer surfaces, and start rotating.
wwwmwww wrote:
And even if you did how do you prove there isn't a shells mechanism out there that could support this puzzle that isn't deep-cut?
That prove is actually quite simple. As soon as we start shaving off a bit of the outer pieces, exposing the hidden ones (e.g. shaving it to a sphere), the resulting 28-exposed-pieces puzzle is unarguably deep-cut. Of course we can shave off more, as any deep-cut puzzle may have a non-deep-cut one inside.
wwwmwww wrote:
Does this puzzle count? Bandaged WitTwo I Magic
No. It is a non-deep-cut puzzle INSIDE a deep-cut one. The challenge is to find an(other) existing example of a non-deep-cut puzzle OUTSIDE a deep-cut one, where only material has been added to the outer pieces of the deep-cut puzzle, and no pieces are added.

I believe there are valid arguments to call Vladimir's Cheese a deep-cut puzzle. It all comes to definitions ...

Oskar

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Feb 18, 2013 3:25 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
That prove is actually quite simple. As soon as we start shaving off a bit of the outer pieces, exposing the hidden ones (e.g. shaving it to a sphere), the resulting 28-exposed-pieces puzzle is unarguably deep-cut. Of course we can shave off more, as any deep-cut puzzle may have a non-deep-cut one inside.

The challenge is to find an(other) existing example of a non-deep-cut puzzle OUTSIDE a deep-cut one, where only material has been added to the outer pieces of the deep-cut puzzle, and no pieces are added.
Ok... if I understand the logic correctly... how about this? Let's call this an N-sided cheese with N=odd where N/2 - 1/2 slices rotate with an edge turn. In this case N=7. But wouldn't any odd N>1 require a core that you define as unarguably deep-cut at some shell level? Or are you wanting a non-cheese example?

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Feb 18, 2013 4:27 pm

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
But wouldn't any odd N>1 require a core that you define as unarguably deep-cut at some shell level? Or are you wanting a non-cheese example?
Yes, I would be looking for a non-cheese example. I guess that my definition of "unarguably" is a bit arguably itself. A non-cheese example could perhaps settle the argument, refine our definition of deep-cut and settle the argument for N=odd cheeses.

Oskar

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Wed Feb 20, 2013 4:10 pm

Joined: Fri Dec 28, 2012 1:50 pm
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Oskar mentioned in his video a challenge to twisty puzzle designers: to design a mechanism for the Vladimir's Cheese with a higher justice factor. I have successfully been able to do it (with a justice factor of 40%) and have made a video demonstrating the mechanism. You can view it here.
I may finalize this design in the future, but I am currently interested in other puzzles at the moment.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Wed Feb 20, 2013 4:35 pm

Joined: Mon Feb 06, 2006 12:52 am
NXTgen wrote:
To answer the question posed on youtube, Jason Smith's first Pentultimate has a visibility factor on 8.7804878048780487804878048780­488%

Are you sure? It seems to me like you counted those layers of the corners and centers as individual parts too. But I don't think you do justice to the idea of the justice factor here. I wouldn't count the edgepieces on a Mf8 Starminx II as two or even three parts because they consist of two plastic pieces with a screw. That's just one way to deal with the manufacturing.
So Jason's first Pentultimate has only 12 center pieces going all the way down into the core and 20 corner pieces. Than it also has these plates. 5 for each center piece (distributed over different layers).

So I would get 32/(32+5*12) = 34.78%

http://farm4.static.flickr.com/3151/244 ... 643692.jpg

here is a picture. The center and corner "stacks" should be counted as one part each. I would say.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Wed Feb 20, 2013 6:32 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
benpuzzles wrote:
Oskar mentioned in his video a challenge to twisty puzzle designers: to design a mechanism for the Vladimir's Cheese with a higher justice factor. I have successfully been able to do it (with a justice factor of 40%) and have made a video demonstrating the mechanism. You can view it here.
I may finalize this design in the future, but I am currently interested in other puzzles at the moment.
NICE!!! I think you've even broken this statement:
Oskar wrote:
That prove is actually quite simple. As soon as we start shaving off a bit of the outer pieces, exposing the hidden ones (e.g. shaving it to a sphere), the resulting 28-exposed-pieces puzzle is unarguably deep-cut.
Also I think I now understand your mechanism better then I do Oskar's. Oskar, do you want to make a video detailing your mechanism maybe indicating which pieces benpuzzles was able to combine or remove?

Oh and thank you benpuzzles for leaving your video public. I noticed from your comments that it was initially only intended for Oskar. But I found it very interesting so I'm glad I had the opportunity to view it.

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Wed Feb 20, 2013 7:22 pm

Joined: Fri Dec 28, 2012 1:50 pm
Location: Near Las Vegas, NV
wwwmwww wrote:
Also I think I now understand your mechanism better then I do Oskar's. Oskar, do you want to make a video detailing your mechanism maybe indicating which pieces benpuzzles was able to combine or remove?

Carl

I'm pretty sure the only difference is that Oskar's mechanism has a third, additional shell which is deeper cut than the second shell in mine (in which the cuts cross through the center of the top) which creates 14 extra pieces (analogous to all of the pieces on a 1-layer version of Oskar's Cheese Block). Therefore I didn't have to "combine" or "remove" any additional pieces. I might be wrong though.

wwwmwww wrote:
NICE!!! I think you've even broken this statement:
Oskar wrote:
That prove is actually quite simple. As soon as we start shaving off a bit of the outer pieces, exposing the hidden ones (e.g. shaving it to a sphere), the resulting 28-exposed-pieces puzzle is unarguably deep-cut.

I actually talk about this at the end of my video which leads to my belief that this puzzle is NOT a deep-cut puzzle, firstly because on the vertical axes less than half of the pieces move for a single turn, and secondly because no inner shell in my mechanism alone has this property either.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Wed Feb 20, 2013 8:39 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
benpuzzles wrote:
I actually talk about this at the end of my video which leads to my belief that this puzzle is NOT a deep-cut puzzle, firstly because on the vertical axes less than half of the pieces move for a single turn, and secondly because no inner shell in my mechanism alone has this property either.
Yes you did. Which remineded me there was something else I wanted to say. I believe the definition of "deep-cut" needs to be a bit stronger then just a turn which moves half of the pieces. The two halfs must also be isomorphic. The best example of a puzzle were a turn does move half of the pieces yet it isn't deep cut that I can think of at the moment is the Complex 3x3x3. It contains 64 pieces and a turn on the right face for example turns 32 pieces yet that isn't a deep cut turn. There exists no isomorphic maping between the pieces in the right face and the rest of the pieces in the puzzle. And even though this is basically the only requirement I put on deep cut, I'm still not in the majority here. The slice-turn Kilominx is a puzzle where each turn DOES move an isomorphic half of the puzzle yet I'm probably the only one here that I think considers it a deep cut puzzle.

Other then that I agree with everything you said... and again I really enjoyed the video. Personally I think I enjoy the mechanisms more then I enjoy the puzzles themselves much of the time so I for one wish more designers would make video introductions of their mechanisms. Its sort of the idea behind Oskar's justice factor, that being that so much of what is interesting and goes into making the puzzle functional can be hidden in the final product. I'm not saying Oskar's metric is the best gauge of that as clearly your mechanism also hides as much ingenuity as Oskar's. But it was a clever way to highlight the concept of all this hidden ingenuity.

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Thu Feb 21, 2013 12:09 am

Joined: Thu Dec 21, 2006 5:32 pm
Location: Bay Area, CA
wwwmwww wrote:
The slice-turn Kilominx is a puzzle where each turn DOES move an isomorphic half of the puzzle yet I'm probably the only one here that I think considers it a deep cut puzzle.
You had me coming around to your point of view here, but after continued thought I think you nailed it for me here by opening up the definition to non-planar divisions. I see I've never posted in that thread since, but consider at least two of us now think of the Slice Kilominx as deep cut.

Dave

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Thu Feb 21, 2013 12:16 am

Joined: Mon Jun 18, 2012 10:56 am
Location: The Netherlands
Thank you so much for sharing this video. I stand in awe both for the design, as for the capability to work with this software (it looks SO very hard and you make it look SO easy).

The video was very informative and provided me with a completely new view on mechanisms and such. Also.. now I FINALLY understand what it means when someone says fudging.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Mar 03, 2013 6:30 am

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
I believe the definition of "deep-cut" needs to be a bit stronger then just a turn which moves half of the pieces. The two halfs must also be isomorphic.
Carl,

Following your requirement for the halves to be isomorphic, then the 6-piece "Skewed Cheese" (sketches below) is not deep cut either. The non-isomorphism is now easy to see as the angles are 30-90-30-90-30-90, instead of 60-60-60-60-60-60 for a regular Rubik's Cheese. But how about 59.9-60.1-59.9-60.1-59.9-60.1? Still not deep cut? In that case a mass produced Rubik's Cheese is not deep cut either, as these happen to be the angles of the mass-produced version. Mass production is not infinitely accurate, you know.

Alternatively, we could agree that the mass-produced Rubik's Cheese and its Skewed Cheese cousin are both deep-cut. But then you have lost the isomorphic argument for Vladimir's Cheese being non-deep-cut.

Oskar
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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Mar 03, 2013 11:43 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
wwwmwww wrote:
I believe the definition of "deep-cut" needs to be a bit stronger then just a turn which moves half of the pieces. The two halfs must also be isomorphic.
Carl,

Following your requirement for the halves to be isomorphic, then the 6-piece "Skewed Cheese" (sketches below) is not deep cut either. The non-isomorphism is now easy to see as the angles are 30-90-30-90-30-90, instead of 60-60-60-60-60-60 for a regular Rubik's Cheese. But how about 59.9-60.1-59.9-60.1-59.9-60.1? Still not deep cut? In that case a mass produced Rubik's Cheese is not deep cut either, as these happen to be the angles of the mass-produced version. Mass production is not infinitely accurate, you know.

Alternatively, we could agree that the mass-produced Rubik's Cheese and its Skewed Cheese cousin are both deep-cut. But then you have lost the isomorphic argument for Vladimir's Cheese being non-deep-cut.
Very interesting argument Oskar!!! And maybe I should defer to a mathematician in this case... but I think I would say that even in the 30-90-30-90-30-90 case the puzzle is still isomorphic. As I understand it isomorphic doesn't mean identical... it just means there exists a one to one mapping of the of the pieces between the two halves of the puzzle. I would say that holds till you get to 0-120-0-120-0-120 as then you start removing pieces. And for this definition I consider a piece any piece that is involved in the solution of the puzzle. Internal pieces, i.e. the mechanism, are totally independent of a puzzles classification as deep cut or not.

Would you consider a bump 2x2x2 isomorphic?

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Mar 03, 2013 11:50 am

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
but I think I would say that even in the 30-90-30-90-30-90 case the puzzle is still isomorphic. As I understand it isomorphic doesn't mean identical... it just means there exists a one to one mapping of the of the pieces between the two halves of the puzzle. I would say that holds till you get to 0-120-0-120-0-120 as then you start removing pieces.
Carl,

Thank you for taking my bait. Then how about 0.1-199.9-0.1-199.9-0.1-199.9? You could still see the "hidden" pieces shine through. And note that even for the 0-120-0-120-0-120, one can clearly see the "hidden pieces" mid turn.
wwwmwww wrote:
And for this definition I consider a [hidden] piece any piece that is involved in the solution of the puzzle.
So I can turn the 0-120-0-120-0-120 into a deep-cut puzzle by stickering the "hidden" pieces? And I can turn a regular Rubik's Cube into a deep-cut puzzle by removing all center and edge stickers?

Oskar

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Mar 03, 2013 5:15 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
So I can turn the 0-120-0-120-0-120 into a deep-cut puzzle by stickering the "hidden" pieces?
I'd say yes. Even in the 0-120-0-120-0-120 case if the "hidden" pieces become a part of the puzzle, be that through the use of transparent parts or parts that are stickered and only seen mid-turn, then I would still consider it a deep-cut puzzle.
Oskar wrote:
And I can turn a regular Rubik's Cube into a deep-cut puzzle by removing all center and edge stickers?
Again I'd say yes. Most 2x2x2 are built up from 3x3x3's where one corner is bandaged to 3 face centers, i.e. a Fused Cube and the corners are simply extended to hide the slice layer. The puzzle you propose, a 3x3x3 with unstickered face centers and edges, IS fundamentally a 2x2x2 and functionally the only difference is the job of keeping the slice layer in alignment is now the responsibility of the player instead of it being taken care of mechanically by the bandaging. But if I can solve your puzzle then I can also solve the 2x2x2 as they are fundamentally the same puzzle. Due to aesthetic differences, I wouldn't recommend Rubik trying to sell such a puzzle as a 2x2x2. But I don't think those aesthetic differences should be considered when discussing the nature of the fundamental puzzle.

Where my definition is weak is I'm not sure I've got a mathematically solid definition of what constitutes a valid one-to-one mapping. Consider the Complex 3x3x3. 32 solvable pieces turn with the R-turn and there are 32 solvable pieces that do not turn with the R-turn. I could make a list of 1 to 32 for each set and then say I map 1 to 1, 2 to 2, 3 to 3, etc, but I can see why this isn't valid for this puzzle (Looks like I'm wrong... see below). It has to do with the nature of the piece types. Pieces must map to their own piece type, or their inverse, as I see it. But from a purely mathematically point of view I can't state the exact reason.

Ok... it appears I've rushed to the conclusion that the Complex 3x3x3 isn't deep cut.

If the above mapping is valid, and I believe it is, then the Complex 3x3x3 IS a Deep-Cut puzzle.

Going back to the Vladimir's Cheese, the equatorial cut is clearly deep cut. So this does bring up another weakness of the definition. Is a Deep-Cut Puzzle defined as a puzzle which contains a deep cut? Or one which ONLY contains deep cuts?

Carl

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Mar 04, 2013 6:10 pm

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
wwwmwww wrote:
Going back to the Vladimir's Cheese, the equatorial cut is clearly deep cut. So this does bring up another weakness of the definition. Is a Deep-Cut Puzzle defined as a puzzle which contains a deep cut? Or one which ONLY contains deep cuts?
You bring up a very good point here, and one that I believe should actually be expanded on. Since we are simply defining characteristics of puzzles here, and not (necessarily) discovering mathematical truths, we are free to slice the definitions as finely as needed to provide clarity.

In your example above, we could certainly say that Vladimir's Cheese is deep cut on one (horizontal) axis, and not on the others. We could also limit the definition of "deep-cut" to slices that are planar and pass through the origin point of the puzzle (or intersection of the axes). Thus a bump 2x2x2 would still be deep cut. Of course any one puzzle might contain some deep cuts, and some shallow cuts. This more limited definition is how I prefer to think of the term "deep cut".

Non-planar cuts could fall into their own categories, which may or may not require new terminology. Limiting the definitions this way might provide some clarity, while broadening the language used to describe various types of puzzles.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Mar 04, 2013 6:17 pm

Joined: Fri Dec 28, 2012 1:50 pm
Location: Near Las Vegas, NV
I think there is also another problem with the definition, and a very important one: is a bandaged version of a deep-cut puzzle still considered a deep-cut puzzle? One could say that the Skewb Ultimate is actually a bandaged version of the Chopasaurus, and everyone can agree that they are both considered deep-cut puzzles. So could we say that since the Vladimir's Cheese is a bandaged version of a 2-layer 14-segment cheese (a deep cut puzzle) that it should also be considered deep-cut? I think this is important to consider.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Mon Mar 04, 2013 6:37 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: San Jose, California
I've been too busy to seriously follow this thread but this topic is one of the oldest theoretical debates we've had. I won't be able to participate in this discussion to the level of detail and attention I'd like but I'll try.

David Pitcher wrote:
[...]we are simply defining characteristics of puzzles here, and not (necessarily) discovering mathematical truths, we are free to slice the definitions as finely as needed to provide clarity.

This is a great distinction. In the past we've have these big debates about the meaning of "deep cut" and other definitional matters, sometimes forgetting that we can tweak the definitions as needed.

Carl, your mapping of pieces is the exact reason I originally called the Complex 3x3x3 deep cut. Not only do half the pieces move in a turn, the pieces that move are nicely paired with a piece that doesn't move. I'm not happy with this definition.

• One property I think deep cut puzzles should have is that all cuts (physical or theoretical) should be "deep". If there are cuts that can be used that are obviously shallow then I don't think the puzzle can be called deep cut.
• Another property I think deep cut puzzles should have is that all piece types in isolation produce puzzles that are deep cut. For example, the 2x2x2 is deep cut because it only has one piece type and those pieces are deep cut. The 3x3x3 isn't because the edges in isolation do not form a deep cut puzzle. If you take any traditionally deep cut puzzle, this property holds true.

A few weeks back Daniel Kwan (dkwan) and I were chatting about adjacent pieces and some grip theory stuff and he came up with a very neat idea to measure deep cuttness that doesn't take geometry into account. His suggested definition is that a piece type is deep cut if it is self-inverse. That is, if you invert all of the grips, you get another piece of the same type. This is true of all pieces in all traditional deep cut puzzles.

The reason I don't think the Complex 3x3x3 is deep cut is that if you only allow for the moves {U, F, R, B, L, D} then all you have to do is look at the regular 3x3x3 edges to see which move was done. There is no ambiguity.

Your table of mapping edges to inverted-edges is fine but to invert every piece in the puzzle like that, I think you also need to "invert" the moves you do. Instead of {U, F, R, B, L, D} you'd need to do the inverse of these moves (what we've called anti-moves in the past). In your table mapping, the R pieces that move on the left map to the anti-R pieces that would move in an anti-R move on the right. Since anti-R is just R + reorientation about R' it should come as no surprise that this forms an ambiguity.

I think what you're getting at is that if you invert every grip of every piece on the Complex 3x3x3, the resulting puzzle is still the Complex 3x3x3. That is also true of traditionally deep cut puzzles. I think it's a necessary but not sufficient property.

Edit: Just so I'm clear, following DKwan's definition for a deep cut piece being self-inverse. A deep cut puzzle is a puzzle made up of only deep cut pieces. If a puzzle isn't made entirely of deep cut pieces then you can look at the non-deep cut pieces to see what move was done.

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sat Mar 09, 2013 12:09 pm

Joined: Mon Nov 30, 2009 1:03 pm
benpuzzles wrote:
since the Vladimir's Cheese is a bandaged version of a 2-layer 14-segment cheese.
Vladimir's Cheese is a shape-modded version of of a 2-layer 14-segment cheese. There is no bandaging.
wwwmwww wrote:
Going back to the Vladimir's Cheese, the equatorial cut is clearly deep cut. So this does bring up another weakness of the definition. Is a Deep-Cut Puzzle defined as a puzzle which contains a deep cut? Or one which ONLY contains deep cuts?
If you wish to make the distinction, then let's call a 2x3x3 "semi deep cut", and reserve the word "deep cut" for a 2x2x2. With this definition, we all agree that Vladimir's Cheese is at least semi deep cut.

Oskar

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Sun Mar 10, 2013 4:01 am

Joined: Thu Dec 21, 2006 5:32 pm
Location: Bay Area, CA
I think the most valuable meaning of "deep cut" is in describing a particular move, not the entire puzzle. Many puzzles have deep cuts but are not made up of exclusively deep cuts, some examples being the 4x4x4 and Smart Alex.

Calling a puzzle deep cut if all of its cuts are deep seems logical, but a fall out of the collection and not the important definition.

Dave

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 Post subject: Re: Vladimir's Cheese by OSKARPosted: Wed Mar 13, 2013 4:46 pm

Joined: Fri Dec 28, 2012 1:50 pm
Location: Near Las Vegas, NV
Oskar wrote:
benpuzzles wrote:
since the Vladimir's Cheese is a bandaged version of a 2-layer 14-segment cheese.
Vladimir's Cheese is a shape-modded version of of a 2-layer 14-segment cheese. There is no bandaging.

Wow, that was REALLY dumb of me to say especially since I designed my own mechanism for this sorry!

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