Many new puzzles have been made and proposed lately and there is even vague talk of the Petaminx. I have a few questions about some of the higher order circle cubes. I believe that all of the contributors on the crazy thread have worked to make solving these types of puzzles accessible. I don’t know the numbers but I assume there is a growing audience for circle and crazy planet puzzles.

First, I have often wondered why the Circle 4x4 III was made (why have the Circle Corners)? Perhaps there was a mechanism being considered for a 4x4 Crazy Planet series? Is this possible (a mechanism for 4x4 Crazy Planets, or even that with a new core the outer pieces of the current puzzle could be utilized)?

Or perhaps it was a prototype for a Crazy Planet 5x5 IV Series? It would seem to me that the turning would be about the same for this series. I have been solving some of these planets on the Bo Hu Sim lately and they are really nice challenges, I think they could be well received (especially among the crazy audience)?

If not, then the 5x5 III (in the tradition of the 4x4 II) would be a fantastic puzzle in which the solution is very accessible. I am sure it has a market.

The next question is getting fanciful. Is the mechanism for the Circle 5x5 II possible? This is one of the most fun symmetrical cubes I have ever solved. It would be such an amazing cube to have mass produced, would the market be there? I think it would.

The next leap from there is nonsense, but hear me out.. If there was an 8 planet 5x5 III series, who would smuggle it to work in their lunch box?

Last. We have seen that the Master Kilominx is coming. Is the Circle Master Kilominx II mechanism possible (in the tradition of the 4x4 II)? Would the circle edges move with the outer layer, or would everything just fall out?

Just something to think about,
Cheers,
Burgo.

Considering the prices of just ONE crazy 4x4 of any type, I highly doubt an entire planet series of 4x4s or 5x5s would be a good business decision for MF8 (Looking at over 300, possibly $400 for 8...).

That being said, the market for the Crazy 5x5s (I, II, III, whatever) would probably be about the same as the market for the Crazy 4x4. I would think it would be a good puzzle for MF8 to produce.

In addition to your examples, why not have just the small circle? That would still provide a unique puzzle. There are two kinds of big circles and two kinds of small circles. What's to say there aren't 6 combinations of Circle 5x5x5s? Then factor in Crazy versus Circle, and you've gotten yourself 48 unique Crazy 5x5x5s to collect and solve.

I'm not seeing a full-on Crazy release of these anytime soon unless it's like the Crazy 4x4x4, which is really just a Circle 4x4x4.

It is Possible!!! see my old post: Double-Zero Cube 5x5
viewtopic.php?p=208980#p208980

I have been solving quite a few of them on the sim lately and I put some thought into which ones I suggested.. I think the smaller circle 5x5s are a bit trivial (for the market that would buy them), and a lot of the other formats are similar to solve.

Then maybe it’s time to make an ingenious push button centre piece that locks and unlocks faces? There has been some talk about this kind of thing. There’s more than one way to solve a 3x3 . Actually, I still don't know if the crazy 4x4 planets are possible, that's why I was wondering if that's why the 4x4 III was made.. I see no other reason for its existance .

Thanks for the link, there are many fascinating puzzles in that thread. The Esoteric Tetrahedron looks amazing, just to name one. I am too young to recall that thread, I just turned 1 a few weeks ago (according to first 3x3 owned and/or solved).

Your Double-Zero Cube 5x5 is the 5x5 II from Bo Hu’s simulator. It is such an interesting puzzle and the solution is not out of reach, it has got to be a target for production then.

I still don’t know if the 5x5 IV is manufacturable because of those thin middle outer edges. I really hope someone says.. "No no, it’s possible!!!" It is the ideal one for a crazy series. I couldn’t resist making a thread to talk about this, I am on the edge of my seat just thinking about it .

Cheers,
Burgo.

Lots of things are possible if you don't restrict yourself to cubic cubies.

What I call the Real5x5x5.

Pillowing would also work in many cases.

Carl

Anything is possible when you eat Reese's Puffs!

and I think it could work, if you look at the Crazy 4x4 II those inner corners and edges are MINISCULE compared to the possible 5x5 IV. But it still works fine.

--JC--

Carl, I have always followed your thinking with amazement. Thanks for revealing the True 5x5 which I believe is the double circle 5x5 IV from the Bo Hu Sim that gave me so much trouble . It's nice to now know what the pieces actually are. I literally cannot believe that you designed an `almost complete` working model for it. I left it off my list because I thought it was the just the stuff of dreams, and maybe it still is? But I now certainly appreciate it much more!

Lately, I think I can feel my brain starting to grow on the outside of my skull. And some of it is your fault .
Cheers,
Burgo.

Hey Carl,

You inspired me to go solve this today (along with all this talk of multipuzzles). It's a hard puzzle, dare you to make it .
As I suspected though, not much interest in a 5x5 crazy planet series (YET) .

Cheers,
Burgo.

Back here: viewtopic.php?f=1&t=21384, I posted asking about double circle cubes. In this current thread, Burgo has posed a similar but broader question.

The thing with me is that I love solving puzzles and I love being stumped (for a short time) only to come up with a solution (more often than not through collaboration with others). My favourites are any type of crazy puzzles. I look at the diagram of the 5x5 IV in Burgo's post and actually almost get sad that it doesn't exist. What I don't have is a lot of understanding of the mechanics or design of these things; that, I freely admit.

But I can't believe the market for something like the 5x5 IV, or a series of crazy 5x5s, would be so small. If such a series were mass-produced, I for one would snap them up in a heartbeat. In fact, I'd snap up any 5x5 circle cube. What will it (or does it) take to get something like this brought to market? The designs are there, just no puzzles yet. Is there really as little interest as there seems in this type of thing?

Hi,
I can't think of a place to ask this.. so I will continue in some of the thoughts from this conversation (I have some other things to add here later too)..
I solved this puzzle tonight and I had trouble identifying the small triangles inside the circles.. does anybody know what they are synonymous with? I understand the rest.. It's basically a pyraminx crystal in which you can see all of the megaminx pieces including centre orientation.. but these little triangles seem to be completely independant? And I can't make them out?
Cheers,
Burgo.

If I understood you right, you want to know to which piece in the Multidodecahedron these small triangles are equivalent to.
To no piece of the Multidodecahedron. Under my current naming scheme these pieces bear the name Tz15. They are an examle of the ZHPs = Pieces which can serve as holding point but have zero volume if only equidistant planar cuts are used. These are also known as virtual pieces, a term I dislike nowadays.
You might want to search for "virtual pieces" in the forum but you will get drowned with theory-heavy threads.

Andreas

Hi Andreas,
I try not to answer without doing the resaerch, but.. it is a long slog through most of that, and I only understand some of it. I wonder, can you give me the short of it?

You are correct with what I want to know.. In my experience a circle is like a window into the puzzle, showing the pieces and/or orientations of pieces within the puzzle (many of us are familiar with this concept by now with the Real5x5 etc). These small triangles are the tips of the pyraminx crystal edges when they are placed. But before they are placed they scatter like a crazy planet's CCs with no relative position to anything (I solved them similar to how I would solve a Mercury 3x3's CCs). Is this how virtual pieces behave?

Now if this is a megaminx and a pyraminx crystal, there is nothing else `in there` that they could be, right? Because a pentultimate etc, is on the `outside`. So ?

Cheers,
Burgo.

I'll take a shot at this. This topic started off as an "Analysis of twistability" which was a way of looking at the pieces in a twisty puzzle.

The first approach was taken by Andreas and went something like this. Let's use a 3x3x3 for our first example. In a 3x3x3 there are 2 independant layers of rotation on each of 3 axes. On each axis we have 3 choices of the 2 layers we can pick as the independant ones. The 3rd layer is fixed and will never rotate. We can do this for each of the 3 axes. So lets pick the 6 face layers and ask yourself what is the piece that never moves. I've called this a "holding point" its a point fixed in space with this choice of turnable layers. With that pick we can see the "holding point" is the core cubie. It never moves. Let's pick 3 slice layers and 3 face layers. There are 8 ways to do this and each way maps to one of the 3x3x3 corners. In fact there are 3^3 or 27 different selections of layers which can be made and each maps to one of the 27 cubies in a 3x3x3. All these are REAL pieces, i.e. they have real volume.

The second approach was taken by Matt Galla and instead of looking at the pieces which didn't move it looked at the way pieces moved with the face layers. In effect the core is always considered as the holding point. There are 6 face layers and any piece can move with ANY combination of all 6 layers. For example, it allows for the existance of a piece which turns with both the left and the right faces for example. Each piece moves with a given layer or it doesn't, a binary choice. And there are 6 layers. So this method defines 2^6 pieces or 64 pieces. Of these 64 pieces, 27 are real and the same 27 pieces identified with the first method above. In general this method is always a superset of the first method. The other 37 pieces are imaginary. The 3x3x3 doesn't contain any virtual pieces.

The above two approaches are detailed in this thread. There is also a third approach I've taken which I view as a more general use of the second approach which allows for an arbitrary pick of a holding point. I go into more detail about it here. And Matthew S may have yet a 4th approach he's detailing here which appears to be yet an even bigger superset. I have yet to fully understand that one myself.

All that said lets focus on just the first two methods and look at the dodecahedron. I AM trying to keep this as simple as I can. First let's count the pieces in Tom's MultiDodecahedron.

1 = Core (Piece Type 0)
12 = Megaminx Face Centers (Piece Type 1)
30 = Megaminx Edges (Piece Type 2)
20 = Megaminx Corners (Piece Type 3)
30 = Master Pentultimate Edges (Piece Type 4)
60 = Master Pentultimate X-Centers (Piece Type 5)
12 = Master Pentultimate Face Centers (Piece Type 6)
20 = Master Pentultimate Corners (Piece Type 7)

You can see all the piece types labeled in this animation.


This is a total of 1+12+30+20+30+60+12+20 or 185 pieces. All of which are playable and solveable in Tom's puzzle. The only one which isn't stickered directly is the core but you DO know its orientation as its directly attached to the Megaminx Face Centers.

Now lets use the first method above to look at the face turn dodecahedron. It, just like the 3x3x3, has 2 independant layers per axis of rotation. On each axis we have 3 choices of the 2 layers we can pick as the independant ones. However we now have 6 axes of rotation so this leads to 3^6 possible holding points. That's 729 pieces. Of these we know 185 are real. The other 544 are virtual. I still like that name. It lends itself to nice naming. Tom's puzzle is the Face Turn MultiDodecahedron. A puzzle with all 729 of these pieces I'd call the Face Turn Augmented Dodecahedron as real + virtual = augmented.

Now let's use the second method to look at the face turn dodecahedron. We hold on to the core so its our holding point. There are 12 faces which can turn. For each face a given piece is either in it or it isn't. Again a binary choice so this method defines 2^12 pieces. That's 4096. Of these 185 are real, 544 are virtual, and the remaining 3367 are imaginary. Or you could include the virtual pieces as a subset of the imaginary if you like. And a puzzle which had all 4096 of these pieces I'd call the Face Turn Complex Dodecahedron as real + imaginary = complex.

The virtual pieces can and do turn up in quite a few puzzles. The one you found above is a great example. The "plus 1" I found in this puzzle is another. They are much easier to bring into existance (be given a real positive volume) then the other imaginary pieces. Here is my Augmented Skewb idea which could be made as a real physical puzzle. Note it has the corners and face centers from a normal skewb (the real pieces) plus both of the virtual cores. To my knowledge there are no imaginary pieces in any of the playable puzzles you can currently find on line. The most talked about complex puzzle is the Complex 3x3x3 and someday I believe we will have an app which will allow us to play with it. Andreas has already calculated its number of permutation (states) here. And I have ideas on how to make it a real playable puzzle. However its starts with first making my Real5x5x5 a working real puzzle which I have yet to do. And it gets even more complicated from there. In short you really have to jump though hoops to give the imaginary pieces a real positive volume that playable... even in an app. I believe this is due to the fact they can't serve as holding points so in that way they are even less real then the virtual pieces.

I hope you could follow that. Basically I just went though counting the pieces. Knowing how to group them into types, corners, face centers, cores, etc. is detailed in all the above threads.

I hope this helps,
Carl

Well if you go IN far enough you come out the other side so yes I could see Pentultimate like pieces turning up in something like this. However there is a whole slew of virtual pieces (FAR more then the real ones in there) that just don't have any volume in a puzzle cut with planes. These circle puzzles aren't simply planes and that does allow some of the more exotic pieces to "become real" and turn up in puzzles like this. Pick one of them and pretend you are holding the puzzle by that point. Now ask yourself which two layers are free to turn on each axis without this piece moving. You can now see its a virtual piece (not imaginary) and you have its signature. As for Andreas' naming scheme which names this piece Tz15 I'm not sure about that myself at the moment. I suspect its some encoding of this signature.

Hi Carl and Andreas,

To me, this is a very interesting conversation, thank you for sharing your knowledge. I’m glad I asked because there’s so much preceding thought on this, it’s nice to have some background in a nutshell. I have had a few hours of interesting reading and only scratched the surface.

I was getting very used to the NxNxN Circle Puzzles where no virtual pieces exist. The dodecahedron & skewb etc, I think, introduce new `non-parallel` angles that cause this.

With the NxNxN puzzles, the larger the diameter of the circle/s the deeper into the puzzle you see. I `was thinking` that with this larger diameter I must be seeing `further in`. Hmm, Virtual Pieces.

In the Circle Crystalminx II, the cuts are on their way `in` from a Megaminx `to` a Pyraminx Crystal; we haven’t really even got to a full Pyraminx Crystal yet, we are `only just` exposing its pieces. If we didn’t have a circle, we would still have all of the `literal` Megaminx pieces- even the Megaminx edges (which with the circle are now super-centres), and of course the Megaminx edges are now only visible through the CCs.

It is a very interesting puzzle then, in a number of ways. I liked solving it too .

Interesting. So.. you think a circle `could` cut 2 planes in such a way that the piece revealed is `beyond` the scope of the puzzle? I wonder if there is an example we can find?

I will include a picture of the puzzle I solved before this (the Circle Crystalminx I) in which the pieces in question do not exist because of the smaller circle cut (it is much more straightforward).. but it shows `the moment before these pieces are exposed` .

Cheers,
Burgo.

Thank you Carl for saving me some time. I would have answered to but that would not have happened before the weekend. Not that complicated.
Some months ago (before I became moderator) I worked on a naming scheme for ZHP's (aka virtual) and NHP (aka imaginary) pieces.
In this case it means:
T: It has the same symmetry group as the real T-pieces.
z: It is a ZHP.
15: 15 pieces of this type are moved by every turn.

This turned out to don't be perfect but these names are better than the ones I used before.

Andreas