Date: Thu, 06 Nov 86 13:41:48 -0500 (EST)
From: Dan Hoey <hoey@nrl-aic.ARPA >
~~~ Subject: Magic

I have three macros for transforming 2x4 rectangles. To solve the
puzzle, I use two of them followed by a seven-flip macro that changes a
2x4 shape into a 3x3-1 shape (beats me how BECK can call this a
``2x3x3''). Took me a couple of half-days to solve it.

I have found 32 different 2x4 rectangles. I think that is all of them,
but I haven't got any proofs, nor even a decent mathematical model for
deciding when a flip is possible.

I am trying to understand how the strings work. First, it looks like
there is twice as much string as necessary; each string is doubled. I
guess that this duplication has no effect on the puzzle except for
durability, but until I can dyke one out can't be sure. I'm concerned
that the string may be one double loop, so I'm looking for a good way
to make sure the thing doesn't unstring entirely when I cut one.

Each side of each piece has four short channel segments and four long;
half are occupied with string. If you continue each segment across
each hinge to the next piece, you get eight channels composed of
alternating long and short segments. Again, four of the eight channels
are occupied. In the positions I've seen, each of the channels
contains eight pairs of segments.

But Magic is more complicated than that--the strings do not always
follow a channel from piece to piece. On half of the pieces, there is
an extra loop of string that wraps back onto the piece without
following the channel to the next piece. I don't know what function
this serves.

If a good model of the string interactions can be developed, we may be
able to make an attack on the doughnut problem based on the length of
string channels. In the doughnut, there are four ten-pair channels and
four six-pair channels. We may be able to show that the string wouldn't
reach one, and would exceed the other. More likely, the model will
prohibit the doughnut more directly.

There is another string-related question I am wondering about. I have
noticed some of the string-pairs getting twisted. I wonder how bad
this can get. Does anyone have an operation that can be repeated to
make the twists tighter and tighter? Are these puzzles built for
obsolescence?

I have been considering Magic metrics, but it's a difficult problem.
Counting flips is easy enough, but how do you count a move that skews a
parallelogram? Are such skew moves necessary?

Dan