Message #752

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Chronicles of a Rubik junkie’s experience with the {5}x{5}
Date: Sun, 01 Nov 2009 18:05:28 -0600

My thanks to Chris for the cool insights about the duoprisms! And thanks to
Melinda for the picture, dramatically showing the way stickers are slaved to
their respective tori. I hadn’t mentally put some of this together as
cleanly as in these emails, even after solving the {5}x{5}-3. Great stuff
:D

One thing that was neat for me was to think about while reading was how the
{4}x{4} fit into it all. Since it has only one type of 2C piece, there
seems to be a succession of symmetry breaking that happens with the
progression of puzzles - {4}x{4} is more symmetrical than general {n}x{n},
which in turn is more symmetrical than general {n}x{m}.

In some ways, it’s hard to say which puzzles are more difficult. The
{4}x{4} might take less sequences, but it’s stickers can move amongst both
tori, so the scrambling result is more complicated (it wouldn’t produce a
neat picture of dual tori like the {6}x{6} did for Melinda). But even if
there is somewhat of a conservation-of-difficulty effect with the tradeoff
between number of piece types and scrambling capacity, I guess it still
feels like difficulty rises more quickly as the duoprisms become more
general. Chris could say better of course.

As an aside: the {3}x{3} (which we aren’t officially supporting yet because
we’re still considering how the twisting works on it) also only has 1 type
of 2C piece, but it is degenerate when it comes to 2C pieces around the
"rings". And so unlike the flexible {4}x{4}, it’s pieces are again slaved
to their respective tori. I always knew the 4^3 was special :)

Chris, congrats to you as well on all the firsts your snagging :) And much
thanks for all the issue updates and feedback along the way!

All the best,
Roice

P.S. I agree with Melinda about the {5}x{5}-5 being a big prize. Not sure
I’ll ever tackle it, but that had been my favorite from very early on.

On Sat, Oct 31, 2009 at 12:42 AM, Melinda Green <melinda@superliminal.com>wrote:

>
> [Attachment(s) <#124a91e52cd6e093_TopText> from Melinda Green included
> below]
>
> 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}
> 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
>
>