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 Post subject: What cuboids can be emulated properly by bandaging a cube?Posted: Tue May 13, 2014 5:34 pm

Joined: Sat Sep 15, 2012 7:42 am
I've recently gotten into modding. I've already completed a huge amount of work on a 3x5x5 from a 5x5x5. My 4x6x6 from a 6x6x6 is not far behind, and I even have a start on a 3x5x7 from a 7x7x7.

My question is: is there a rule or pattern to the cuboids that can be made in this manner? I see in particular that the 4x4x4 can pretty much emulate everything with dimensions < 4. Is this the case with higher cubes? For instance, can I make a 3x4x5 out of a 5x5x5? What about a 5x6x7 from a 7x7x7? Proportionality is not a concern, obviously, but properly emulating a fully-functional cuboid is. So "asymmetric" bandagings like a Camouflage 3x3x3 are not part of this conversation [though I find them fascinating in their own right!].

For the record, here are the cuboids that can be simulated by a 4x4x4. If any of these are not proper emulations, please let me know:

- 1x1x_ [degenerate]
- 1x2x2
- 1x2x3
- 1x2x4
- 1x3x3
- 1x3x4
- 1x4x4
- 2x2x2
- 2x2x3
- 2x2x4
- 2x3x4
- 2x4x4
- 3x3x3
- 3x3x4
- 3x4x4
- 4x4x4 [obviously]

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Named 3x3x3 Bandaging Patterns

Last edited by themathkid on Thu May 15, 2014 8:53 am, edited 1 time in total.

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 Post subject: Re: What cuboids can be emulated properly by bandaging a cubPosted: Thu May 15, 2014 8:53 am

Joined: Sat Sep 15, 2012 7:42 am
I take it from the lack of responses that this was a stupid question?

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Named 3x3x3 Bandaging Patterns

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 Post subject: Re: What cuboids can be emulated properly by bandaging a cubPosted: Thu May 15, 2014 5:28 pm

Joined: Tue Jun 11, 2013 12:48 pm
No! It is a very good question.
Maybe only interesting for really advanced solvers.
After I solved so many massproduced puzzles, I slowly start to think about these things.
(Until now I dont have any idea about "building-rules").

But I found a talented builder on youtube. He explained how to mod certain cubes and also some building rules. In case you didnt know him already I recommend to have a look. His youtube name is "badrobot32".

Lately I bandaged my 6x6x6 into a chimera cube.
I am still working on completing the last two 3x3x3 cubes. A really interesting challenge.
Also on youtube I hardly couldnt find any tutorial about these kind of challenges. So I am not wondering that there isnt much feedback. But as I said the topic is well woth a discussion (at least in my eyes).

cheers!

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 Post subject: Re: What cuboids can be emulated properly by bandaging a cubPosted: Thu May 15, 2014 9:33 pm

Joined: Fri Feb 20, 2009 6:38 pm
Just saw this - I'm really interested in this question for n=5 because I would like to make some cuboids. The way I would like to do this is by bandaging, then extending the pieces. Having a list of appropriately functional cuboids would reduce the number of options for me...

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 Post subject: Re: What cuboids can be emulated properly by bandaging a cubPosted: Fri May 16, 2014 9:52 am

Joined: Sat Sep 15, 2012 7:42 am
I've been playing around with the 5x5x5 today. After seeing Nathan Wilson post a picture of his 2x5x5 on the Puzzle Photography group on Facebook, I had some revelations. First, it's very easy to implement cuboids where all the dimensions have the same parity [even / odd] by bandaging from the outside in. Failure to bandage from the outside in can result in some weird things that look like cuboids but aren't [for example, this 3x3x5 lookalike]. Second, to change parity of a single axis, hide [for odd puzzles] or expose [for even puzzles] the middle layer. For 5x5x5, hiding a middle layer can be countered by extensions to help make layers more proportional.

The "easy" cuboids I mentioned above are as follows for the 5x5x5:

Attachment:

Easy 5x5x5 Cuboids.PNG [ 149.1 KiB | Viewed 588 times ]

Below are all cuboids with largest dimension 5, with the layer hiding process explained for each cuboid.

1x1x5 - Picture above
1x2x5 - Bandage like a 1x3x5 and hide the middle layer on the 3 axis
1x3x5 - Picture above
1x4x5 - Bandaged like 1x5x5 and hide a middle layer on the 5 axis
1x5x5 - Picture above

2x2x5 - Bandage like a 3x3x5 and hide a middle layer on both 3 axes
2x3x5 - Bandaged like a 3x3x5 and hide a middle layer on one of the 3 axes
2x4x5 - Bandaged like a 3x5x5 and hide middle layers on the 3 axis and a 5 axis
2x5x5 - Bandage like 3x5x5 and hide the middle layer on the 3 axis

3x3x5 - Picture above
3x4x5 - Bandage like a 3x5x5 and hide a middle layer on a 5 axis
3x5x5 - Picture above

4x4x5 - Hide a middle layer on two axes
4x5x5 - Hide a middle layer on any axis
5x5x5 - Picture above

Unfortunately, I think these ideas need to be fleshed out in greater detail. There are some unintended consequences that I don't currently know how to deal with. In particular, the 4x4x5 and 4x5x5 made in this way would allow 90-degree turns on a 4x5 side, which should not be possible on a true cuboid. Instead, it would function more like Chino's Cubic 4x4x5 [which I happen to own ].

Can anyone shed some light on this process of hiding / exposing layers?

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Call me Seth

Named 3x3x3 Bandaging Patterns

Last edited by themathkid on Fri May 16, 2014 12:38 pm, edited 3 times in total.

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 Post subject: Re: What cuboids can be emulated properly by bandaging a cubPosted: Fri May 16, 2014 12:16 pm

Joined: Sat Sep 15, 2012 7:42 am
Okie dokie. I've been thinking about this all day [I have a lot of free time at my job, thankfully ]. To further explain the problem, I made this picture. I started with a 10x10x10. I bandaged it to look like a 5x5x5, then unbandaging and rebandaging to remove the middle layer. Basically, layers 2 and 4 extend halfway into layer 3 to cover it up.

Attachment:

Hidding a Layer.png [ 20.97 KiB | Viewed 574 times ]

This, unfortunately, allows the weird turn described in the previous post. A proper 4x5x5 should not allow this turn.

Attachment:

weird turns.PNG [ 16.25 KiB | Viewed 574 times ]

To remedy this, we need to play around with layers 1 and 5. By extending these layers into layers 2 and 4 respectively, we end up with the "outside in" style that I talked about in the previous post.

Attachment:

double hide.png [ 37.71 KiB | Viewed 574 times ]

This prevents the weird turn from occuring:

Attachment:

blocked turn.PNG [ 14.71 KiB | Viewed 574 times ]

My take away from this is that hiding a layer is a much more complicated process than I initial envisioned. It requires manipulation of EVERY layer on that axis, not just the 2 layers adjacent to the center. With this process, though, the proper functioning of the cuboid is achieved. I venture to guess that every desired cuboid can be achieved in this manner. That is, I believe an NxNxN cube can make any cuboid with largest dimension < N. Odd cuboids seem preferable, as hiding a layer looks easier than exposing a hidden layer. As I have not yet attempted either, this is just speculation.

I can just hear someone screaming "but Seth, people already know this stuff." And I'm sure that's true. But this post is aimed at novice to intermediate modders. People like me who are still relying on tutorials to make their mods, wondering what else might be possible. It is not aimed at the Traiphums and Tony Fishers of the world, or the CAD cuboid designers.

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Call me Seth

Named 3x3x3 Bandaging Patterns

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 Post subject: Re: What cuboids can be emulated properly by bandaging a cubPosted: Fri May 16, 2014 7:29 pm

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
I don't think anyone has done a comprehensive mapping of this. It's definitely an interesting topic, because there are so many variants and possible ways to do the bandaging. Pillowing has often been unnecessarily used, and I think it probably adds some confusion to the topic. What I'd like to see is a comprehensive list of the most efficient ways to bandage the current crop of Mass Produced puzzles to mimic all the `Real Cuboids`, without worrying about pillowing or aesthetics in modding.. that would be very useful. From what I can see, you're well and truly on your way.. thanks.

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1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

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 Post subject: Re: What cuboids can be emulated properly by bandaging a cubPosted: Sun May 18, 2014 11:44 am

Joined: Sat Sep 15, 2012 7:42 am
I went ahead an made mockups of all cuboids with largest dimension 5 as they would look if they were made from a 5x5x5:

Attachment:

All 5x5x5 Cuboids.PNG [ 375.04 KiB | Viewed 422 times ]

Burgo wrote:
I don't think anyone has done a comprehensive mapping of this. It's definitely an interesting topic, because there are so many variants and possible ways to do the bandaging. Pillowing has often been unnecessarily used, and I think it probably adds some confusion to the topic. What I'd like to see is a comprehensive list of the most efficient ways to bandage the current crop of Mass Produced puzzles to mimic all the `Real Cuboids`, without worrying about pillowing or aesthetics in modding.. that would be very useful. From what I can see, you're well and truly on your way.. thanks.

What do you classify as "real cuboids"? Perhaps this is doable. Seems the problem lies in the brick cuboids. There are just not enough mass produced E-E-O or O-O-E cuboids. With a few big ones, say 6x6x7 and 6x7x7, we could do a loooot more.

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Call me Seth

Named 3x3x3 Bandaging Patterns

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 Post subject: Re: What cuboids can be emulated properly by bandaging a cubPosted: Tue May 20, 2014 3:11 am

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
By `Real Cuboids` I meant simulating puzzles with proportional cubies, without creating other bandaged (or unbandaged) situations.. the kinds of problems you are already identifying.

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

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