Date:  3 September 1980 2108-EDT
From: James.Saxe at CMU-10A (C410JS30)
Subject: Orbit classification revisited
To:  cube-lovers at MIT-MC
Message-ID: <03Sep80 210846 JS30@CMU-10A>

Hi, folks.

Having read Vanderschel's msg of Aug. 6, it appears to me that the
explanations of the orientation parities are unnecessarily complex,
though the material on permutation parities is presented in a way
that should be immediately convincing to anyone familiar with the
notion of even and odd permutations from elementary group theory (and
anyone who isn't should be!).  Here's my attempt at a more elegant
demonstration.

In what follows, I will use the term "facelet" to denote any (visible)
face of a cubie.  Thus, each edge cubie has two facelets and each
corner cubie has three facelets.

I will address Edge Orientation Parity (EOP) first.  Consider the
following diagram:

		+-------+
		|   1	|
		|0  U  0|
		|   1	|
	+-------+-------+-------+-------+
	|   1	|   0	|   1	|   0	|
	|0  L  0|1  F  1|0  R  0|1  B  1|
	|   1	|   0	|   1	|   0	|
	+-------+-------+-------+-------+
		|   1	|
		|0  D  0|
		|   1	|
		+-------+

The numbers label absolute positions, not facelets, and therefore
remain in the same configuration when the cube is manipulated.
Imagine that a mark is placed on each facelet that occupies a
position labeled "0" when the cube is in the solved configuration.
Thus, each edge cubie will have one marked and one unmarked facelet.
(Unlike the numbers, the marks are attached to the facelets and will
move as the cube is manipulated).  The parity of an edge cubie in an
arbitrary configuration is defined as the number labelling the
position occupied by its marked face, and the EOP is defined as the
sum of the parities of all edge cubies modulo 2.  A quarter turn of
any face reverses the parity of 4 edge cubies, thus leaving the EOP
fixed.  By induction, no sequence of manipulations starting from the
solved configuration can produce a configuration EOP = 1.  So much
for EOP.

[Just for grins, here's a cute way to determine the parity of an edge
cubie without consulting (or reconstructing) the diagram:  Assign
each of the cubie's two facelets a number in {0,1,2} according as it
is oriented parallel to its home position ("Self" -> 0), parallel
to the other facelet's home position ("Other" -> 2) or perpendicular
to both home positions ("Neither" -> 1); add these two numbers and
reduce modulo 3 (yes, three!) giving the parity of the cubie.  The
sum can never be 2 because that would imply that the two facelets
were parallel to each other.  Here's an addition table with
double-ended arrows showing the possible transitions:

	      (S) 0   (N) 1   (O) 2
	    +-------+-------+-------+
	    |	    |	    |no     |
      (S) 0 |	0 <-+-> 1   |	+   |
	    |	^   |	^   |	 way|
	    +---+---+---+---+-------+
	    |	v   |no |   |	    |
      (N) 1 |	1 <-+---+---+-> 0   |
	    |	    |	|way|	^   |
	    +-------+---+---+---+---+
	    |no     |	v   |	v   |
      (O) 2 |	+   |	0 <-+-> 1   |
	    |	 way|	    |	    |
	    +-------+-------+-------+

]

Now for Corner Orientation Parity (COP).  Consider the diagram below:

		+-------+
		|0     0|
		|   U	|
		|0     0|
	+-------+-------+-------+-------+
	|2     1|2     1|2     1|2     1|
	|   L	|   F	|   R	|   B	|
	|1     2|1     2|1     2|1     2|
	+-------+-------+-------+-------+
		|0     0|
		|   D	|
		|0     0|
		+-------+

Once again, imagine that we mark all (corner) facelets which occupy
positions labeled "0" when the cube is in the solved configuration.
The parity of a corner cubie in an arbitrary configuration is defined
as the number which labels the position of the cubie's marked face.
The COP is the sum of all the parities of all corner cubies modulo 3.
Inspection of the diagram will reveal that the twists U, u, D, and d
leave the parities of all corner cubies unchanged.  Any of the other
possible quarter twists will increment (modulo 3) the parities of two
corner cubies and decrement the parities of two others, thereby
leaving the COP unchanged.

[As Vanderschel pointed out, one way to compute the parity of a
corner cubie by looking at it is to note the number of clockwise (as
viewed from outside the cube) 120 degree twists of the cubie that it
would take to bring the marked facelet parallel to its home position.
Note that the parity of a corner cubie, unlike that of an edge cubie,
depends on the selection of a particular pair of opposing colors for
the marked facelets.  While this lack of symmetry may be considered
unfortunate, it is an inevitable result of the fact that four is not
divisible by three.  It is easy to show that the COP as a whole is
independent of the choice of distinguished faces.]

Vanderschel also mentions the extended problem, but does not quite
make it clear that the FOP changes every time a qtw is done.  This
constrains all three of {FOP, EPP, CPP} to be the same, so that only
two of the eight plausible states of the <FOP, EPP, CPP> vector are
actually achievable by twisting.

				Jim