# 180 Cubes, classified according to permutability and orientability



## guzman (Aug 21, 2010)

Hi,

I had some free time and I decided to classify some 3x3x3 cubes according to the permutability and orientability of their edges, corners and centers.

My final product is a series of pages with:

*180 Virtual Cubes according to permutability and orientability*,
you can play with anyone of them by just clicking them.

Let me be more clear.

Consider the edges of a cube: they could be all the same or all different or divided in two subsets containing 6 equal pieces each, etc ...
Moreover: the edges could be orientable or non-orientable or half orientable and half non-orientable, etc ...

I decided to use the following notations, where the first number (or sequence of numbers) refers to the permutability and the second number (or sequence of numbers) refers to the orientations:

E(12;1): 12 equal edges, only 1 orientation.
E(12;2): 12 equal edges, 2 orientations each.
E(1;1): all different edges, only 1 orientation.
E(1;2): all different edges, 2 orientations each. 
Edges(4 4 4, 1): three kinds of edges, 4 edges of each kind; 1 orientation
Edges(4 4 4, 2): three kinds of edges, 4 edges of each kind; 2 orientation for each edges ...

Same thing goes for corners and centers.

You can find a motivation and a more detailed explanation of this notation here

Consider the possibilities (read notations above):
C(8;1) C(8;3) C(1;1) C(1;3) C(4 4; 1) C(4 4; 3)
E(12;1) E(12;2) E(1;1) E(1;2) E(4 4 4; 1) E(4 4 4; 2) E(3 3 3 3; 1) E(3 3 3 3; 2) E(6 6; 1) E(6 6; 2)
F(*;1) F(*;2) F(*;4)
and consider all their possible combinations.
They give rise to 180 cubes. I've designed a cube for each of those categories. 

I don't know if this work has any particular interest but at least I had some fun with it.
There are various cubes that are quite fun and there are lots of cubes that force you to solve all kind of parities.
Of course the list could grow up a lot more by considering more categories [such as 
C(*; 1), C(*; 3), C(*; 1^4 3^4 ), C(8; *), C(1; *), C(4 4; *),
E(*; 1), E(*; 2), E(*; 1^6 2^6), E(12; *), E(1; *), E(4 4 4; *), E(3 3 3 3; *), E(6 6; *), E(9 3; 1), E(9 3; 2) ...
F(6; *) ] 
but, for now, I'm done ... maybe in the future if I have some more time.

Hope you find it somehow interesting.

guzman.


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## robindeun (Aug 21, 2010)

cool


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## BigGreen (Aug 21, 2010)

robindeun said:


> cool



story


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## Anthony (Aug 21, 2010)

BigGreen said:


> robindeun said:
> 
> 
> > cool
> ...



bro


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## hawkmp4 (Aug 21, 2010)

guzman said:


> There are various cubes that are quite fun and there are lots of cubes that force you to solve all kind of parities.


Like...?


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## guzman (Aug 21, 2010)

Anthony said:


> BigGreen said:
> 
> 
> > robindeun said:
> ...



Sorry, but I'm italian and I'm not sure I understand the meaning of your comments ... I'm also afraid to ask.



hawkmp4 said:


> guzman said:
> 
> 
> > There are various cubes that are quite fun and there are lots of cubes that force you to solve all kind of parities.
> ...



You mean, which one is fun? What about this one that doesn't have orientations at all

Or you mean, which ones do force you to solve parities ? most of them.

I know this classification may sound somewhat wierd,
but it was actually fun for me to look for cubes satisfying the constraints.

I also think many of them can be used for explanatory pourposes or as simplified cubes to start with ... dont know. 
Anyway, I thought I could share them since I made them ...

guzman.

PS: if you find mistakes or better cubes to any of the cases I'll be glad to know.

PS: Perhaps it could be fun to look for optimized algs in some cases such as for the orientation-free cube (or sub-cube as I call it)


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## Anthony (Aug 21, 2010)

guzman said:


> Anthony said:
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> 
> > BigGreen said:
> ...


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## hawkmp4 (Aug 21, 2010)

The terminology you're using is odd. What do you mean by 'orientability' and 'permutability?' Also, what situations are you referring to as 'parity?'


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## guzman (Aug 21, 2010)

Anthony said:


> guzman said:
> 
> 
> > Anthony said:
> ...



Now I understand, thank you   (for the explanation at least)




hawkmp4 said:


> The terminology you're using is odd. What do you mean by 'orientability' and 'permutability?' Also, what situations are you referring to as 'parity?'



You are right, sorry, I'm italian and I'm quite new to cubing.
By orientability I mean: are the cubies orientable? that is, do they have more than 1 orientation? and, if so, how many cubies are orientable ?
By permutability I mean that two cubies can be swapped without affecting the the solved state, so, in more simple words, they're basically just equal.

By parity situations I mean those situations where, for instance, only two edges are apparently swapped.
This can happen in many of the cubes I listed since in some cases they have some equal edges that might inadvertedly be swapped leading to the final situation where two different edges are swapped.
Hope it is clear.

guzman

PS: for instance solving this simple cube you might reach a situation where only two edges of the last layer need to swapped. Of course this isn't possible and you actually have to swapped those two ant two more equal egdes. But this may force you to learn new algoritms. 
In this sense, even though they are all simpler than the super-cube, they may in some cases require more knowledge (just like the void cube).


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## guzman (Aug 22, 2010)

Two questions.

Consider this cube:
all the edges are equal and all the corners are equal,
but edges, corners, and centers need to be oriented.
So this cube has the maximum number of orientations but only 1 permutation. Of course you can solve it with any method, but what are the best algs for solving a cube like this ?

On the contrary.
Consider this cube:
all the edges are different and all the corners are different so it has the maximum number of permutations, but it only has one orientation (cause all the cubies are non-orientable).
You can solve it with any method you want, but what are the best algs to solve a cube like this ? For instance, if we use Fridrich, can we simplify the PLLs using the fact that pieces are always oriented ?

I think these are mathematically and "cubistically" meaninful questions.

And that's the reason I made a catalog like that (so that one can actually try those cubes). 

[I hope I also answered to the "Cool story bro" comments]

guzman.


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## RCTACameron (Aug 22, 2010)

I like this idea. This cube is nice to play with. I had thought of the idea of a cube with the same colour on opposite sides, but this is the first time I have seen it. I like this idea.


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