I was finally able to finish a couple of Helicopter Cube sticker mods

The ultimate one looks really hard!

You know there are a LOT more positions in the super helicopter cube? Actually there are about 1.65 million more positions to it!!!! Also, I was wondering where you bought them... Thanks Alex

I got them at http://www.thetwistystore.com/

[strike]Only if you jumble [/strike] Cancel that. Even without jumbling it adds something. Interesting, I didn't expect that.

Actually isn't it a lot more than 1.65M? I'm thinking it's 4!^6 (191 million) because that's what they is taken out on the calculation I see for the 4x4x4. Or is that 1.65M for the non jumbling version?

They're very beautiful. I'd love to have a sticker variant of the Helicopter Cube. But first I need a normal one!

Very interesting! And not obvious at all ... it took me a while to see what you were talking about! (I am trying not to give it away, for those who want to think about it themselves.)

I don't see how the super helicopter cube adds anything without jumbling. Only the centers are (effectively) different from normal, and if you scramble a normal helicopter cube without jumbling:

1. All the centers are unique, since there are four orbits which aren't interchangeable, and each of the four centers of one colour are in a different orbit.
2. There is no center orientation (centers cannot be rotated in place).

Please correct me if I'm wrong though.

Matt

PS. Oh, and on the original point of the thread, these mods look awesome

Great Work
I love the 'tiled' effect on the left one.

Wow, I didn’t consider this effect either until it was brought up here. I solve centers first so if I don’t jumble and correctly position the 4 centers of one of the sides the rest will fall into place. But if you do a corners first approach you would only have a 1 in 6 chance of having the centers correct.

Now once you jumble then these puzzles really get interesting

Here's an example showing that it does add something. My first instinct was that because the 4 orbits never intersect, you would never have to worry about them switching. However, the orbit's don't line up the same way on every face so super stickering does matter. This example shows moving the white center to another face.


As for permutations for non-jumbling super helicopter cubes, I believe this is the equivalent of adding the square center pieces on a shallower cut helicopter cube. It gives the puzzle an absolute rotation. This would mean the number of positions is increased by a factor of 24. Difficulty isn't increased to much though imo.

Didn't think of that . Thanks for pointing it out . I agree, it doesn't really add any difficulty to the puzzle.

I also had not thought about this issue, that moving a group of 4 centers to another face (may) interchange their relative positions. Cool concept! How about picture stickers instead?? Sure would be quite painful to solve

Per

When I saw the title, I was thinking, "Oh My Gosh you built a 3 and 4 layer Helicopter Cube?!?!?"

Very nice sticker mods. They look hard.

Oh, I guess your right because I screwed up the multiplication...

also, the corners must be matched up to the correct center, rather than just some random one... plus, once it's jumbled... god...
I'd say that the super helicopter may actually have the most difficulty difference of any puzzle versus its super version... and almost definitely of the order 2 puzzles(or order 3, depending on your definition...)

The super stickers are a good idea. My first reaction was the same -- because of the orbits these stickers would have no effect. It's neat that they actually matter.
I can't speak for jumbling but for non-jumbling scrambles, I don't think it would be all that bad. Gelatinbrain's "Helicopter Dodecahedron" (1.4.8) has a neat orbit orientation problem too. Basically, you can solve one or two centers completely but unless you solved the centers with the overall orientation of the puzzle correct, you wont be able to solve the remaining centers. There was discussion about it here.

For those of you that solve the helicopter cube corners first (I do), to solve the super version would require you to first collect all of the white stickers onto the right face, using their relative positions to determine which face is the 'correct' white face as well as the other color of all of the other faces. This extra step shouldn't take more than 30-60 seconds.

Is there any puzzling difference between the two sticker variations? I think they are the same as puzzles; please correct me if I am wrong about this.

Starting from the left sticker-mod, it's easy to get the one on the right: just extend the small regions on the edges of the stickers until they cover the big triangle with the original color. To go from the sticker-mod on the right to the one on the left: note that there is exactly 1 corner sticker with red/blue color pair (there is also 1 with the colors reversed). The third color on that corner is yellow, so place a large yellow triangle over most of the red/blue corner sticker, and do the same to the 1 middle piece that matches it. This gives the sticker-mod on the left.

OK, so I don't understand the names "Super" and "Ultimate" in this context.

BTW, I asked in another thread if it is possible to rotate a single corner on the helicopter cube by using jumbling turns. (Has this been answered definitively? Certainly I've never had a single corner twisted after jumbling and unjumbling.) Is it possible to switch exactly 2 center pieces with jumbling twists? With the normal stickers you wouldn't notice that! Hmmm, I just may have to buy some more helicopter cubes.

Nice sticker-mods!

Yes it has been definitively answered by TomZ in the same thread. No you cannot rotate a single corner.

Of course, I saw that reply! But I'm looking for the analysis that shows that a single corner can't be rotated.

Ultimate cube and Super cube

OK, so the stickers have similar style to the "Ultimate Cube" and "Super Cube" respectively. I thought it should somehow refer to the puzzle it creates. For instance, Volitar's avatar (would that be Volavatar???) shows a gigaminx with ultimate-style stickers. But the puzzle is not "ultimate" in that several pieces (the trapezoidal pieces adjacent to face centers) are not distinguishable.

I suppose that's an OK way to name them, but it's not what I expected.

Can you point me to an analysis that shows a single corner can't be rotated on a 3x3x3 so I know what you're looking for?

I just checked my copy Winning Ways vol. 4 (2nd edition, 2004). They have an analysis (pretty terse) on pages 869-870. I know you're probably looking for something online; the best I can do at the moment is this (30 year!) old cube-lovers message by David Vanderschel. It may be a bit cryptic, because it's before notation and terminology was standardized. His analysis is very close to the analysis in Winning Ways, maybe with different terminology.

While I'm at it, has anyone mapped out all the jumbled configurations of the helicopter cube? This would give an effective way to answer both questions (about twisting a single corner, swapping 2 centers). First, make a graph with vertices for every jumbled configuration (ignoring the colors) and an edge between two states that differ by a single edge rotation (whether 180 degrees or not). For every loop in this graph that starts and ends at the cube configuration, check whether or not that loop changes total cornertwist, or changes the parity of center permutations. Of course, you only need to do this for a finite set of loops, e.g. if they form a basis of the fundamental group of the graph.

But I think there should be a more intrinsic way to analyze it. Has that been done already?