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 Post subject: More Jumbleable Gibberish
PostPosted: Tue Sep 04, 2007 12:54 am 
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Location: Marin, CA
I've put yet even more thought into possible jumbleable-only deep cut puzzles. They appear to be hard to come across, if they exist at all, but I have some reasonable candidates (or more properly, best candidates).

There are two families left out of the kepler-poinsot solids - specifically, the prisms and the antiprisms. The most likely to be useful are the antiprisms, because the three triangles produce a rather large number of possible ways things can be positioned, and two of the positions allow three of the four slices of adjacent vertices to still be sliced, with one allowing all of them.

Since the square antiprism doesn't contain pairs of opposite vertices, the deep cut version of it has eight slices. This is a big problem - with jumbleable puzzles, the more pieces there are, the more likely everything is to get blocked and for them to not be properly scrambleable. You want as few slices as possible, especially for deep cut.

Going for minimum number of slices, and removing the ones which irritatingly wind up being equivalent to the ordinary slicing things we're used to (there are a lot of those in three dimensions) basically yields two candidates, both with five slices - the pentagonal antiprism and the pentagonal crossed antiprism. So if you were to make slices exactly midway through each pair of opposite vertices of one of those, you'd get a puzzle which was deep cut with five slices and quite possibly could be scrambled via jumbling.

So, uh, any idea if those are actually scrambleable, or is their motion too limited?


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 Post subject:
PostPosted: Tue Sep 04, 2007 4:34 pm 
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With your suggestion of the pentagonal antiprism, where would you want the cuts to be / what kind of motion are you talking about?

I ask, because I was considering scrapping the pentagonal antiprism from a 3x3 mod that I've half done and is annoying me, in favor of a different approach with the same shape...

Please elaborate on what you have in mind, as I think we may be on the same page ;)

Cheers!

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я буду нуждаться пиво скоро.


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 Post subject:
PostPosted: Tue Sep 04, 2007 5:31 pm 
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The cuts would be through the center on a planes perpendicular to lines going from the center to the vertices. I would be very surprised if you've ever thought about this, because it isn't even vaguely a shape mod and would require that the center spindle mechanism be made completely from scratch, if it even works at all. I don't generally do shape mods, so in general you should assume that any mechanism idea of mine needs some fundamentally new pieces, and probably needs to be made from scratch. 8-)

My question is whether this would work at all, it's related to the probably rather opaque discussion of jumbleable mechanisms which came up in the thread about the bevel cube.

Did you get my email, by the way?


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 Post subject:
PostPosted: Tue Sep 04, 2007 5:51 pm 
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Indeed I did! We're on the same page ;)

What I meant to ask, I guess, is where the cuts would "go" in the sense that, yes, the petagonal face would get cut up in the same manner as a fece of a Pyraminx Crystal / Aleh's Brilic, but where would they go? Vertically down to the opposite face, bisecting the triangles on the "sides" of the antiprism? With the resulting puzzle having 10 pieces total, five sets of two mirror image bits (not too interesting that way, in my opinion), or something else?

Cheers!

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я буду нуждаться пиво скоро.


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 Post subject:
PostPosted: Tue Sep 04, 2007 6:55 pm 
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Okay, I'd be very surprised if you'd ever thought about the pentagrammic crossed antiprism :-)

Since all the slices go through the center, and no three go through the same line, there will be 22 pieces. If there is one such slice, there are two pieces, with two such slices, there are 4 (2 + 2) with three, eight, as in the 2x2x2 (4 + 4), with four, fourteen as in the skewb (8 + 6) and with five, twenty-two (14 + 8). The amount added with each successive slice is always two more than the last slice added.

The best way to visualize these is probably drawn on a plain old sphere. Since it's deep cut, any outer surface will just be a shape mod and not change the working of the puzzle any.


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 Post subject:
PostPosted: Tue Sep 04, 2007 7:28 pm 
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Ah, I see--I very-much mis-understood you the first time round.

However, I have actually considered a related item, in that you can consider a pentagonal antiprism as an icosahedron that has been "flattened" to remove 10 faces via truncation. If you use your five-plane version on the original icosahedron, you end up with 10 deep cuts. So basically what you're putting out there would essentially be a bandaged version of that concept, correct? Specifically, try visualizing the puzzle 2.13 here squished into a pentagonal antiprism, with the appropriate cuts removed.

Is that what you're talking about?

Also, I think I'm mis-understanding your use of the word "jumbleable". What exactly do you mean by that?

Cheers!

_________________
я буду нуждаться пиво скоро.


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 Post subject:
PostPosted: Tue Sep 04, 2007 8:01 pm 
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Argh, you're absolutely right. That makes the pentagonal antiprism a whole lot less interesting, since it's basically a bandaged shrapnel ball. That leaves the pentagrammic crossing antiprism as the only deep cut puzzle with a real hope of being jumble-only.

Jumbling is discussed in the helicopter cube thread. It came to me as quite a surprise, and now I'm trying to do a thorough study of it.


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 Post subject:
PostPosted: Tue Dec 04, 2007 4:38 am 
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Argh, I just realized that the crossed pentagrammic antiprism is just the great icosahedron with two vertices removed, so it's busted too.

That basically leaves the smallest possible deep cut design which isn't just a bandaged version of something more symmetrical as the dual of the hexagonal prism. I'm a little leery of anything with that many pieces though, it's got 32 right off the bat. It also happens to be a deeper cut version of the basic jumbleable triangular prism, which I'm already working on, so perhaps the way to go is simply to continue down that path and do build-ups to deeper cut puzzles later.


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