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 Post subject: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 1:33 am 
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Over in the puzzle building forum the Polaris thread has been a bit hijacked by a discussion of what a deep cut puzzle is. I had some comments and thought it deserved its own thread, outside of Puzzle Building.

See that thread for some of the discussion (as well as the development of a *very* cool puzzle). I won't repeat it here, but for the last post to which I will respond. Noah gives a pretty simple and good definition, but I think it goes deeper.
Noah wrote:
Easy definition of a deep cut puzzle.

All the planes of rotation intersect at a single point.

I am not sure there is an easy definition.

This doesn't cover the 2x2x1 because the lack of a third cut means the two planes meet on a line, not a single point. On a 2x1x1 (Boob cube) it is hard to see how this even applies, as there is no second plane to intersect, but I suppose we consider this to be trivially deep cut and won't want to pollute our definition just to fit it in.

I think "All planes of rotation intersect the center point" might do it, but perhaps the concept of a center point is ill defined on shape mods that can take a deep cut puzzle off center.

Perhaps "All planes of rotation intersect at least one common point" covers an arbitrary number of planes. Or does this get disqualified because in some geometries (perspective) parallel lines meet at infinity (and parallel planes at a line at infinity), which would mean by this definition a 3x3x3 is deep cut? Can't have that. But I think we can rule out perspective geometries as silly for this definition.

But next it may get confusing with puzzles like a 4x4x4 or 6x6x6 (this point already mentioned in the other thread) where some cuts are deep but others not. Clearly not all planes intersect at a common point, but clearly the puzzle is deep cut. (Or is it?)

Perhaps it is better to speak of "deep cut" in the context of the plane of rotation, rather than the puzzle itself. A cut may be a deep cut of a puzzle, even if not all cuts on the puzzle are.

Can we perhaps define a deep cut not relative to other cuts on the puzzle? On a 2x1x1 *any* cut is deep cut, regardless of shape or where the cut is made. Likewise on a 2x2x1, but for the caveat that the planes not be parallel.

Can we perhaps define a deep cut as a rotational plane dividing two isomorphic groups of pieces? Does this definition, if correct (and it has been some time since I have done any of this sort of thinking, so I am not confident it is), inescapably lead to puzzles who's deep cut planes must intersect at a common point? If not, are any exceptions of interest for puzzle building?

What of non-symmetric deep cuts? A 2x2x1 with cuts not meeting at 90 degrees is perhaps not an interesting puzzle as you won't get very far after the first turn. Likewise a 2x2x2 with non-90 degree cuts.

I can't help but think of the Unscrambled which might seem deep cut but doesn't divide isomorphic groups. But since the slice intersects the puzzle center it certainly feels like a deep cut. Is it? Similarly, is the main dividing plane of a Square-1 deep cut? If so, the concept of isomorphic groups won't hold up unless we throw bandaging into the mix. And perhaps we need to, as deep cut puzzles can be bandaged and still be deep cut (think of a 2x2x2 with two corners fused).

I'll let the more mathematically adept minds on the forum correct me. It seems for every proposal I put out, I find a way to possibly invalidate it.

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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 2:50 am 
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DLitwin wrote:
Can we perhaps define a deep cut as a rotational plane dividing two isomorphic groups of pieces?


As long as we make sure that the puzzle is not "jumbled"
(and if it has been "jumbled", it should be returned to its normal position)
then in my opinion, the definition is excellent.

And then, we could differentiate puzzles which have no deep cuts (e.g. Megaminx),
only deep cuts (e.g. 2x2x2), or a combination (some of which are jumbable puzzles)
of them. This concept can become even more interesting (and complex)
the more different (angle) cases we add.

Perhaps there should be some Atlas of them?

:)


Pantazis

PS How about magics, are they deep-cut? I think they are deep-cute! :P

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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 3:05 am 
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Would puzzles with planes that go through a central point all be archimedian and platonic solids, or is there some other group of solids that would have this property?


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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 4:55 am 
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How about a slightly different definition. All planes of rotation go through the center of the puzzle.

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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 5:13 am 
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I just thought about this idea, though the definition derived from it it may differ from what the majority of you guys think:

Take a puzzle, imagine what the cuts would look it after it has been shaped modded into a sphere with the centre of the mechanism as the centre of the sphere, and see if any of the cuts would be the equator(Is it what it's called?), if there is one of more cuts, then it is deep cut.

One special example which I think is bizarre is the Square-1(The vertical cut)


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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 7:42 am 
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It seems that people are thrown off my the Square-1 because the "deep cut" of the puzzle doesn't meet an edge, it meets the middle of a face.

OOOH!!! [Sorry, majorly off topic, but I don't wanna loose the idea]
Shape-mod a Square one in such a way that you do have the "deep cut" meeting a newly made edge!!!


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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 11:53 am 
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TBTTyler wrote:
Would puzzles with planes that go through a central point all be archimedian and platonic solids, or is there some other group of solids that would have this property?

It depends on your definition of "central point".

Center of mass? You can create any number of odd shapes built up on a deep cut puzzle (let's say a skewb, as people love this for mods) where the center of mass is the deep cut intersection point yet the outside is quite asymmetric. So I don't know that that works.

If by central point you mean all points and all faces are equidistant from a common point then perhaps you head towards the platonic and archimedian solids. I believe you can throw in the concept of face normals intersecting the same point. But this central point even by this definition doesn't need to be the deep cut plane intersection point, think of an offset mod like Tomas' 2x2x2 bump cube.

But back to the original question, a 2x2x2 sphere meets all of your criteria but isn't a platonic or archimedian solid. As long as you keep symmetry there are infinite shapes outside of the platonic and archimedian set that match as well.

Dave :)

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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 12:23 pm 
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Much of what you described as not adhering to my suggestion are shape mods. The original geometry still supports my conclusion.
As for the sphere, Marco described it earlier and I will try and paraphrase: If all cuts can be described as great circles on a sphere, the puzzle is deep cut.
This completely removes shape from consideration. If a puzzle is deep cut, then any shape mod is deep cut by this definition (which is what you're looking for)
I think this is closer to a universal definition.

Then, perhaps, the most visually pleasing and complex puzzles (imo) happen to fall into the categories of the platonic and archimedian solids as references.

Oops! I made a mistake. I didn't fully understand the definition of "Archimedean solid" and was under the misguided impression that rhombic polyhedra were included. Does anybody know what group they fall into?

Some Deep cut puzzles based upon polyhedra with one type of face:
Tetrahedron/octahedron = skewb
Cube = 2x2
Dodecahedron = Pentultimate
Icosahedron = grigr called it a half minx, but I really don't like that name for some reason.
rhombic dodecahedron = 24 Cube (aka Little Chop)
rhombic triacontahedron = Big Chop

Any others?


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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 12:41 pm 
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DLitwin wrote:
Over in the puzzle building forum the Polaris thread has been a bit hijacked by a discussion of what a deep cut puzzle is. I had some comments and thought it deserved its own thread, outside of Puzzle Building.

See that thread for some of the discussion (as well as the development of a *very* cool puzzle). I won't repeat it here, but for the last post to which I will respond. Noah gives a pretty simple and good definition, but I think it goes deeper.
Noah wrote:
Easy definition of a deep cut puzzle.

All the planes of rotation intersect at a single point.

I am not sure there is an easy definition.
My definition is that every plane of rotation cuts the puzzle in half. Let's define a "move" as "holding one side of a plane/cut still, while rotating the other side of the plane/cut, where the side that stays still has an equal or greater number of moveable pieces than the side that rotates". If the puzzle has x different possible "moves", how many distinct puzzle states are created by those x "moves", not counting re-orienting the entire puzzle? Answer: x/2. For example, UL and DR are the same move on a Skewb, just with the puzzle oriented differently after the move. "Every turn cuts the puzzle in half" -- in my opinion, if you can't say that then you don't have a deep cut puzzle, although some puzzles may have "some deep cuts" without being a bonafide/purely "deep cut" puzzle.


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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 3:56 pm 
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Eliminating jumbling and bandaging moves for a minute...

And assuming the circles on a sphere approach.

I wonder if it's also useful to say that a deep cut puzzle should have only rotation moves less than 360 degrees that return the puzzle to a new position, but the same configuration of circles on the sphere. I think this is useful because it eliminates some trivial examples like the boob cube, and a puzzle cut by only 2 non-perpindicular deep planes.

Here are some interesting examples:

Imagine a masterball without any horizontal cuts. Would you call that deep cut? I think I would. Rotations of 180 return the puzzle to a new position, but the same configuration of slices.
Imagine a Rubik's UFO without the horizontal cut. Deep cut? I think so. 180 degree rotations get us to a new position, but the same configuration again. Now, remove one slice. Still deep cut? 180 degree rotations are dead ends.
Another example would be a vertex turning dodecahedron that only has 5 cuts - between vertices on 2 opposite faces. It's deep cut by one definition, but only 360 degree moves get you back to the same configuration, so this puzzle doesn't deserve to be called deep cut..


Also, I think I would define a deep cut puzzle as any union between a deep cut puzzle defined above, and an optional other puzzle.

In other words, it's deep cut if the puzzle can be made to meet the more strict criteria above by removing zero or more (shallow) slices. This would allow the union of a Pentultimate and a Megaminx to be called deep cut. It would also allow the masterball to be called deep cut.



So here's my opinion at this moment:

1. Let S be a sphere placed at a position such that all axes of rotation for our puzzle intersect the sphere and are perpindicular where they intersect.
2. Has 2 or more cuts C that make great circles on the sphere S
3. All axes for cuts C allow rotational moves less than 360 degrees that produce the same configuration of great circles made by C in (2)


I think this definition allows additional shallow cuts without specifically stating so.

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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 4:22 pm 
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io wrote:
1. Let S be a sphere placed at a position such that all axes of rotation for our puzzle intersect the sphere and are perpindicular where they intersect.
2. Has 2 or more cuts C that make great circles on the sphere S
3. All axes for cuts C allow rotational moves less than 360 degrees that produce the same configuration of great circles made by C in (2)

Your definition of #1 confused me at first as I thought "perpendicular where they intersect" referred to the planes intersecting other planes, rather than the planes intersecting the sphere. Clearly we can't restrict planes to 90 degree intersections or we cut out the skewb and pentultimate and other non-2x2x2 puzzles. But assuming that plane/sphere intersections is what you mean, is this not just a more complicated way of saying the plane intersects the center point, or cuts a great circle? I think sticking with the center point definition is most understandable.

I like #3. It is a more specific way of getting at what I meant by "not an interesting puzzle" regarding non 90 degree 2x2x1 puzzles.

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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 4:47 pm 
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You're right, #1 was very poorly worded.

EDIT : It was an attempt to describe WHERE the sphere is instead of just saying "center if the mechanism".

The difference between a cut and an axis (the line perpindicular to the cut) is confusing.



Here's another attempt:


1. Let S be a sphere placed at a position such that all axes of rotation, where each axis is visualized as an infinite LINE, intersect the sphere and are all perpindicular to the sphere where they intersect it.

or maybe equivalently and much shorter:

1. Let P be the single intersection point for all rotation axes of a puzzle. Let S be a sphere centered at P.


(These imply a non-siamese-ness.)

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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 5:41 pm 
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I always thought that a deep cut puzzle is one that for every possible move, all the faces are affected.
This obviously excludes the 2^3 cube.


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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 5:49 pm 
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ppeccin wrote:
I always thought that a deep cut puzzle is one that for every possible move, all the faces are affected.
This obviously excludes the 2^3 cube.


well the that discludes just about everything except the skewb

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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 6:56 pm 
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ppeccin wrote:
I always thought that a deep cut puzzle is one that for every possible move, all the faces are affected.
This obviously excludes the 2^3 cube.

Well we can't exclude the 2x2x2 or what sort of deep cut definition do we have? :)

I think your definition gets to the point of my "rotational plane divides two isomorphic groups". Isomorphic meaning equivalent in structure and relationship although because of extension or truncation they may not be exactly the same. If the rotational plane divides the two groups evenly (hence requiring them to be "equivalent") then one can argue that the changes made (by the rotation) to the groups are equivalent in magnitude (and inverse?). I suppose this is a way of saying the faces are equally "affected".

Dave

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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 7:38 pm 
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ppeccin wrote:
I always thought that a deep cut puzzle is one that for every possible move, all the faces are affected.
This obviously excludes the 2^3 cube.

The only puzzle with this property is the Skewb. Even puzzles with the same mechanism but different shapes (Skewb Diamond, Skewb Ultimate) have unaffected faces after a turn.


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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 8:11 pm 
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DLitwin wrote:
ppeccin wrote:
I always thought that a deep cut puzzle is one that for every possible move, all the faces are affected.
This obviously excludes the 2^3 cube.

Well we can't exclude the 2x2x2 or what sort of deep cut definition do we have? :)

I think your definition gets to the point of my "rotational plane divides two isomorphic groups". Isomorphic meaning equivalent in structure and relationship although because of extension or truncation they may not be exactly the same. If the rotational plane divides the two groups evenly (hence requiring them to be "equivalent") then one can argue that the changes made (by the rotation) to the groups are equivalent in magnitude (and inverse?). I suppose this is a way of saying the faces are equally "affected".

Dave
I think your "rotational plane divides two isomorphic groups" hits the nail on the head. Every turn of the 2x2x2 twists 4 cubies against 4 other cubies. Every turn of the Skewb twists 4 corners and 3 square faces against another 4 corners and 3 square faces. Every turn of the Pentultimate twists 10 corners and 6 pentagonal faces against another 10 corners and 6 pentagonal faces -- always equivalent groups either side of the cut. A natural consequence of this -- and a feature unique to deep cut puzzles -- is that it's possible to achieve all possible permutations of the puzzle using a set of exactly half the number of possible different moves as defined by an minority or equal number of pieces rotating against an majority or equal number of pieces. Thus you can reach any position with a 2x2x2 turning only 3 of its faces (9 distinct moves), any position with a Skewb using only 4 of its corners (8 distinct moves), and any position with a Pentultimate using only 6 of its faces (24 distinct moves).


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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 10:10 pm 
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isnt the pyraminx crystal a deep cut puzzle? the slices do not intersect over the central axis


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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 10:19 pm 
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I wouldn't say the Pyraminx Crystal is deep cut.

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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 10:21 pm 
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QUINBLZ wrote:
isnt the pyraminx crystal a deep cut puzzle? the slices do not intersect over the central axis

It's not a deep cut puzzle. It's a deepER cut puzzle than a megaminx, but it's not deep cut. The deep cut face turning dodecahedron is the pentultimate.


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 Post subject: Re: Deep cut puzzles
PostPosted: Mon Sep 22, 2008 11:11 pm 
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After reading thoroughly this extemely interesting thread,
I have thought that we may just trying to classify puzzles
the wrong way. The complexity of all different puzzles
does not allow for much flexibility.

But my comment above wouldn't sound creative if I didn't
have a suggestion:

To define a "cut number" C as a number equal or less than 1,
which projects the analogy of:
the changable puzzle pieces which are being moved, and
the changable puzzle pieces which are not being moved.
(and if the number is C>1, we simply swap it with 1/C).

Here, a puzzle piece is changable, if it is visible and could swap places
with another puzzle piece, while its neigborhood remains the same.
(Therefore, centers are included!)
Each puzzle will be represented by the name of those cuts.

So if we had a puzzle with a number of different cuts C1, C2, ..., Ck,
this puzzle should be defined as (C1. xN1, C2. xN2, ..., Ck. xNk).
where Ni represents the amount of cut numbers with value Ci (where i=1,...,k).

Some examples here:
(the second number of the multiplication shows the number of cuts
which have the same "cut number" C. Thus the decimal dot is placed
to differentiate real numbers from integers):

Boob Cube will be called (1. x1)
2x2x1 will be called (1. x2)
2x2x2 will be called (1. x3)
3x3x3 will be called (9./17 x6)
4x4x4 will be called (1. x3, 2./5 x6)
Skewb/Ultimate Skewb will be called (1. x4)
Megaminx will be called (11./81 x12)
Pyraminx Crystal will be called (1./7 x12)
Square-1 will be called (4./5 x2, 1. x1)

(I hope I have not made a calculation mistake above, but you know what I mean!)


That way, we will even get the non-trivial pieces in the picture,
and in my opinion, a classification of puzzles becomes easier.
Based on this, definitions of whether a puzzle is "clear deep cut"
(i.e. it only has 1.'s), "mixed deep cut", or "non-deep cut",
could be easily given. After this, I can already envision of other
possible puzzles which have never been thought before.

I might use this notation on my website, because I kind of like it.

:)


Pantazis


PS. Using [exact Nr of pieces in part A] over [exact number of the rest of the pieces]
may be redundant as we would get a similar puzzle, but I am still not sure about this.
But by using the smallest angles between different cuts with the same cut number
may help, e.g. in the case of the Rubik's UFO which should not be mixed
with a Skewb!!!
PS2. Attention: Do NOT mix up a "cut number" with a "cat number"!!! ;)

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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 2:12 am 
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See, I knew someone with an advanced degree would jump in ;) I am also still waiting for a comment by Bram, from whom I first heard the concept of deep cuts.

I like your notation, but have a few comments:
1.) Are the decimal places really needed? Do we really need it to be a cut number as opposed to simply two integers? If expressed as anything other than distinct integers it invites one to convert it to another notation (decimal, for example) which would lose much of the descriptive value.

2.) I like the idea of your first PS, i.e. keeping the value (or values, if you will) unreduced. While not as easy to spot deep cuts, it isn't exactly hard and more information about the puzzle is retained.

So without it we have:
Boob Cube will be called (1/1x1)
2x2x1 will be called (2/2x2)
2x2x2 will be called (4/4x3)
3x3x3 will be called (9/17 x6)
4x4x4 will be called (28/28x3, 16/40x6)
Skewb/Ultimate Skewb will be called (7/7x4)
Megaminx will be called (11/81x12)
Pyraminx Crystal will be called (15/35x12) (I think you meant 3./7 here)
Square-1 will be called (8/10x2, 9/9x1)
UFO will be called (6/6x4)

(I wonder if the Square-1 is better defined not bandaged as (12/14x2, 13/13x1), as otherwise turns of the central slice can be described as even 7/13x1 because of bandaging)

So the UFO is distinguished from the Skewb without the need (in this case) for angular information.

But that would be nice too, although I can't figure a nice way to include it without cluttering up the entire system. It seems Jaap's Sphere app has addressed many of these points in a completely different way (Here's a Skewb for those who haven't seen the app before, it is specified by "sym=1&blue=150).

But this does not seem to cover non-symmetric puzzles which your notation nicely does.

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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 2:25 am 
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DLitwin wrote:
So the UFO is distinguished from the Skewb without the need (in this case) for angular information.


Two thinking brains are better than one, so I appreciate your comment. Spot on! :)
I also agree 100% with your other suggestions.

Regarding similar cases to the one of the Square-1 having pieces interchanged,
it would always be safe to use this notation only when the puzzle is in its solved state.
I take no responsibility to what will happen to the "cut numbers universal paradox"
once such a puzzle is horribly mixed up.
(Or we could set some conditions for how to choose "variable" cut numbers)

:mrgreen:


Pantazis

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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 5:11 am 
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Okay, here is my thought on the matter. Does the Deep cut apply only to the center, or can it be applied to a face as well?

In that case, the Pyraminx has intersections centered on the faces. The skewb diamond has the same geometry but is deep cut via the center.

The junior cube is both of those, with intersections at the face and center.

The Megaminx is analogous to the standard Rubik's cube, and likewise has a max of 3 planes that intersect a single point inside the puzzle. The Pyraminx Christal (also known as a brillic) has points in the center of each face at which 5 out of 12 planes intersect at a single point. One can definitely argue that the Pyraminx Chrystal is more deep cut than a Megaminx, as each twist translates 10 edge pieces instead of 5. Finally, we have the Pentultimate, which has all planes intersecting at a center point. I am skeptical that it would even be possible to construct such a monster, but there are virtual simulations.

Actually it has been constructed - Congrats IO!!! - and it is one hell of a beast:
http://www.puzzleforge.com/main/index.p ... &Itemid=54

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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 10:06 am 
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How about this. N = Planes of rotation. When N = ...

N = 1 ... A single plane of rotation is present, the puzzle is deep cut.
N = 2 ... If the planes, the planes should be a single line, ergo still a deep cut puzzle.

So from this, we can derive that if N ≤ 2, the puzzle is deep cut.

If the puzzle is...

N = 3 ... the puzzle is only deep cut if the puzzle has all the planes of intersection intersecting at a single point.
N = 4 ... Same as N = 3

So we understand that if N ≤ 2, the puzzle is deep cut, but if N ≥ 3 the puzzle is only deep cut if all of the planes of rotation intersect at a single point.


As for puzzles like the 4x4 and 6x6, I consider them to have deep cut planes, however the full puzzle is not deep cut, like the 2x2. They are a combination of deep and shallow cuts. A hybrid if you will.


Dunno, my two cents.

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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 10:30 am 
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Noah wrote:
N = 2 ... If the planes, the planes should be a single line, ergo still a deep cut puzzle.


Not necesarily.......... it could be a 3x1x1 cuboid! Ok that is rather trivial.

But can a non-deep cut puzzle have deep cuts? E.g. the 4x4x4? 3 planes meet at a point, the other six meet at other points.

I like Kastellorizo's idea:
kastellorizo wrote:
Boob Cube will be called (1. x1)
2x2x1 will be called (1. x2)
2x2x2 will be called (1. x3)
3x3x3 will be called (9./17 x6)
4x4x4 will be called (1. x3, 2./5 x6)
Skewb/Ultimate Skewb will be called (1. x4)
Megaminx will be called (11./81 x12)
Pyraminx Crystal will be called (1./7 x12)
Square-1 will be called (4./5 x2, 1. x1)

(I hope I have not made a calculation mistake above, but you know what I mean!)


That way, we will even get the non-trivial pieces in the picture,
and in my opinion, a classification of puzzles becomes easier.
Based on this, definitions of whether a puzzle is "clear deep cut"
(i.e. it only has 1.'s), "mixed deep cut", or "non-deep cut",
could be easily given. After this, I can already envision of other
possible puzzles which have never been thought before.

I might use this notation on my website, because I kind of like it.


That is quite playful and interesting. I can't think of anything that couldn't be defined in this way, although I am sure that it would be possible to think of situations where this does not uniquely identify a puzzle, i.e. two different puzzles could have the same category number.

What fun!


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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 10:48 am 
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contrabass wrote:
How about a slightly different definition. All planes of rotation go through the center of the puzzle.


This would exclude a whole bunch of puzzles, every size regular nxnxn cube included :shock:

Per

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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 11:01 am 
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perfredlund wrote:
This would exclude a whole bunch of puzzles, every size regular nxnxn cube included


Exactly. I do not believe that regular nxnxn cubes (other than the 2x2) are deep cut, though the evens do contain deep cut slices.
It's OK to have an extremely limited definition of a type or property of a puzzle.


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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 1:20 pm 
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Julian wrote:
I think your "rotational plane divides two isomorphic groups" hits the nail on the head.

Hmmm... I am no longer sure this definition completely satisfies me.

I have two concerns:

1.) Consider the Brain Ball. You can't quite see from the picture, but when you rotate some pieces others on the other side of the puzzle rotate as well (four on one side, three on another).

Here we have a plane of rotation that is complicated by rotating disjoint groups. Pieces rotating and pieces not rotating are found on both sides of the rotational plane. If we consider a Brain Ball that was symmetric with 16 pieces, and a set of four and four turning on the 180 degree rotation we might even say that the groups divided by the plane are isomorphic (maybe...?). But I hesitate to call this a deep cut puzzle.

2.) This brings up the issue of isomorphic groups that are identical, but not half the entire puzzle. The BrainBall is my only current example of a puzzle of this sort, but I'm sure one could construct many others where you have planes of rotation dividing isomorphic groups that do not comprise the entire puzzle. [Edit: Imagine a Flip-Side with an even number of columns and even columns in the swap area for an other puzzle example].
I think to be a true deep cut we have to add this further qualification.

I guess what I am getting at is a plane of rotation can be anywhere along the axis of rotation. The concept of "dividing isomorphic groups" needs to be strengthened to imply that the plane's location divides groups that comprise exactly half of the puzzle pieces and those groups move as one unit.

Dave

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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 3:51 pm 
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Noah wrote:
How about this. N = Planes of rotation. When N = ...

N = 1 ... A single plane of rotation is present, the puzzle is deep cut.
N = 2 ... If the planes, the planes should be a single line, ergo still a deep cut puzzle.

So from this, we can derive that if N ≤ 2, the puzzle is deep cut.

If the puzzle is...

N = 3 ... the puzzle is only deep cut if the puzzle has all the planes of intersection intersecting at a single point.
N = 4 ... Same as N = 3

DLitwin wrote:
Julian wrote:
I think your "rotational plane divides two isomorphic groups" hits the nail on the head.

Hmmm... I am no longer sure this definition completely satisfies me.

I think to be a true deep cut we have to add this further qualification.

I guess what I am getting at is a plane of rotation can be anywhere along the axis of rotation. The concept of "dividing isomorphic groups" needs to be strengthened to imply that the plane's location divides groups that comprise exactly half of the puzzle pieces and those groups move as one unit.

Dave
I think that Noah's and your posts together have now given us a decent definition (assuming a word accidentally got missed or deleted, as I'm pretty sure Noah meant to type "N = 2 ... If the planes intersect, the planes should be a single line, ergo still a deep cut puzzle"). So my suggested definition, pinched from Noah and you and now offered back for further peer review, is:

Provided that all moving pieces of the puzzle move together either side of each "cut", and defining the term "cut" to mean a complete cut through the puzzle that acts as a plane of rotation for all its moving pieces: If a puzzle has one cut, or two cuts that intersect within the puzzle, or three or more cuts that intersect at a single point within the puzzle, the puzzle is a (pure) deep cut puzzle.

I would add my personal interpretation of what has already been suggested by Kastellorizo: If some cuts can be removed to make the puzzle a (pure) deep cut puzzle, it is a mixed/hybrid/partial deep cut puzzle. If just "deep cut puzzle" is stated, it can be assumed that "pure deep cut puzzle" is meant. But of course that last bit is following my own personal bias!

[Edit - Made a couple of word substitutions in the definition: "defining" for "regarding" and "acts" for "serves".]


Last edited by Julian on Tue Sep 23, 2008 7:56 pm, edited 1 time in total.

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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 3:59 pm 
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I think Julian hit the nail on the head there. Seems perfect to me.

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 Post subject: Re: Deep cut puzzles
PostPosted: Tue Sep 23, 2008 5:33 pm 
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I'm still a little uneasy about the fact that cuts are allowed that can only hit another moveable position after 360 degrees. (Like a UFO without the horizontal slice, and with one vertical slice removed.)

May I suggest this line?

All cuts in question must allow meaningful rotations less than 360 degrees.

Or this more restrictive one?

All cuts in question must allow meaningful rotations less than 360 degrees that restore the original layout of cuts and prepare the puzzle for another meaningful move.

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 Post subject: Re: Deep cut puzzles
PostPosted: Thu Sep 25, 2008 2:51 am 
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DLitwin wrote:
[Edit: Imagine a Flip-Side with an even number of columns and even columns in the swap area for an other puzzle example].


One moment. This is entering another field. ;)

So far, I had been thinking of puzzles which are rotatable, using one full circle at a time.

i.e. puzzles like the Octo, the Braintwist, or even the Gripple(!) can be classified with the
"cut number" definition. But a Rubik's clock, Great Gears, or Rubik's Rings cannot.
Thus, number of cirlces rotated at time is needed here.

Now, if we add to the picture a different type of movement, i.e. sliding, which we then
wish to combine it with rotations, then and only then, we would be able to define puzzles
such as the Flip-Side, the Trillion, and even the Hexadecimal Puzzle!

So, should we try to define now the sliding bit too? (Elementals included!)

;)


Pantazis


PS. This is just some fast thoughts, I have been un-humanely busy lately!

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 Post subject: Re: Deep cut puzzles
PostPosted: Thu Sep 25, 2008 8:13 pm 
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Seeing as everyone is making a huge debate out of this (though I don't think the topic deserves it), I thought I would just throw in this question for fun.

Is a pyraminx a deepcut puzzle????????????

:lol: enjoy!


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 Post subject: Re: Deep cut puzzles
PostPosted: Thu Sep 25, 2008 10:02 pm 
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Allagem wrote:
Seeing as everyone is making a huge debate out of this (though I don't think the topic deserves it), I thought I would just throw in this question for fun.

Is a pyraminx a deepcut puzzle????????????

:lol: enjoy!

Thanks, I've been dying to ask that question!

Most people don't intuitively think they're deep cut, yet most of the definitions offered so far would classify the Pyraminx and/or Tetraminx as deep cut.

From the perspective of puzzle analysis (and design) there's an insightful gem in figuring out why.

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 Post subject: Re: Deep cut puzzles
PostPosted: Fri Sep 26, 2008 2:09 am 
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VeryWetPaint wrote:
Most people don't intuitively think they're deep cut, yet most of the definitions offered so far would classify the Pyraminx and/or Tetraminx as deep cut.

Interesting.

The planes don't meet in a common point (each set of two planes meets at the line formed by the inside edge of an edge piece, all surrounding the core).

I think the plane dividing isomorphic groups definition rules it out quite handily as well.

I agree that because it has no center piece near slice boundaries it might intuitively seem deep cut, but what given definition supports it?

Dave

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 Post subject: Re: Deep cut puzzles
PostPosted: Fri Sep 26, 2008 4:45 am 
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Sorry, "most" was a poor choice of words. While following this thread (and the Polaris thread) I've occasionally noticed well-intentioned conjectures that could inadvertently classify Pyraminx or Tetraminx as deep-cut. (io's axis-based conjecture, for example)

Bear with me, I'm not really suggesting Pyraminx should be classified as deep-cut. Not purely, anyway. I'm just suggesting it has a key connection to the deep-cut puzzle attribute. That connection provides some insight into the whole group of puzzles that includes Pyraminx, Tetraminx, Skewb Cube, Skewb Diamond, and Skewb Ultimate. Oh, and Cubominx.


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 Post subject: Re: Deep cut puzzles
PostPosted: Fri Sep 26, 2008 11:09 am 
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Pyraminx? What Pyraminx? I only see the (5/9x4, 1/13x4) puzzle.
And the Tetraminx is the (4/6x4) puzzle, isn't it?
(The Tetraminx values actually reveal a greater... "deep-cutiness"!!! - i.e. fraction is closer to 1)

:P


Pantazis


PS On a serious note, I might have some nice definitions for the the sliding moves too! :)

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 Post subject: Re: Deep cut puzzles
PostPosted: Fri Sep 26, 2008 11:55 am 
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VeryWetPaint wrote:
Sorry, "most" was a poor choice of words. While following this thread (and the Polaris thread) I've occasionally noticed well-intentioned conjectures that could inadvertently classify Pyraminx or Tetraminx as deep-cut. (io's axis-based conjecture, for example)

Bear with me, I'm not really suggesting Pyraminx should be classified as deep-cut. Not purely, anyway. I'm just suggesting it has a key connection to the deep-cut puzzle attribute. That connection provides some insight into the whole group of puzzles that includes Pyraminx, Tetraminx, Skewb Cube, Skewb Diamond, and Skewb Ultimate. Oh, and Cubominx.



I must win an award for "worst communication of ideas". My axis-based statement (#1 in my list) was only about placing a theoretical sphere inside the puzzle. You know all the stuff about great circles on the sphere? I was just trying to define what sphere you need to answer that question about a puzzle.

So, #1 says where in space to place an imaginary sphere, and nothing more. It doesn't say anything about the deepness of a puzzle.
#2 says that cuts forming great circles on that sphere are deep.
#3 is my pet thing about a need for the puzzle to be scrambleable.

The pyraminx and tetraminx aren't deep cut by my definition, because when you imagine a sphere centered at the intersection of all the axes (#1 gives you the sphere), the cuts don't form great circles on the sphere (#2 says this). So it's not deep cut.

Does that make more sense? I didn't do so well in English class as you can tell. :)

Time to start drawing some pictures this weekend. :D

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 Post subject: Re: Deep cut puzzles
PostPosted: Fri Sep 26, 2008 3:26 pm 
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You're right, I goofed. I was mentally visualizing the cuts of a Halpern-Meier Pyramid not a Pyraminx. (It was after midnight so my mind was foggy.)

But it's still an interesting observation if you substitute the right puzzle: the H-M Pyramid seems deep-cut in some ways and non-deep-cut in other ways. The cuts seem unbalanced because and each turn divides the puzzle into 1 vertex on one side, and 3 vertexes on the other. Could these groups be isomorphic? (Upon reflection I think they could, depending on how you define the group operations.)

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 Post subject: Re: Deep cut puzzles
PostPosted: Fri Sep 26, 2008 5:59 pm 
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The Halpern-Meier is a skewb shape mod. If the Skewb is deep cut, then the pyramid should be as well.

The turning groups are isomorphic even if the pieces don't look the same on the facade.
A center face = vertex.


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 Post subject: Re: Deep cut puzzles
PostPosted: Fri Sep 26, 2008 6:07 pm 
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I wasn't even thinking about the fact that it was a Skewb mod, but if we define it in Pantazis notation: (7/7x4) it becomes clear that it is. The groups are isomorphic even though they look quite different, and it meets all the other definitions for deep cut that seem to be rising to the surface of this conversation :)

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 Post subject: Re: Deep cut puzzles
PostPosted: Fri Sep 26, 2008 7:17 pm 
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Ah, so you readily accept that the vertexes and centers belong to the same group. Naturally.

I approached that topic timidly because I initially thought the group operations might depend on how you chose to define their properties (being operated on): the vertex pieces have two conspicuous properties, orientation and position, but I thought it might be valid to define the centers as having the same two properties or just the position property, depending on your whim. If this was allowed it would break group isomorphism.

Upon reflection that was perfectly absurd. Orientation is an intrinsic property of the faces, even if it isn't conspicuously observable. Duh!

The hazard of being a latecomer...

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 Post subject: Re: Deep cut puzzles
PostPosted: Sat Sep 27, 2008 9:33 am 
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Thinking about the "isomorphic groups" and "great circles" approaches this morning.... I started wondering if they're equivalent in this context?

Given a theoretical sphere divided by some number of arbitrary great circles, I have a question:

1 Does each and every circle divide the sphere into two isomorphic groups of pieces?
2 Can you think of an arrangement of great circles that doesn't have this property?

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 Post subject: Re: Deep cut puzzles
PostPosted: Sat Sep 27, 2008 11:05 am 
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io wrote:
Thinking about the "isomorphic groups" and "great circles" approaches this morning.... I started wondering if they're equivalent in this context?

Given a theoretical sphere divided by some number of arbitrary great circles, I have a question:

1 Does each and every circle divide the sphere into two isomorphic groups of pieces?
2 Can you think of an arrangement of great circles that doesn't have this property?


Here's a thought on that:
When any two great circles are on a sphere, they intersect in a line that goes through the center of the sphere, and therefore the center of the circle. (A diameter)
No matter how many other circles exist, they always intersect each other in diameters.
These intersection lines obviously go through the center point of the circles.
Because all of these lines intersect at a central point, they always make vertical angles.
Vertical angles are always equal.
Thus, for every arc cut on a sphere by great circles, there is another arc that is equal and directly opposite.

I'm pretty sure this forces symmetry, and isomorphism.


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 Post subject: Re: Deep cut puzzles
PostPosted: Sat Sep 27, 2008 3:32 pm 
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In rubik-type puzzles, the axes of rotation generally all go through a single point. A slice is 'deep cut' if it goes through that point.

The Unscrambled is deep cut but bandaged. Puck puzzles are deep cut but have a funny mechanical doohickey. Things like the brain ball are hard to classify in a particularly meaningful way. At that point the real question becomes what fraction of all pieces get moved at once. In the case of the brain ball the two groups are of size 7 and 6, which is as close to even as you can get with 13 pieces. Geared puzzles can divide the puzzle into multiple groups, thus raising the 1/2 barrier to 2/3.


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 Post subject: Re: Deep cut puzzles
PostPosted: Sun Sep 28, 2008 7:30 am 
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Bram wrote:
Things like the brain ball are hard to classify in a particularly meaningful way. At that point the real question becomes what fraction of all pieces get moved at once. In the case of the brain ball the two groups are of size 7 and 6, which is as close to even as you can get with 13 pieces.


Good point. For the Brain Ball, we may assume that there are
13 (imaginary) slices instead of one (all possible moves should be included).
The next step is to ensure that we include the spherical parts,
i.e. the middle one and the two side pieces of the sphere which can be seen as
as one connected piece or two pieces. But since there is not change between
them (including orientation), it should be regarded as one piece.

Therefore, accroding to the notation, its name should be (7/8 x13)

:)


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 Post subject: Re: Deep cut puzzles
PostPosted: Thu Oct 08, 2009 5:34 pm 
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io wrote:
Thinking about the "isomorphic groups" and "great circles" approaches this morning.... I started wondering if they're equivalent in this context?

Given a theoretical sphere divided by some number of arbitrary great circles, I have a question:

1 Does each and every circle divide the sphere into two isomorphic groups of pieces?
2 Can you think of an arrangement of great circles that doesn't have this property?


I hate to bring up an old topic but I just spotted a puzzle that I really think is relevant to this topic. Look at this puzzle...

http://www.shapeways.com/model/49957/slice_kilominx.html

It is a slice turn Kilominx. In many many ways... it is to a Megaminx what a 2x2x2 is to a 3x3x3. Is a Slice-turn Kilominx a deep cut puzzle? Each cut divides the puzzle into 10 equal pieces... granted one of them is displaced physically into 2 pieces but they move together as one unified whole. It also only has 6 valid rotations, half what is available on the Megaminx. Just like the 3 valid rotations on a 2x2x2 which is half the rotations a 3x3x3 has. This is one property of deep cut puzzles is it not? If it is... then notice the Pentultimate isn't the ONLY face-turning Dodecahedron based puzzle that is deep cut.

I'm really quite curious what others here think?

I've also just completed a rather abstract analysis of the Pentultimate under the assumption that it was the only Order=1 Face-Turning Dodecahedron and found a VERY big clue in that analysis that it wasn't the only one... in fact it pointed me strait to the Kilominx. This analysis was me copying something Andreas Nortmann had done so I'll sit on that for now as I'm not sure he wants that shared but from a mathematical point of view... it would be VERY nice to be able to include the Slice-turn Kilominx in the same class as the Pentultimate.

Order=2 Face-Turning Dedecahedrons would include the Megaminx, Pyraminx Crystal, Starmix, etc.

Carl

P.S. The reason I copied io's post is because he missed one question tied to his two. Can an arrangement of non-Great circles divide the the sphere into two isomorphic groups of pieces?

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Last edited by wwwmwww on Thu Oct 08, 2009 6:17 pm, edited 2 times in total.

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 Post subject: Re: Deep cut puzzles
PostPosted: Thu Oct 08, 2009 6:01 pm 
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wwwmwww wrote:
I hate to bring up an old topic but I just spotted a puzzle that I really think is relevant to this topic. Look at this puzzle...

http://www.shapeways.com/model/49957/slice_kilominx.html

It is a slice turn Kilominx. In many many ways... it is to a Megaminx what a 2x2x2 is to a 3x3x3. Is a Slice-turn Kilominx a deep cut puzzle? Each cut divides the puzzle into 10 equal pieces... granted one of then is displaced physically into 2 pieces but they move together as one unified whole. It also only has 6 valid rotations, half what is available on the Megaminx. Just like the 3 valid rotations on a 2x2x2 which is half the rotations a 3x3x3 has. This is one property of deep cut puzzles is it not? If it is... then notice the Pentultimate isn't the ONLY face-turning Dodecahedron based puzzle that is deep cut.

It's very perceptive of you to notice that Oskar's Slice Kilominx divides the pieces into two identical groups, but I think you've jumped at the wrong metaphor. It's probably more accurate to say this puzzle is to a Megaminx what an Equator is to a 3x3x3. Although that's not an isomorphically perfect comparison, it emphasizes that pieces on the Slice Kilominx always move around its equator.

Since all the pieces move trivially on the surface of a sphere Oskar's Slice Kilominx certainly doesn't embody the intent of the term "deep cut".

Your observation emphasizes the difficulty of devising a clear, concise, easily-understood definition.


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 Post subject: Re: Deep cut puzzles
PostPosted: Thu Oct 08, 2009 6:24 pm 
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It's not deep cut. It may share some properties with deep cut, but it isn't deep cut
Bram wrote:
In rubik-type puzzles, the axes of rotation generally all go through a single point. A slice is 'deep cut' if it goes through that point.


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