I start this thread to not take Gelatibrain's off topic further. The multiquote is used to make everyone understand the beginning of this topic.

Stefan Schwalbe wrote:

Andreas Nortmann wrote:

Stefan Schwalbe wrote: I came more and more to the conclusion, that a twisty puzzle is a combination (or composition) of piecetypes. That helps me also solving. A puzzle is a combination of aspects: piecetype 1 permutation, piecetype 1 orientation, piecetype 2 permutation, piecetype 2 orientation and so on.

gelatinbrain wrote:I agree with you on basis. But more I discover new puzzles, more they makes me confused.
Anyway, the definition and classification of puzzles should be based on their abstractions(graphical stracture, group configuration, piece attributes, etc) and not on their physical or geometrical aspect. It is not that easy even under very limited conditions...

Great minds think alike:
http://twistypuzzles.com/forum/viewtopi ... 96#p190396
Based on what Stefan outlined there are 18 derivates of the 3x3x3, including the 1x1x1. Examples for all of them are shown here: viewtopic.php?f=14&t=17517
Based on what Stefan outlined above there are 972 diferent faceturning dodecahedrons, including the (dodecahedral) 1x1x1. Even more variants are possible, if the pieces from the circleMegaminx's are considered.

Thank you, Andreas. Very interesting links. I could even play with orientation-only pieces i.e. a set of identical pieces, wich show orientation. This discussian is maybe worth a new thread. I have found 17 piecetypes on the singlecut icosahedron set. I posted that before on this thread.

I allowed myself to correct my link in the quote. Please look it up again.
There you can see that in class DC2 CarlHoff's animation shows 18 different types of pieces because I count the core as the 0th type. All your pieces are related to mine:
Your faces are my corners and vice versa.
Your edges are my edges.
Your corner-edges and edge-corners are my T-Faces
Your corner-faces and face-corner are my X-Faces
Your pieces with index 7.x and 13.x are my obliques.
=> same result.

That leads for DC2 in:
2239488 different puzzles if orientations are impossible if positions are impossible too.
16777216 different puzzles if orientations are allowed without positions.
Here I assume that a piecetype is somewhat uniform: all are identical or all are different.

This could be even higher:
A) You could consider duplicates pieces like here:
http://twistypuzzles.com/forum/viewtopi ... 57#p239057
B) There are 591 other theoretical piece types which could serve as HoldingPoints in a puzzle of DC2.
C) There are even more theoretical pieces which could be represented in 3D but which can't serve as holding point.

schuma wrote:Today I made an illustration of a single-cut octahedron with internal pieces and posted it here. It's partly inspired by your work. I think such an illustration would be very interesting and very complicated for an icosahedron. But one can find all the pieces that you have classified in it. And, it would be very cool if anyone makes a program like this. It should be possible to solve if there is an option to make outer layers transparent. I guess the interface will be like that of MC4D. I need to learn a lot in order to make such a program by myself.

You show a beautiful Multi-HC2.
There are
72 different puzzles if orientations are impossible if positions are impossible too.
128 different puzzles if orientations are allowed without positions.
Considering duplicated pieces is somewhat difficult in HC2 since the pieces are "naturally partitioned".

Wow, good work, Andreas. I'm not sure that I understand everything..
Yes, I forgott to count the core, but it is an important piecetype, I agree.
When I started with my own puzzle-simulations I counted 3 axles for the face turning cube as you. Then I found Jaap's Sphere-Applet and it convinced me of counting 6 axles. Now after reading some of your posts I must do both, not sure, what is better. You do the classification of puzzles. I will need some time, to understand everything, but I'm very interested .
Hi schuma, I have seen your single-cut octahedron with internal pieces. It's beautiful. Yes, the first pieces get burried and forgotten. If we don't forget them, we get a multi-puzzle. You have done that very nice.
I must say, that the whole Multidodecahedron thread is very interesting. The same for the Classification of Cubes thread.
Andreas, what do you mean with theoretical piece types ?
Regarding the singlecut face turning icosahedron wich my article was about (it's DC2: dodecahedron cornerturning 2 cuts on 10 axles): in the classification of Cubes thread:

Andreas Nortmann wrote: That means the puzzles of DC2 are made out of:
Core (1 piece)
Index 0: O
Corners (20 pieces)
Index 1: C1 (Inner Corners)
Index 4: C4 (Middle Corners)
Index 14: C10 (Outer Corners)
Edges (30 pieces)
Index 2: E3 (Inner Edges)
Index 8: E9 (Middle Edges)
Index 12: E12 (Outer Edges)
Faces (12 pieces)
Index 6: F3 (Inner Faces)
Index 16: F6 (Outer Faces)
X-Faces (60 pieces)
Index 3: X9 (Inner X-Faces)
Index 9: X18 (Middle X-Faces)
Index 11: X24 (Outer X-Faces)
T-Faces (60 pieces)
Index 5: T12 (Inner T-Faces)
Index 10: T21 (Middle T-faces)
Index 15: T27 (Outer T-Faces)
Index 17: T30 (Far Outer T-Faces)
Obliques (120 pieces)
Index 7: L30 (Inner Obliques)
Index 13: L54 (Outer Obliques)

I agree with it. It's well done .

I'm surprised to see there are so many different DC2 puzzles. Since there is no hope to solve all of them, one may want to solve the hardest one. So here is the question, what is the hardest puzzle in that category?

An naive answer (or natural guess) would be: The one with all the pieces distinct and all orientation-sensitive is the hardest one. Is it true?

The one with all different and orientation-sensitive pieces requires the most amount of work to solve. But at the same time, the pieces give you a lot of information as reference. For example, many people who can solve the original Rubik's cube cannot solve the void cube, which has no centers, (or equivalently, the "centers" are identical holes), because there is a 50% chance to run into a parity issue. I think a puzzle with this kind of traps is hard, and fun to solve. I had an experience in a small 4D puzzle, where I thought I could solve it, then I ran into parity issue three times.

Is there such puzzles with a lot of traps in, say, all DC2 puzzles or HC2 puzzles?

Stefan Schwalbe wrote:When I started with my own puzzle-simulations I counted 3 axles for the face turning cube as you. Then I found Jaap's Sphere-Applet and it convinced me of counting 6 axles. Now after reading some of your posts I must do both, not sure, what is better.

Still pending question. Very true. I prefer the lower numbers because it seems akward to label the Pentultimate as a 12-axis puzzle.

schuma wrote:I'm surprised to see there are so many different DC2 puzzles.

I think I should explain how I came to those numbers. I will use DC2 as example:
There are 18 types of pieces.
Type O (the core) is always solved.
Type C1 allows at most orientations, since it is fixed to the core.
Types C4, C10, E3, E9, E12, F3, F6 allows position and orientations.
Types X9, X18, X24, T12, T21, T27, T30, L30, L54 allow only positions.

Pieces of types C4 and others in that category could be

• absent or all identical
• in need for positioning
• in need for positioning and orientating
• in need for orientating only

Case A: We assume that orientations are possible only when positioning is possible.
2 * 3^7 * 2^9 = 2239488
Case B: We assume that orientations are possible without the need for positioning.
2 * 4^7 * 2^9 = 16777216

Calculating the numbers in case of duplicated pieces is a complete different story.

schuma wrote:For example, many people who can solve the original Rubik's cube cannot solve the void cube, which has no centers, (or equivalently, the "centers" are identical holes), because there is a 50% chance to run into a parity issue.

I have to confess that I haven't solved the problem schuma brings up. There are puzzles where some kind of "global orientation" is visible. On the 3x3x3 and the megaminx you can see the stationary faces and therefore see the orientation of the core to avoid the void cubes parity problem. As far as I know there is not such parity problem on the void megaminx and the pyraminx chrystal which both lack global orientation, too.
Please note: I don't consider parity issues in the edges of the 4x4x4 as problematic here.

Stefan Schwalbe wrote:Andreas, what do you mean with theoretical piece types ?

To bring you on the current position of this discussion I have to drown you in several links to older, very extensive threads. If you are non-tired and have some hours time you could start with this one:
http://twistypuzzles.com/forum/viewtopi ... =1&t=15667
Please note:
A) I consider the names virtual / imaginary pieces as outdated.
B) In most puzzles virtual / imaginary pieces are useless but to force e.g the CircleMegaminx into the classification scheme they are needed.

schuma wrote: So here is the question, what is the hardest puzzle in that category?

An naive answer (or natural guess) would be: The one with all the pieces distinct and all orientation-sensitive is the hardest one. Is it true?

No one is better qualified than yourself to answer this question. AFAIK, no one in the world has solved more puzzles in this cathegory than you.
This is this kind of naive thought which drove me to make 5 sticker variants of 1.2.6. But you solved all of them almost within same move counts...

Andreas, an alternative way of classification:
Instead of "number of cuts per axis", I would propose "numbers of abstract layers per axis".
For example, the 3x3x3 Circle Cube consists of 3 layers in my definition, the pair of opposite circles and the slice forming together one layer.
My 3.1.15 is a 4-layer puzzle(opposite circles and the slice forming 2 separate layers).
This could cover a wider range of symmetric puzzles including those with circular or any other non-planar but still symmetrical cuts.
It's also useful to understand how physically separate pieces form together one unique virtual piece.
But maybe I'm missing some of your points and not understanding the profound reason of adapting your method, in which case my excuses...

Andreas Nortmann wrote: I have to confess that I haven't solved the problem schuma brings up. There are puzzles where some kind of "global orientation" is visible. On the 3x3x3 and the megaminx you can see the stationary faces and therefore see the orientation of the core to avoid the void cubes parity problem. As far as I know there is not such parity problem on the void megaminx and the pyraminx chrystal which both lack global orientation, too.
Please note: I don't consider parity issues in the edges of the 4x4x4 as problematic here.

I think the reason void megaminx and pyraminx crystal dont have parity issue is only because five is an odd number and and the odd-number cycle is an even permutation.

gelatinbrain wrote: This is this kind of naive thought which drove me to make 5 sticker variants of 1.2.6. But you solved all of them almost within same move counts...

Same reason: in 1.2.6 the corner turning is many three cycles and three is odd. So all permutations are even. No parity issue also.

Although it is not a parity problem, in 1.2.2, I did end up in a trap of turning only one corner pieces, which took a lot of effort to resolve. It's also because of lacking stationary faces. I don't call it parity, because this issue is not a two-state thing like even-odd, but a three-state thing, the orientation of a corner of three faces. I think the phenomenon of 1.2.2 makes it a "hard" puzzle coz it has a trap.

gelatinbrain wrote:Instead of "number of cuts per axis", I would propose "numbers of abstract layers per axis".

Accepted.

gelatinbrain wrote: It's also useful to understand how physically separate pieces form together one unique virtual piece.
But maybe I'm missing some of your points and not understanding the profound reason of adapting your method, in which case my excuses...

I guess you have an example in mind. Would you name it so I can try to explain it from there on?
See this anyway:
http://twistypuzzles.com/forum/viewtopi ... 15&t=15003
In this puzzle the sets of 4 edges of a single slice behave connected although they are physically seperate. Physical connections are ignored.
The example I gave is a bad one since I presented a subgroup of a puzzle. The main point is: Only behaviour matters.

Andreas Nortmann wrote:The main point is: Only behaviour matters.

I totally agree with you.
I would say farther "only the behavior visible from the exterior matters". Solvingwise we should ignore the interior mechanism and invisible pieces.

Maybe the terms I used in my previous post are not good. Instead of "abstract layer" and "virtual piece", I should call them "logical layer" and "logical piece".
A "logical layer" is a set of layers turning together at an operation. Visually they can be separate layers. But it dosen't matter.
In the same manner, a "logical piece" is a set of pieces turning always together and their relative position and orientation never change.
Here too, I think the simplest example is the 3x3x3 circle cube.
The 6 center pieces of this puzzle turns always together. They reflect always the orientation of the core piece. So they are logically considered as a single core piece.
An edge piece and two pie-shaped pieces of this puzzle too, they form together a logical edge piece.
I think logical pieces are automatically determined from the configuration of logical layers.
But I still need to elaborate...

gelatinbrain wrote:An edge piece and two pie-shaped pieces of this puzzle too, they form together a logical edge piece.
I think logical pieces are automatically determined from the configuration of logical layers.

So far we have an agreement. No I need someone with whom I can make up a website about all the theory stuff I have gathered last two years.

Dear Andreas, can you help me with the signatures of 4.1.12 and 4.1.13?Today I had the idea of an piecetype table where pieces are identified with their signature, wich contains usefull move-sequences for each piecetype. This table could contain piecetypes from all known puzzles. It would be easy, to find a usefull movesequence for a piecetype, because of the signatures. It would contain the best and shortest move-sequences.

Andreas Nortmann wrote:So far we have an agreement. No I need someone with whom I can make up a website about all the theory stuff I have gathered last two years.

What kind of help would you need for that?

Stefan Schwalbe wrote:Dear Andreas, can you help me with the signatures of 4.1.12 and 4.1.13?

You pick always the hard ones, don't you?
4.1.12 is HC2 [F3 X9 VC3] [F3]
4.1.13 is HC2 [F3 X9 VC3 VE3] [F3 VE3] where
I => VE3
II => VC3
III => E3
IV => X9
V => F3
VC3 is the second of three ZHP's which exist in HC2.

Before you start searching: The third ZHP in HC2 can be found in HC1 as well. There are two of these pieces which I named VO as provisorium. One is visible in 5.1.1

By the way: 6 of 12 VE-pieces are visible in MasterPyraminx which I consider as member of HC2 as well.

Not to throw a monkey wrench into the problem, but have you looked at all at the difficulty of the Polaroid Cube. There are 4 possible corner piece types on that puzzles. Oskar and I had several discussions on what set of pieces would provide the most difficult solving experience. I only looked at a few possibilities and couldn't think of any good way to account for there being multiple solve states.

GuiltyBystander wrote:Not to throw a monkey wrench into the problem, but have you looked at all at the difficulty of the Polaroid Cube.

I would say, the polaroid cube is one of the 746 2x2x2-variants you have when duplicated pieces are considered. Difficulties to identify which pieces is which are so far not considered.

Hi Andreas, thank you for the signatures Very illuminating.

Andreas Nortmann wrote:III => E3

That means, they are the dino-cube edges. They were not easy to solve.

Andreas Nortmann wrote:You pick always the hard ones, don't you?

I don't know, really. I just solved that two puzzles. They didn't frighten me enough. I thought, I can do that. Why seeking for easier puzzles when I can solve that 2 puzzles.
But with the signatures I should really start with easier ones!
I wondered myself about the right signatures. They are both deep cut, but with an additional circle. Is it HC3? - No it is HC2. And the middle slice appears inside the circles. So the deep cut is a trap!?
I really feel, that these signatures can help understanding puzzles.
By the way, what means ZHP? Ok, maybe I find it myself.

Stefan Schwalbe wrote:I wondered myself about the right signatures. They are both deep cut, but with an additional circle. Is it HC3? - No it is HC2. And the middle slice appears inside the circles. So the deep cut is a trap!?

Exactly. Both octahedra might look deep cut but they aren't HC1. Gelatibrain suggested to replace the interpretation of xxN:
"N=number of cuts per axis" is easy to understand but not that precise.
"N=number of logical layers per axis minus one" is way more precise.
In these case we have two layer which seem to be separated by a deep cut and in addition we have the cylindrical part of the puzzle which represents the slice in an "ordinary" puzzle of HC2.

Stefan Schwalbe wrote:By the way, what means ZHP? Ok, maybe I find it myself.

See the end of this posting: viewtopic.php?p=231977#p231977
ZPH = ZeroVolumePieces under equidistant cuts that can act as a holding point.
If you have read this monstrous posting (http://twistypuzzles.com/forum/viewtopi ... 90#p191090) you have already learned about that concept which I named "virtual pieces" back then.