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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 6:50 pm

Joined: Thu Jun 11, 2009 9:09 pm
Location: New Hampshire
What ever happened to the Polaris project?

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 7:01 pm

Joined: Thu Jan 06, 2005 8:53 pm
Location: Los Angeles
squabpuzzles wrote:
What ever happened to the Polaris project?

Kind of off topic, but:
Still in the works ... Got the masters all ready, and a couple molds made about a month ago. Unfortunately, I'm working 12 hours a day 7 days a week at the moment so I don't have the time, energy, or motivation to work on it.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 7:10 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
TBTTyler wrote:
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.

Well the slice of the Kilominx isn't planer... but you can think of it as conical if you'd like. It explains the two disjoint areas in one piece of the cut and that conical slice DOES go through that point.

Carl

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 7:51 pm

Joined: Thu Jan 06, 2005 8:53 pm
Location: Los Angeles
wwwmwww wrote:
Well the slice of the Kilominx isn't planer... but you can think of it as conical if you'd like. It explains the two disjoint areas in one piece of the cut and that conical slice DOES go through that point

Counterexample: Make a 3x3 into a slicing sphere, and you can use the same argument.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 8:22 pm

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
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 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.
Stating that these two regions are isomophically alike is like saying that the Arctic and Antarctic circles are isomorphic with the Tropics of Cancer and Capricorn. Just because they contain the same ten identical pieces, the configuration is vastly different (not to mention that the Tropics are generally very hot whereas the poles are generally very cold )

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cisco wrote:
Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 8:26 pm

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
Sorry to double post, but I feel that what I have to say warrants it:

Here is my opinion of what defines a deep cut puzzle:

#1 - All axis of rotation intersect at a single point, which I will refer to as the zero point. The vast majority of standard twisty puzzles fit this description, whether Platonic, Archemedian , radial (puck, square 1), or otherwise. Shape mods, extended cuboids, bandaged puzzles, etc fit this as well. The shape, size, and other measurable dimensions of the puzzle are not taken into consideration at all. We will only analyze the cut planes and the axis of rotation. For example, this places three functionally different puzzles, the Megaminx, Pyraminx Crystal, and Starminx, in the same configuration because each has 12 congruent cutting planes on six axes with dodecahedral symmetry.

#2 - Any cutting plane that intersects the zero point is a deep cut plane. Any cutting plane that does not intersect the zero point is a shallow cut plane.

#3 - Puzzles whose cut planes do not intersect the zero point are shallow cut puzzles. Puzzles whose cut planes intersect the zero point are deep cut puzzles. Many puzzles, such as the 4x4x4, are both shallow cut and deep cut.

#4 - higher order puzzles (5x5x5, gigaminx, etc) can have multiple layers of shallow cuts per axis, but are still not considered deep cut unless the cutting planes intersect the core (zero point)

Examples:

Cubic (face rotating - 3 axis):
1x1x1 - N.A. - not a puzzle
2x2x2 - purely deep cut
3x3x3 - purely shallow cut
4x4x4 - deep cut and shallow cut
5x5x5 - multilayer shallow cuts
nxnxn (n>5) - large even cube contain deep cuts in addition to multiple layers of shallow cuts; odd cubes contain only multiple layers of shallow cuts

Cuboids:
1x1x2 - N.A. - no intersecting planes
1x2x2 - purely deep cut
1x3x3 - purely shallow cut
2x2x3, 2x3x3, 3x3x4 (even & odd) - deep cut and shallow cut
2x2x4 (even) - deep cut, with additional shallow cutting planes
3x3x5 (odd) - purely shallow cut, with additional shallow cutting planes

Octahedral, Tetrahedral (vertex rotating cubic - 4 axis):
All Skewb mechs - deep cut
Pyraminx - shallow cut
Octahedron - shallow cut
Rainbow Cube - shallow cut
Dino Cube - shallow cut

Dodecahedral (6 axis):
Kilominx, Impossiball - shallow cut
Megaminx - shallow cut
Gigaminx, etc. - multiple shallow cut
Pyraminx Crystal - shallow cut
Starminx - shallow cut
Pentultimate - deep cut

Isosahedral (10 axis):
???

Rhombic Dodecahedral (edge rotating cubic - 6 axis):
Little chop - deep cut
Helicopter cube - shallow cut

Rhombic Triacontahedral (15 axis):
Big chop - deep cut

Radial Puzzles have multiple deep cut vertical planes (with horizontal axes) and optionally may have one or more horizontal planes (with a central vertical axis) as well:

Square 1, 21
Super Square 1
Puck Puzzle
UFO
Masterball
Etc.

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cisco wrote:
Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.

Last edited by stardust4ever on Thu Oct 08, 2009 8:55 pm, edited 3 times in total.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 8:38 pm

Joined: Mon Jan 26, 2009 9:00 pm
Stardust4ever took the cake with that post.

Now what we need is a classification system for "deeper, shallow cut" puzzles. Like a Starminx to a Pyraminx Crystal to a Megaminx.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 8:48 pm

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
EMarx wrote:
Stardust4ever took the cake with that post.

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cisco wrote:
Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 8:53 pm

Joined: Thu Jan 06, 2005 8:53 pm
Location: Los Angeles
EMarx wrote:
Now what we need is a classification system for "deeper, shallow cut" puzzles. Like a Starminx to a Pyraminx Crystal to a Megaminx.

Perhaps use a number 0-180 indicating the angle of the cone that would make that puzzle on a sphere. 180 would be deep cut.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 8:54 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
TBTTyler wrote:
Counterexample: Make a 3x3 into a slicing sphere, and you can use the same argument.

Not quite... you'd have to link the opposite sides such that all the pieces in the cone were linked together. And even then it doesn't cut the puzzle into into two isomorphic groups of pieces.

Maybe try that on a 4x4x4 and link U to D and MU to MD ect but if you do that you are back to a puzzle that I think is equivalent to a 2x2x2.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 9:00 pm

Joined: Thu Jan 06, 2005 8:53 pm
Location: Los Angeles
wwwmwww wrote:
TBTTyler wrote:
Counterexample: Make a 3x3 into a slicing sphere, and you can use the same argument.

Not quite... you'd have to link the opposite sides such that all the pieces in the cone were linked together. And even then it doesn't cut the puzzle into into two isomorphic groups of pieces.

Maybe try that on a 4x4x4 and link U to D and MU to MD ect but if you do that you are back to a puzzle that I think is equivalent to a 2x2x2.

Also, it was a counterexample. It wasn't supposed to work.
Restating: If you make a 3x3 into a slice turning sphere and apply the conical cut argument, the pieces aren't in isomorphic groups. Therefore saying that conical cuts go through the center is not enough to call a puzzle deep cut.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 9:07 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
stardust4ever wrote:
Stating that these two regions are isomophically alike is like saying that the Arctic and Antarctic circles are isomorphic with the Tropics of Cancer and Capricorn. Just because they contain the same ten identical pieces, the configuration is vastly different (not to mention that the Tropics are generally very hot whereas the poles are generally very cold )

If I followed that correctly then you couldn't have a Tetrahedral Face-Turn deep cut puzzle... one end would be very pointy... and the other rather flat yet you can have a Tetrahedral Skewb which appears on your list.

http://en.wikipedia.org/wiki/Isomorphism

Isomorphic doesn't mean identical... it just means you can map from one set to the other. If I could map each point that was X degrees below zero in the artic to a spot that was X degrees above zero in the tropic then those points ARE isomorphic.

Carl

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Last edited by wwwmwww on Thu Oct 08, 2009 9:28 pm, edited 1 time in total.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 9:11 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
TBTTyler wrote:
Also, it was a counterexample. It wasn't supposed to work.
Restating: If you make a 3x3 into a slice turning sphere and apply the conical cut argument, the pieces aren't in isomorphic groups. Therefore saying that conical cuts go through the center is not enough to call a puzzle deep cut.

In that case we are in agreement. You can't turn a 3x3x3 into a deep cut puzzle. However I believe the cuts on a Slice Kilominx do cut the puzzle into isomorphic groups so it's conical cuts are different then any you could apply to a 3x3x3.

Carl

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 9:16 pm

Joined: Tue Aug 11, 2009 2:44 pm
stardust4ever wrote:
Here is my opinion of what defines a deep cut puzzle:

But your rules assume that rotation is always along cutting planes, which is manifestly not the case. In the slice kilominx, the rotational surface is conical. There are plenty more examples of non-planar rotational surfaces.

A more general definition of a deep-cut puzzle might require that (some) cutting *surfaces* intersect a point which is also the intersection of all rotational axes.

However, when deep-cut puzzles are discussed here, the mechanical issues that are the reason for the discussion tend to relate to cutting planes bisecting the puzzle. So perhaps this argument over semantics is a bit beside the point.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 9:24 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
stardust4ever wrote:
Here is my opinion of what defines a deep cut puzzle:

#1 - All axis of rotation intersect at a single point, which I will refer to as the zero point. The vast majority of standard twisty puzzles fit this description, whether Platonic, Archemedian , radial (puck, square 1), or otherwise. Shape mods, extended cuboids, bandaged puzzles, etc fit this as well. The shape, size, and other measurable dimensions of the puzzle are not taken into consideration at all. We will only analyze the cut planes and the axis of rotation. For example, this places three functionally different puzzles, the Megaminx, Pyraminx Crystal, and Starminx, in the same configuration because each has 12 congruent cutting planes on six axes with dodecahedral symmetry.

#2 - Any cutting plane that intersects the zero point is a deep cut plane. Any cutting plane that does not intersect the zero point is a shallow cut plane.

#3 - Puzzles whose cut planes do not intersect the zero point are shallow cut puzzles. Puzzles whose cut planes intersect the zero point are deep cut puzzles. Many puzzles, such as the 4x4x4, are both shallow cut and deep cut.

So... only puzzles with planar cuts can be deep cut puzzles? A planar cut that goes through your zero point cuts the puzzle into isomorphic groups by definition. So it would seem to generalize that a cut that divides the puzzle into two isomorphic groups planar or otherwise could be considered deep cut.

Carl

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 9:32 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
The slice kilominx isn't a deep cut puzzle, it's a slice puzzle. It is, as someone already said earlier, somewhat analogous to the equator puzzle. But it does have the property that when you slice it the puzzle is divided into two groups of ten pieces, which is an interesting property. It also has the property that two pieces which start out opposite always stay opposite, and I suspect has a whole bunch of other strange properties, but I haven't figured out a solution method myself and haven't read any by other people.

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 Post subject: Re: Deep cut puzzlesPosted: Thu Oct 08, 2009 11:50 pm

Joined: Thu Dec 21, 2006 5:32 pm
Location: Bay Area, CA
wwwmwww wrote:
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.
Interesting.

At first I thought you were confusing equal numbers with isomorphism. I had composed a message saying such, and delving into why the Slice Kilominx groups weren't isomorphic. But then I realized perhaps I wasn't looking at things correctly.

In a deep cut puzzle the two groups are isomorphic, which means the same in their relations to each other. In a deep cut puzzle one can arbitrarily choose one side to be considered "at rest" and the other side is permuted in a particular cycle. When considering the other half at rest, you should find the same cycle in the pieces of the opposite half.

While the groups of the slice Kilominx seem on the surface completely different, if we consider them non-visually we can see the connections. Consider the "top" half of the central band of corners as distinct from the "bottom" half. Once you have, we now have our two groups themselves divided into two groups. Each cycles five pieces in a consistent direction. In this sense they are identical, and the physical proximity of the groups in the central band lends a visual connection not seen in the top/bottom group, but that connection isn't relevant in terms of isomorphism.

Up above I noted:
DLitwin wrote:
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.
I think the issue here is that there is no plane of rotation that separates the two groups, because the central band is in between the disjoint top / bottom group.
So I think it meets all the definitions of deep cut other than the planar division. And I think this is something that will keep people from viewing it as deep cut.

But it is deliciously close

Dave

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 Post subject: Re: Deep cut puzzlesPosted: Fri Oct 09, 2009 2:02 am

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
wwwmwww wrote:
stardust4ever wrote:
.....

So... only puzzles with planar cuts can be deep cut puzzles? A planar cut that goes through your zero point cuts the puzzle into isomorphic groups by definition. So it would seem to generalize that a cut that divides the puzzle into two isomorphic groups planar or otherwise could be considered deep cut.

Carl
I am well aware of the fact that while puzzles appear to have planar cuts on the outside, those cuts inside the mechanism may actually be conical, spherical, paraboloids, or any other convoluted shape that one can conjour up. In fact, all puzzles need some kind of core to function whether deep cut or not. Usually, the core itself is not even bisected by the deep cuts, but is either bound to some specific part in the puzzle (ex: most 2x2x2s, ES 4x4x4) or selectively floats between the hemispheres (ex: Rubik 4x4x4, V-Cube 6x6x6). There are some puzzles where the core is directly bisected (ex: the split spindle domino), as well as many puzzles with non-polyhedral geometry (Square 1, Puck, UFO), but that's beyond the point. I'm not concerned with the shape or internal mechanism of the puzzle, only that its movements can be represented in 3D space as flat planes. So what the actual mechanism inside the puzzle looks like, or the real world contours of the cutting planes is irrelevant.

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cisco wrote:
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 Post subject: Re: Deep cut puzzlesPosted: Fri Oct 09, 2009 2:58 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Here come my 2*N cents:

I think we can make things way easier if we ignore the actual physical puzzle and think of the "mother"-puzzles inside of them.
bhearn wrote:
But your rules assume that rotation is always along cutting planes, which is manifestly not the case. In the slice kilominx, the rotational surface is conical. There are plenty more examples of non-planar rotational surfaces.
wwwmwww wrote:
However I believe the cuts on a Slice Kilominx do cut the puzzle into isomorphic groups so it's conical cuts are different then any you could apply to a 3x3x3.
Dealing with non-planar planes: Find the equivalent puzzle with planar planes even if it is theoretical. If that puzzle is deepcut (by the definition of stardust4ever) then the shape variant with non-planar cuts is deepcut too. Example: The Rex cube is a shape variant of the MasterSkewb.

Dealing with the Slice-Kilominx:
The Slice-Kilominx is a Megaminx the inventor has
1. given another shape so that edges and faces are hidden.
2. built with conical cuts
3. bandaged in a fashion comparable to the B12C111; Funny coincidence!
Another explanation:
Take a Megaminx + Build bridges between all 10 pairs of truly opposite corners + Hide the edges and faces => Slice-Kilominx
BTW: A great achievement!

If you just build the bridges and don't hide edges and faces you don't cut the puzzle into two isomorphic groups any longer.
stardust4ever wrote:
Pyraminx - shallow cut
You have convinced me here. The Deepcut tetrahedron with tetrahedral axis is the HMT, not the Pyraminx.

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 Post subject: Re: Deep cut puzzlesPosted: Fri Oct 09, 2009 6:11 am

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
Bram wrote:
The slice kilominx isn't a deep cut puzzle, it's a slice puzzle. It is, as someone already said earlier, somewhat analogous to the equator puzzle. But it does have the property that when you slice it the puzzle is divided into two groups of ten pieces, which is an interesting property. It also has the property that two pieces which start out opposite always stay opposite, and I suspect has a whole bunch of other strange properties, but I haven't figured out a solution method myself and haven't read any by other people.
Simple: just solve half the puzzle as you would a normal Kilominx. The other half will solve itself

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cisco wrote:
Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.

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 Post subject: Re: Deep cut puzzlesPosted: Fri Oct 09, 2009 6:31 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
stardust4ever wrote:
I am well aware of the fact that while puzzles appear to have planar cuts on the outside, those cuts inside the mechanism may actually be conical, spherical, paraboloids, or any other convoluted shape that one can conjour up. In fact, all puzzles need some kind of core to function whether deep cut or not. Usually, the core itself is not even bisected by the deep cuts, but is either bound to some specific part in the puzzle (ex: most 2x2x2s, ES 4x4x4) or selectively floats between the hemispheres (ex: Rubik 4x4x4, V-Cube 6x6x6). There are some puzzles where the core is directly bisected (ex: the split spindle domino), as well as many puzzles with non-polyhedral geometry (Square 1, Puck, UFO), but that's beyond the point. I'm not concerned with the shape or internal mechanism of the puzzle, only that its movements can be represented in 3D space as flat planes. So what the actual mechanism inside the puzzle looks like, or the real world contours of the cutting planes is irrelevant.

I'm not talking about the mechanism of the puzzle. I fully agree that the actual mechanism inside the puzzle and the real world contours of the cuts planes are irrelevant. That is why I don't agree with the below comment. It may not be a deep cut puzzle but if not it isn't due to the mechanism.

VeryWetPaint wrote:
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".

What I'm talking about is simple geometry. The cuts in the Slice Kilominx can't be thought of as planar as a planar cut divides 3-space into 2 halfs... neither of which is disjoint. A conical cut also divides 3-space into 2 pieces, the area inside the cone and the area outside the cone. The area inside looks like this:

So you can see one of those two areas is disjoint. In the case of the Slice Kilominx each axis of rotation is cut with ONE cut surface that goes through what has been called the zero point by stardust4ever. I'm NOT talking about how this was accomplished by the mechanism at all... that is just what it is. If the cuts were planar you'd need two cuts per axis and the puzzle that would be created would be a Kilominx and NOT a Slice Kilominx as along each axis 3-space is cut into 3 pieces and NOT 2.

Carl

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 Post subject: Re: Deep cut puzzlesPosted: Fri Oct 09, 2009 6:40 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Andreas Nortmann wrote:
Here come my 2*N cents:

I think we can make things way easier if we ignore the actual physical puzzle and think of the "mother"-puzzles inside of them.

Can we? If I take a 3x3x3 and build it up to hide the edges and faces then I have a 2x2x2. The "mother" puzzle isn't deep cut so... I see a problem.

Carl

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 Post subject: Re: Deep cut puzzlesPosted: Fri Oct 09, 2009 7:08 am

Joined: Mon Mar 30, 2009 5:13 pm
Deep cut puzzle: "All planes and axes of rotation pass through the same point"?

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 Post subject: Re: Deep cut puzzlesPosted: Fri Oct 09, 2009 9:07 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Kelvin Stott wrote:
Deep cut puzzle: "All planes and axes of rotation pass through the same point"?

Since I see the "?" I'll counter with...

Deep cut puzzle: "All cut surfaces and axes of rotation pass through the same point"?

Yes, mine as writen has a problem that yours doesn't have. Let me add this:

Deep cut puzzle: "All cut surfaces (that divide the puzzle into two isomorphic groups) and axes of rotation pass through the same point"?

Without the addition you'd have a problem with B12C111. Though with that addition the bit about passing through the same point as all the axes of rotation I think is implied and redundant. So maybe one could say...

Deep cut puzzle: "All cut surfaces divide the puzzle into two isomorphic groups"?

Or maybe we should be more specific and say "Deep planar cut puzzle"? Then does the Slice kilominx become a "Deep isomorphic cut puzzle"?

Carl

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 Post subject: Re: Deep cut puzzlesPosted: Fri Oct 09, 2009 9:38 am

Joined: Mon Mar 30, 2009 5:13 pm
wwwmwww wrote:
Kelvin Stott wrote:
Deep cut puzzle: "All planes and axes of rotation pass through the same point"?

Since I see the "?" I'll counter with...

Deep cut puzzle: "All cut surfaces and axes of rotation pass through the same point"?

Yes, mine as writen has a problem that yours doesn't have. Let me add this:

Deep cut puzzle: "All cut surfaces (that divide the puzzle into two isomorphic groups) and axes of rotation pass through the same point"?

Without the addition you'd have a problem with B12C111. Though with that addition the bit about passing through the same point as all the axes of rotation I think is implied and redundant. So maybe one could say...

Deep cut puzzle: "All cut surfaces divide the puzzle into two isomorphic groups"?

Or maybe we should be more specific and say "Deep planar cut puzzle"? Then does the Slice kilominx become a "Deep isomorphic cut puzzle"?

Carl

No problem. In fact I have to confess I only read the first couple of posts in this thread ( ), so I was only trying to summarize what I had understood in a concise way to help stimulate further thinking...

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 Post subject: Re: Deep cut puzzlesPosted: Fri Oct 09, 2009 12:20 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
stardust4ever wrote:
Isosahedral (10 axis):
???
Gelatinbrain 2.1.5

A shape mod produces this Megaminx-Look-Alike

Carl

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 Post subject: Re: Deep cut puzzlesPosted: Sun Oct 11, 2009 7:18 am

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
Kelvin Stott wrote:
No problem. In fact I have to confess I only read the first couple of posts in this thread ( ), so I was only trying to summarize what I had understood in a concise way to help stimulate further thinking...
There's a lot of intense intellectual debate going on thread, moreso compared to the usual "oh, new puzzle, when can I get one," or "check out my new shape mod," kind of stuff. If you just read the first couple of posts, you're missing the most of the dialog.

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cisco wrote:
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 Post subject: Re: Deep cut puzzlesPosted: Mon Oct 12, 2009 8:36 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Ok.... I wanted to make some pictures to help me visualize a Slice Kilominx made using conical cuts versus a normal Kilominx made using planar cuts.

This is the Slice Kilominx. Notice there are no interior pieces. And you can see the two isomorphic groups.

This is the normal Kilominx. It does have interior pieces which belong to the Megaminx.

By the way... looking more at my abstract analysis of the Pentultimate I believe I may see why its pointing to the Kilominx. And that leads me to this question.... but first let me define a puzzles order. An order=1 puzzle is a puzzle with one cut plane (yes, I mean plane and not surface) per axis of rotation. An order=2 puzzle is a puzzle with two cut planes per axis of rotation, etc. So a 2x2x2 is order=1 and a 3x3x3 is order=2.

Now the question is... does order=1 always equal "deep cut"? In other words... can there be a shallow cut order=1 puzzle?

I think the answer may be yes... look at a face-turn cube with 1 offset cut plane per axis. I think you get <U L F> Bandaged (C222) using the notation Andreas presented here.. Notice no slice turns are allowed. Is it fair to call this an order=1 puzzle? If so, what do you think the shallow cut order=1 face-turn dodecahedron looks like? If I'm right it is NOT a Slice Kilominx.

Carl

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 Post subject: Re: Deep cut puzzlesPosted: Mon Oct 12, 2009 11:53 pm

Joined: Thu Jan 06, 2005 8:53 pm
Location: Los Angeles
wwwmwww wrote:
Now the question is... does order=1 always equal "deep cut"? In other words... can there be a shallow cut order=1 puzzle?

I think the answer may be yes... look at a face-turn cube with 1 offset cut plane per axis. I think you get <U L F> Bandaged (C222) using the notation Andreas presented here.. Notice no slice turns are allowed. Is it fair to call this an order=1 puzzle? If so, what do you think the shallow cut order=1 face-turn dodecahedron looks like? If I'm right it is NOT a Slice Kilominx.

Carl

I think this question ties in deeply with the jumbling vs. bandaging (viewtopic.php?p=126916) discussion.
I would call the c222 a bandaged puzzle, and not a deep cut puzzle because the intersection of the slicing planes != intersection of the axes.
And because it's a bandaged puzzle, and the slices can be completed into a full closed group (the 3x3x3), I would say that it's an order 2 puzzle (using your definition of order).

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 Post subject: Re: Deep cut puzzlesPosted: Tue Oct 13, 2009 12:54 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
TBTTyler wrote:
I think this question ties in deeply with the jumbling vs. bandaging (http://twistypuzzles.com/forum/viewtopic.php?p=126916) discussion.
I would call the c222 a bandaged puzzle, and not a deep cut puzzle because the intersection of the slicing planes != intersection of the axes.
Agreed. I'd call this a shallow cut puzzle.
TBTTyler wrote:
And because it's a bandaged puzzle, and the slices can be completed into a full closed group (the 3x3x3), I would say that it's an order 2 puzzle (using your definition of order).
Maybe I should first ask is there another definition of order? Either way, don't the set of operations <U L F> form a full closed group by themselves? I am aware that this group is a subset of the 3x3x3 but I think its full and closed by itself too.

This is all coming from something very odd going on with the Pentultimate and something different does happen with the face-turn dodecahedron. I don't know a good naming scheme for the faces of a dodecahedron so I've numbered them like this:

The faces/axes of rotation are numbered like a dice. Face C (or 12) is opposite 1. Face B (or 11) is opposite 2. Face A (or 10) is opposite 3. Face 9 is opposite 4. Etc. Now bandage the puzzle such that there is only one allowed cut that can rotate per axis. This can be done using this set of operations <4 6 8 A B C> and if so the core of the Megaminx above that is seen inside the planar cut Kilominx becomes bandaged to 6 Megaminx Face Centers, 6 Megaminx Edges, and 1 Kilominx Corner. This animation shows you the Kilominx and then shows you the bandaged core pulled from the puzzle. The blue area is the original unbandaged core of the puzzle and the 6 faces of it that are seen relate to <4 6 8 A B C>.

Now ask yourself if this bandaged Kilominx that doesn't allow slice rotations is a subset of the original unbandaged Kilominx? I'm told no... both have the same number of permutations and are therefor equivalent puzzles. Mathematically it is precisely this bandaged Kilominx that I'm finding inside a Pentultimate which is order=1 and created with 1 deep cut per axis. So I'm really surprised to find what I thought was an order=2 puzzle inside. It certainly appears to perform as an order=1 puzzle and as its equivalent to the unbandaged Kilominx I've been wondering if the Kilominx should actually be classified as an order=1 puzzle. But how? It sure doesn't fit the definition I've been using. I'm not even sure the bandaged one (which is eqivalent) fits the definition as its very similiar to C222.

Carl

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 Post subject: Re: Deep cut puzzlesPosted: Tue Oct 13, 2009 2:54 am

Joined: Thu Jan 06, 2005 8:53 pm
Location: Los Angeles
wwwmwww wrote:
Maybe I should first ask is there another definition of order? Either way, don't the set of operations <U L F> form a full closed group by themselves? I am aware that this group is a subset of the 3x3x3 but I think its full and closed by itself too.

No, not another definition of order, I just hadn't heard it defined and was making myself clear that I was using the definition you set forward.
And the group for ULF is closed, but there are partial slices that need to be completed to flesh out the underlying puzzle (hence the nod to the jumbling thread).

wwwmwww wrote:
Now ask yourself if this bandaged Kilominx that doesn't allow slice rotations is a subset of the original unbandaged Kilominx? I'm told no... both have the same number of permutations and are therefor equivalent puzzles. Mathematically it is precisely this bandaged Kilominx that I'm finding inside a Pentultimate which is order=1 and created with 1 deep cut per axis. So I'm really surprised to find what I thought was an order=2 puzzle inside. It certainly appears to perform as an order=1 puzzle and as its equivalent to the unbandaged Kilominx I've been wondering if the Kilominx should actually be classified as an order=1 puzzle. But how? It sure doesn't fit the definition I've been using. I'm not even sure the bandaged one (which is eqivalent) fits the definition as its very similiar to C222.

I'll need to do a bit of extra thinking about this, but I'm pretty sure that the "order one-ness" comes from the layers on top of the kilominx and bandaging in the kilominx.
Also, calling the bandaged kilominx and the free kilominx the same puzzle seems a bit hasty. The operations that can be performed on the puzzle define the puzzle as do the possible positions. I would still contend that it's an order 2 puzzle, but bandaged for mechanical reasons.

Kinda like how you can still hit all permutations on a 3x3 with one face anchored. Being a subgroup of a puzzle, or having a subset of operators does indeed make it a different puzzle (or at the very least a non-equivalent puzzle) IMO.

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 Post subject: Re: Deep cut puzzlesPosted: Tue Oct 13, 2009 6:27 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
TBTTyler wrote:
Kinda like how you can still hit all permutations on a 3x3 with one face anchored.

Well I think that's only true for the normal 3x3x3. If you anchor of face of a super 3x3x3 it should have one-fourth as many permutations... I believe. This Kilominx has all pieces with a specified position and orientation. But you do make a good point... this bandaged Kilominx would solve differently then a free one as you couldn't perform slice rotations so I guess that does make it non-equivalent. I still need to do some thinking on this problem myself.

Thanks,
Carl

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 Post subject: Re: Deep cut puzzlesPosted: Wed Oct 14, 2009 8:36 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
wwwmwww wrote:
Now the question is... does order=1 always equal "deep cut"? In other words... can there be a shallow cut order=1 puzzle?
The Pyraminx (without trivial tips) is a good example. It is not deepcut according to the definition I agreed above but it is order=1 even more because you can think of it as a HMT (really deepcut and order=1) with 4 pieces hidden.
Obviously this is only true if you allow non-symmetric planar cuts. Symmetric planar cut are only possible in order=1-puzzles if they are deepcut.

I agree with TBTTyler. The C222 is just a bandaged order=2-puzzle although it is a group.

wwwmwww wrote:
Now ask yourself if this bandaged Kilominx that doesn't allow slice rotations is a subset of the original unbandaged Kilominx? I'm told no... both have the same number of permutations and are therefor equivalent puzzles. Mathematically it is precisely this bandaged Kilominx that I'm finding inside a Pentultimate which is order=1 and created with 1 deep cut per axis. So I'm really surprised to find what I thought was an order=2 puzzle inside. It certainly appears to perform as an order=1 puzzle and as its equivalent to the unbandaged Kilominx I've been wondering if the Kilominx should actually be classified as an order=1 puzzle. But how? It sure doesn't fit the definition I've been using. I'm not even sure the bandaged one (which is eqivalent) fits the definition as its very similiar to C222.
The bandaged Kilominx wwwmwww described is indeed similar to the C222-variant of the 3x3x3. The non-different numbers of permutations shouldn't drive you away from that. You already mentioned that <U L R F B> is isomorphic to <U D L R F B> only on a 3x3x3, not on the Super3x3x3. Something similar works on the Kilominx: On the Kilominx <4 6 8 A B C> is isomorphic to <1 2 3 4 5 6 7 8 9 A B C> but not on the Megaminx. The no-slices Kilominx you define has the same number of permutations as the unbandaged Kilominx only because in the Kilominx everything from the Megaminx that disturb that figure has been hidden.

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 Post subject: Re: Deep cut puzzlesPosted: Wed Oct 14, 2009 9:32 am

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
Andreas Nortmann wrote:
The bandaged Kilominx wwwmwww described is indeed similar to the C222-variant of the 3x3x3. The non-different numbers of permutations shouldn't drive you away from that. You already mentioned that <U L R F B> is isomorphic to <U D L R F B> only on a 3x3x3, not on the Super3x3x3. Something similar works on the Kilominx: On the Kilominx <4 6 8 A B C> is isomorphic to <1 2 3 4 5 6 7 8 9 A B C> but not on the Megaminx. The no-slices Kilominx you define has the same number of permutations as the unbandaged Kilominx only because in the Kilominx everything from the Megaminx that disturb that figure has been hidden.
Are you suggesting that every permutation that is possible on an unbandaged Kilominx is also possible on the slices-only Kilominx? How can that be possible if the mechanism does not allow opposite corners to move independently
stardust4ever wrote:
Bram wrote:
....but I haven't figured out a solution method myself and haven't read any by other people.
Simple: just solve half the puzzle as you would a normal Kilominx. The other half will solve itself

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 Post subject: Re: Deep cut puzzlesPosted: Wed Oct 14, 2009 10:27 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
stardust4ever wrote:
Are you suggesting that every permutation that is possible on an unbandaged Kilominx is also possible on the slices-only Kilominx? How can that be possible if the mechanism does not allow opposite corners to move independently
No. I didn't want to suggest that. I wanted to say that every permutation of the unbandaged Kilominx is still reachable if only the 6 mentioned sides are allowed.

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 Post subject: Re: Deep cut puzzlesPosted: Fri Oct 16, 2009 7:05 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
It looks like any solution to the kilominx which only uses the three faces about one corner will apply just fine to the slice kilominx, so it isn't all that hard to solve, although following through on a solution of that from is likely to be a bit disorienting.

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 Post subject: Re: Deep cut puzzlesPosted: Sun Jul 14, 2013 12:18 am

Joined: Thu Dec 21, 2006 5:32 pm
Location: Bay Area, CA
Sorry for the bump but I realized my post here wasn't enough to lend the support to Carl that he deserved, it really belongs in this thread as the place where this very interesting topic is discussed.

In summary, Carl's post above describing non-planar cuts compels me to consider the Slice Kilominx deep cut, even though I don't know if it will ever be universally agreed. The vast majority of twisty puzzle cuts are planar although recently we have seen conical and spherical cuts. For the purpose of generality it seems we should now consider including even extended conical (hourglass) cuts reasonably when considering the question of deep cut.

Dave

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 Post subject: Re: Deep cut puzzlesPosted: Sun Jul 14, 2013 2:09 am

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
That is one EPIC bump!

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 Post subject: Re: Deep cut puzzlesPosted: Sun Jul 14, 2013 12:57 pm

Joined: Thu Dec 21, 2006 5:32 pm
Location: Bay Area, CA
stardust4ever wrote:
That is one EPIC bump!
Ah, the privileges of being an Admin...

The main point was to bring that thought to this thread, rather than it being dispersed through other threads, so I could point the new Twistypedia entry here.

Dave

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 Post subject: Re: Deep cut puzzlesPosted: Sun Jul 14, 2013 4:06 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
DLitwin wrote:
stardust4ever wrote:
That is one EPIC bump!
Ah, the privileges of being an Admin...
True... and even then note he blames it on ME!!!

LOL!!!!
Carl

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 Post subject: Re: Deep cut puzzlesPosted: Mon Jul 15, 2013 2:39 pm

Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
Aaaah... to me it was a nice breeze to remember this discussion from 2009.
And four years later, I still love the "cut number" method!

It also allows to automatically differentiate different types, i.e. we may define
with excellent accuracy:

1. the change of positions of the movable parts, and
2. the change of the orientation of each part, movable or... not!

For example, a 3x3x3 cube has eight corners, six centers, and twelve edges.
Each corner can change both position and orientation, and it has three stickers.
Each center can only change orientation (four positions).
Each edge can change both position and orientation, and it has two stickers.

Therefore using the cut numbers (as defined before) which represent the possible moves,
we have that the 3x3x3 can be written as:

9/17 x6, or more analytically, [1(4*)+4(2)+4(3) / 5(4*)+8(2)+4(3)] x6

Translating the numerator: we have a center which is non-interchangeable (hence the
asterisk *) with four orientations, plus four edges which have two orientations, plus
four corners with three orientations.

Yes, you may think that this looks a bit longer than before, but I assure you, you won't
find another way of describing all known puzzles, simply because it described all the
provided features of a puzzle in an extremely elementary level.

I mean, now, even the Boob Cube has a fairer description of 1(4*)/1(4*) x1, doesn't it?
That is, if we had a triangular Boob Cube (God, what am I saying again?) it can now be
easily distinguished to the description of 1(3*)/1(3*) x1.

Pantazis

PS. Now... I feel tempted to even provide a second generation effort for re-defining
(even arbitrary) the "difficulty of a puzzle"!

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 Post subject: Re: Deep cut puzzlesPosted: Mon Jul 15, 2013 4:07 pm

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
Yeah, it's definitely possible for a puzzle to be deep cut but also non-isomorphic. I mentioned this in another thread a long time ago, but many consider the cheese to be deep cut. Yet I own Timur's constellation Six which is basically a triangular cheese with three pyramid-shaped corners. The Constellation six allows for 90 degree turns and exhibits some fascinating fudged geometry with near-miss angles based loosely on the Fibbonacci sequence. The circular centers would actually have been mod-13 if they weren't free to rotate. The three slice moves which divide the puzzle all intersect on the central axis at 60/120 degree angles in the center of the puzzle, so it is clearly deep cut. Yet when you look at the puzzle, there are two pointy rotatable corners on one side of the slice, with one pointy rotatable corner on the other side, so the sides are clearly not isomorphic to each other.

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