Message #844

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Re: Introducing "MagicTile"
Date: Thu, 04 Feb 2010 00:33:52 -0800

Chris Locke wrote:
>
>
> Congratulations Roice on another jaw-dropping program! The community
> of beyond physically realizable Rubik cubers I’m sure is both shocked
> and pleased by the recent flurry of new puzzles. First we get tons
> new cool 4D shapes to play with, and now hyperbolic puzzles?? What’s
> next, 4D equivalents of these hyperbolic puzzles? :D
>
> First thing I would note is that you should move the "donate" button
> to a more visible location. As you can see by Alexander’s post it’s
> not easy to find. I didn’t even know you had it on the page until
> recently when I actually scrolled all the way to the bottom!
>
> As for the puzzles themselves. I find that from my limited playing
> around with the hexagonal and heptagonal tilings, that low color
> counts completely baffle me, but as the number of colors increases,
> the amount of coupling between the twists decreases, which makes it
> much easier to work with. The 3 and 4 color hexagonal tilings are
> just weird to try and solve :P. But in the cases with many colors,
> it’s feels like it’s just a matter of patient to reduce it down to a
> last layer. From there, Megamix LL algorithms/commutators can be
> generalized to finish them up.

I think that this phenomenon is the same one that makes the megaminx
easier to solve than the cube because there’s more "room" on the surface
of the puzzles with more faces in which to squirrel away some parts
while working on others. The smaller, puzzles are "tighter" and cause
every action to affect just about everything else. I have a funny
feeling that there is some sort of natural difficulty metric in which
the original 3^3 Rubik’s cube will turn out to be the hardest of all the
similar puzzles in all dimensions, but I can’t quite see how to define
that metric. I only feel it.

> As for the double bottoms thing, Melinda gave a pretty good
> explanation as to how it’s a result of the topology. In the case of
> the 3 holed torus (Klein’s quartic) it’s pretty easy to see how
> spreading out from one position can leave with with a case of two
> disconnected unsolved faces. Furthermore, I don’t think it’s possible
> to end up with 3 disconnected faces in this case, and also if you
> understand the topology from the tiling, it should also always be
> possible to solve it such that you end up with just one unsolved
> face. But that in itself is a challenge too :D.

You’d need a genus 4 or higher surface to end up with 3 or more
disconnected unsolved faces.

I’m not sure if you are right that one should be able to follow a
modified layer-by-layer method on any of these puzzles and always end up
with a single unsolved face. The key would be to keep track of the
"outer edge" of your solved patch as you grow it. Never let one part of
that edge connect with any other part. In other words, you make sure to
keep that edge a simple closed curve. Any time a part of it wants to
connect with another part, just leave that area and work on some other
part of the border that has room to spread. The question is whether the
initially growing edge will eventually shrink back down to a single face.

At first I thought yes, but as I started to write the above, I’m now not
so sure. This is reminding me a lot of the short Wikipedia article that
Roice cited on simply connected
<http://en.wikipedia.org/wiki/Simply_connected> spaces. Imagine trying
the above solving strategy with a {4,4} puzzle defined on the surface of
a torus. I think you’ll end up with a seam that you can’t get around. I
just can’t quite see it in my head without some good pictures.

> […] Less colors is too hard, and more colors takes too long.

Each color is a single face. Each time that you see a red face, it’s
always the same face just seen from a different perspective. Kind of
like how gravitational lensing can let you see multiple images of the
same distant galaxy in a single photograph. I agree that tighter puzzles
take more brain work and less tedium but when they become extremely
tight, they seem to get easier again in a way as you can begin to get
your head around the whole thing. In addition to wanting to know which
puzzle is hardest for it’s size, I also want to know is which puzzle out
of all the twisty puzzles different people think has the challenge that
is "just right" for them.

-Melinda