From hoey@aic.nrl.navy.mil  Wed Jun 24 15:45:56 1992
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Date: Wed, 24 Jun 92 15:45:51 EDT
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Message-Id: <9206241945.AA09989@Sun0.AIC.NRL.Navy.Mil>
To: MONET01@mizzou1.missouri.edu, cube-lovers@life.ai.mit.edu
Subject: Re: Ultimate cube

MONET01@mizzou1.missouri.edu writes of a cube that

    ``looks like someone took a knife to a normal solved cube and cut
    a diagonal 'x' through each face and folded the flaps back down
    the sides.  This leads to a cube where opposing centers have an
    'x' that has four colors in a mirror image.  (It is hard to
    describe, sorry.)''

I would appreciate a few more details.  I think the color scheme of
each face you describe is something like

+-----+-----+-----+
|.1111|11111|1111.|
|44.11|11111|11.22|
|4444.|11111|.2222|
+-----+-----+-----+
|44444|.111.|22222|
|44444|44.22|22222|
|44444|.333.|22222|
+-----+-----+-----+
|4444.|33333|.2222|
|44.33|33333|33.22|
|.3333|33333|3333.|
+-----+-----+-----+

where 1,2,3, and 4 are distinct colors, but there are still several
ways to make the colors on different faces match up.  Look at a
corner, where the colors are

    +-------+
   /a.bbbbb/c\
  /aaa.bbb/ccc\
 /aaaaa.b/ccccc\
+-------+.......+
 \fffff.e\ddddd/
  \fff.eee\ddd/
   \f.eeeee\d/
    +-------+

That is, one corner is colored a/b, another c/d, and the third e/f,
where I expect some of a,b,c,d,e,f will be the same color.

One possibility was pictured in Hofstatder's Scientific American
article of February, 1981.  It had b=c,d=e,f=a and used twelve colors.
Jim Saxe and I were impressed by its wasteful use of color and its
failure to exhibit edge orientation.  From your remarks about turning
it over, I suspect this isn't what you mean.

You may be talking about the cube in which a=d,b=e,c=f which uses six
colors.  I would say it is as if you cut an 'x' on a cube and
exchanged each triangle with the other triangle on the same edge of
the cube.  That is a reasonably good coloring.  It isn't really
necessary to solve it twice, though.  To find out whether a given
corner goes on the top or bottom, look at the two colors that the
corner shares with the top face center.  Either the corner will have
the two colors in the same order as the top, or they will be reversed,
and that determines whether that corner goes on the top or bottom.
That tells you where the third color on that corner goes, and the
last color is determined by elimination.

There is an even more interesting coloring that uses only four colors.
In this coloring a=c=e and the other three colors are distinct.  Jim
Saxe and I came up with this coloring in our discussions of
Hofstatder's article.  It isn't quite symmetric enough, since its
reflection is a coloring in which b=d=f, a slightly different pattern.
Our discussions then led to the Tartan coloring we talked about in our
article of 16 February 1981.

The only cube in the archives called the Ultimate Cube is the one that
has ``over 43 quintillion solutions.''  It has all six sides colored
the same.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil