Message #750
From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Chronicles of a Rubik junkie’s experience with the {5}x{5}
Date: Fri, 30 Oct 2009 22:42:47 -0700
Chris Locke wrote:
> Hello Roice, and congrats on solving the {5}x{5} 3, and for sharing your
> story with us!
>
> I took inspiration from the fact that last night, when you uploaded your
> solution to {5}x{5} 3, that there was still puzzles other than the {5}x{4}
> which I failed to finish first, and as such, was able to start a second
> puzzle. I was quite fascinated by how the uniform duoprisms worked,
> especially how whereas the {5}x{4} has multiple kinds of pieces based on
> which faces they are touching (like there are 3c pieces that touch two 5 and
> one 4 block, and 3c pieces that touch one 5 and two 4 blocks, each requiring
> different sequences), the uniform duoprisms only have one kind of piece
> since the two torii that make up the duoprism are formed by the same kinds
> of blocks (fun trying to ‘visualize’ it as two interlocking torii :D). This
> meant that I would only need algorithms for 2c(within torus), 2c(between
> torus), 3c, 4c.
>
I hadn’t noticed it before but you’re totally right that there are two
different kinds of 3c pieces! That means that sometimes we may need to
be clear on which type of 3c piece we’re talking about. That’s very cool
and I bet you’re right that this will be why the non-uniform duoprisms
will be harder to solve than the uniform ones.
> So I decided to try to solve {6}x{6}, it being the next largest uniform
> duoprism. From my experience with the {5}x{4}, I was able to rather quickly
> solve all 2c pieces without macros, and for 3c and 4c, was able to rather
> quickly find new algorithms. Basically, all I ended up needing was a
> 3-cycle for 3c pieces, and a 3-cycle for 4c pieces. Also, along the way you
> notice that the colors of each torus, stay on it’s respective torus. So if
> you have a white face on one torus, you won’t find a white sticker on the
> other torus.
>
Yes, I’d noticed that when I was reimplementing scrambling. I just
happened to end up with mostly yellow/red colors on one torus and
blue/green colors on the other. Then when I performed a full scramble, I
ended up with two beautifully speckled toruses, one in each color
scheme. [See attached screen shot.] At first I was sure that I simply
had a bug in my scrambling algorithm. It took me a while before I
understood what you just pointed out.
Here’s a 12-color specification in case anybody wants to reproduce this.
Just be sure to unwrap the formatting and put this all on one line in
your facecolors.txt file:
255,0,204 153,0,153 255,51,0 255,102,102 255,153,0 255,255,0 102,153,0
0,255,0 0,204,153 0,255,255 0,0,255
102,102,255
> Helps to keep this in mind when you are working away. You are
> able to solve all kinds of parity situations with just these moves by
> careful usage of conjugation. For instance, if on one face you have all 3c
> in place except you need to swap two, you can bring down one from an
> unsolved layer, 3 cycle it into the face you’re working on (my 3-cycle macro
> only does swaps in a plane, but conjugation can make it do almost anything),
> then put that piece back up into the face it came from but in an adjacent
> position, then take the piece you want to bring back down, and pull it
> down. The result being you do two pair swaps, one in the face you’re
> working on, one in the other face you don’t care about. For orientation,
> you can do a similar thing, but by commuting with a twist of one of the
> surrounding torii’s faces (it temporarily messes up a couple 2c(within
> torus) pieces, but the commutation fixes that right up). It really helps to
> also have scrap paper to use and carefully keep track of where you move
> pieces and whatnot when you are trying to find the proper conjugations
> needed, but after a while you can start to see the bigger picture and do
> these fixes on the go.
>
> Interestingly, the exact same methodology applied for the {5}x{4} puzzle
> too, only I needed separate algorithms for the two kinds of 3c pieces each.
> And upon solving the {5}x{4} 3 and {6}x{6} 3, I have the feeling that the
> same algorithms can be adapted to solve {n}x{n} 3 for any size duoprism of
> length 3, it would just obviously take more time. So to answer your
> question Roice, I think that while these are definately parity cases we run
> into, since they can be solved by 3-cycles and conjugation for both the
> uniform and non-uniform duoprisms I solved, they are probably a prevalent
> feature of all duoprisms. But unlike {4}x{4} 3, I never ran into the case
> where a single 4c piece needs to be flipped. It might be that I was lucky,
> but the whole puzzle just felt like complicated parity cases like that are
> non-existent. Again, I could be wrong, but that’s just what I felt (hard to
> draw accurate results from a single trial though :D). As for 4 length
> puzzles… the parity possibilities with that scare me!
>
This suggests to me that one of the next big prizes up for grabs by
those patient cubists among you who like big puzzles will be the {5}x{5}
- It has the same sort of notational appeal as the 5^5, and the parity
issues it brings should make it suitably frightening in that dimension
as well. To the couple of masochist among you who have been asking us to
implement 6^4 and larger: even though you now finally have those cubes
available, I suggest that you focus on this plumb new prize first! ;-)
> Another note, Roice, I two typically find my double swap 4c macro first
> also, but it’s fairly simple to take that and make it into a straight-up
> 3-cycle by putting it into a commutator I discovered.
>
> In conclusion, most of these length 3 puzzles I feel can be solved fully
> once you isolate a 3-cycle for each of the corresponding pieces. I had more
> algorithms for when I did {5}x{4} 3, but if I did it again I’d probably do
> it similar to how I solved the {6}x{6} 3 and cut down on my algorithms
> greatly (I had one realllly crazy 174 move algorithm just to do a 3-cycle of
> one kind of 3c pieces, but it can probably be greatly shortened if I start
> from scratch :D).
>
> Anyway, hopefully my train of thought put into text makes sense ^^
The thing that I really like is that even though I’m a better coder than
a cubist, I found that that actually did make sense to me! Either I’m
learning, or you’re a good writer. Probably both!
Thanks for the report and the instruction. Oh, and a huge
congratulations on snagging the first {6}x{6} 3 solution, Chris!!
-Melinda