Message #475

From: Roice Nelson <roice3@gmail.com>
Subject: checkerboard fun
Date: Fri, 11 Apr 2008 00:16:30 -0500

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 if anyone wants to see what they
look like.

Take care all,

Roice