Message #476

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] checkerboard fun
Date: Fri, 11 Apr 2008 18:24:19 -0500

heh, I should have known to do that. I promise I am trainable. It just
takes a long time. (Sarah can vouch that both of those statements are true
:))

Anyway, here ya go!

www.gravitation3d.com/mc4d/checkerboards/

On Fri, Apr 11, 2008 at 1:21 AM, Melinda Green <melinda@superliminal.com>
wrote:

> Screenshots!!
>
> Roice Nelson wrote:
>
> > Hey guys,
> > I’m avoiding doing my taxes and so I’ve had some interesting
> > investigations into MC4D checkerboard patterns I thought I’d share :) I was
> > curious about how the uncommon checkerboards on the 3D cube extended to four
> > dimensions. If you aren’t familiar with those, it is possible to make a
> > checkerboard that has two 3-cycles of colors instead of three 2-cycles (the
> > latter being the most familiar and most easily produced pattern that
> > exchanges opposite colors). As far as I know, there is no easy sequence to
> > make the 3-cycle checkerboard, so you have to manually place all the pieces.
> > There is also a checkerboard pattern on the 3D cube with one 6-cycle! You
> > can quickly make it if you already have the 3-cycle checkerboard by then
> > applying the sequence of moves that normally makes the 2-cycle checkerboard
> > from the pristine state. While the 6-cycle is a "superposition" of a
> > 2-cycle and a 3-cycle, these two seem to be sort of basic checkerboard
> > patterns for the 3D puzzle in that neither can be created by superpositions
> > of the other. Also of note in 3D is that you can do a single 2-cycle that
> > only checkerboards 2 opposite faces and leaves the remaining 4 faces solid.
> > Puzzle states that are superpositions of this valid state are also valid
> > (in fact, maybe it is better to consider this a basic unit with which to
> > develop more complicated checkerboards over the ‘three 2-cycle pattern’
> > since the latter is just 3 of these single guys).
> > With that background I wondered what types of checkerboard cycles can
> > be produced on MC4D and what the lowest level patterns are that can be
> > superimposed to make the more complicated ones. I figured some of these
> > might only be be creatable by manually placing pieces like in the 3D case,
> > which would be a lot of work, so I decided to edit log files by hand and use
> > Don’s cool solve feature in MC4D to check whether certain puzzle states were
> > valid. Much of what I found was surprising and against what I might have
> > guessed. Here is a rundown…
> > *Full Checkerboards:*
> > puzzle states that were possible:
> > four 2-cycles of opposite colors (standard checkerboard like the ones
> > in the hall of fame)
> > four 2-cycles of adjacent colors
> > one 6-cycle and one 2-cycle
> > two orthogonal 4-cycles (the cycle "direction" of the first did not
> > force a direction of the second cycle. I can describe more about this
> > interesting case if anyone wants me to.)
> >
> > puzzle states that were not possible:
> >
> > two 3-cycles and one 2-cycle (various arrangements tested)
> > four 2-cycles (2 opposite, 2 adjacent)
> > one 8-cycle
> > *Partial Checkerboards:*
> > puzzle states that were possible:
> > two 2-cycles of opposite colors (4 solid faces)
> > two 3-cycles (2 solid faces)
> >
> > puzzle states that were not possible:
> >
> > one 2-cycle! (contrast this with 3D case)
> > two 2-cycles of adjacent colors
> > one 4-cycle (whether through 2 sets of opposite faces or 4 adjacent
> > faces or a combination)
> > one 4-cycle and one 2-cycle
> > three 2-cycles (whether opposite cycles or not)
> > …
> > Without going too far into the last category, it was clear that
> > superpositions of the possible partial checkerboards led to the possible
> > full checkerboards I found. Superpositions of impossible partial states
> > could lead to valid states, but didn’t necessarily do so. I also noticed
> > one basic unit could produce all the valid full checkerboards I found. This
> > was the partial checkerboard with two 3-cycles (the valid 2-cycle partial
> > checkerboard can be made by superimposing that one, yet another contrast to
> > the 3D case). Since this was a trial and error approach vs. an enumeration
> > proof, I certainly may have missed some valid checkerboards, but this seems
> > like it might be it. I wondered (but doubted) if any of these less common
> > checkerboards could have shorter solutions than the current record.
> > I found it interesting that it was not possible to do a full
> > checkerboard pattern having 3-cycles. I tried this in various ways (having
> > the 2-cycle that would go along with it be of opposite faces or adjacent
> > faces and playing with the twirl direction of the two 3-cycles).
> > I put files of the possible full checkerboards at
> > www.gravitation3d.com/mc4d/checkerboards <
> > http://www.gravitation3d.com/mc4d/checkerboards> if anyone wants to see
> > what they look like.
> > Take care all,
> > Roice
>
>