Having read the CUBE-LOVERS mail, I note a recurring theme of dismay
that there is no clearly stated procedure for deciding whether a given
configuration is reachable (or solvable) or not. Furthermore, I share
Dyer's view that there has been no persuasive presentation of the fact
that there are precisely 12 equivalence classes of permutations under
the transformations permitted by the cube. In this note, I intend to
clear up this situation for the benefit of the less sophisticated
readers of this material.
I should apologize at the outset for my ignorance. I have not read
Singmaster's pamphlet, nor have I communicated with any of the
knowledgeable people around here at Stanford on the subject of cubes.
My knowledge stems entirely from my own personal efforts at solving
cubes and what I have gleaned from CUBE-LOVERS mail. So I hope
readers will be tolerant of any lengthy explanations of well-known
COMMENTS ON NOTATION
I will start by offering my two-bits-worth on notation. It is
relevant because I will use some of it later.
I prefer to call the cubies in the centers of the faces "face cubies",
because "center" and "corner" both start with a "C". (Also, I have
worked with other 3X3X3 cubes and tended to use "center" for the one
in the middle that you cannot see.)
I do not agree that there is any need for notation to describe
orientation of the whole cube or reorientation of it. For discussion
purposes, you should pick one orientation of the cube (based on the
direction in which its face cubies are oriented) and leave it that
way. All transformations can be described relative to that
Rotating a center-slice should not be considered to be a move, since
it changes the orientation of the cube.
Since it is not the case that all cubes have the same color pattern,
no direct reference should be made to color. Instead the exposed
faces of any cubie should be identified in terms of the direction
(LRFBUD) they will face when the cubie is in its home (or at START)
position relative to the chosen orientation of the cube.
When I was learning to work cubes, the concept of complementarity
played a critical role. (I do not claim my methods are good. It takes
me about 20 min.) It is certainly handy to have a way of talking
about pairs of opposing faces. I think most would agree that the top
and bottom can be referred to as "horizontal" faces, and left and
right can be called "lateral" faces. For front and back, I really do
not know a good word; I suggest calling them "extremal", since their
face cubies are the closest and most distant from the observer's point
ORBIT CLASSIFICATION (for the uninitiated)
It is possible to define some "parity" concepts that simplify stating
the characterization of the equivalence classes of cube
configurations. The precise definitions will follow; but to start
with, we will name them:
Edge Permutation Parity (EPP = 0 or 1)
Edge Orientation Parity (EOP = 0 or 1)
Corner Permutation Parity (CPP = 0 or 1)
Corner Orientation Parity (COP = 0, 1, or 2)
A configuration is reachable if COP and EOP are zero and CPP=EPP.
More generally, setting TPP=CPP+EPP mod 2, we can define
Total Permutation Parity (TPP = 0 or 1).
Then the Parity Vector, defined by (TPP, EOP, COP), can be used to
represent the equivalence class of a configuration.
Permutation Parities Defined -
Background: A permutation of [1,n] is considered to be even (0) or odd
(1) depending on whether an even or odd number of pair swaps is
required to restore the set to original order. You can compute it by
counting the number of reversals in the sequence - ie., the number of
pairs, (p,q) such that p>q and p precedes q.
For the cube, you can assign a position number (1-8) for each of the
corners and also a number (1-12) for each of the edge positions. For
each cubie, you can the speak of its home position number and its
current position. Writing down home position numbers in order of
current position gives a permutation of natural numbers in which you
can count reversals to see if it is odd or even. This is done without
regard to orientation.
Properties: A quarter turn is an odd permutation of the four edges
involved and also of the four corners. Thus a quarter turn changes
both EPP and CPP but leaves TPP unchanged. TPP is preserved by all
Edge Orientation Parity Defined -
For each edge cubie consider its Oriented Distance from Home, defined
to be the smallest number of quarter turn twists required to put it in
its home position with correct orientation. It is no bigger than 4.
As an example, an edge cubie at home with the wrong orientation is at
an oriented distance of 3. EOP is the sum modulo 2 of the Oriented
Distances from Home for all edge cubies.
Properties: For each edge cubie affected, a quarter turn either
increases or decreases its Oriented Distance from Home by 1. Since 4
edge cubies are affected, the net effect must be even. Thus all
twists preserve EOP.
Corner Orientation Parity Defined -
Looking head-on at the apex of any corner you can consider twisting it
to any one of three positions. For any given corner cubie we define
its individual orientation parity to be the number of 120 degree
counter-clockwise twists required to bring the horizontal face of the
cubie into horizontal position relative to the whole cube. COP is the
sum modulo 3 of the individual parities. (Since it is three-valued,
it could be argued that COP ought not to be called a "parity", but
that fouls up the parallelism in the discussion.)
Properties: Twisting a horizontal face does not change the orientation
of any corner. Twisting an extremal or lateral face does alter
orientations. Consider a counter-clockwise quarter turn of either
kind of vertical face. For each of the two corner cubies that remain
in the same horizontal face, the orientation parity decreases by 1
modulo 3. For the other two, it increases by 1. Again, the net
effect is zero. COP is preserved by all twists.
Equivalence Classes -
There must be at least 12 orbits, since all transformations preserve
the Parity Vector, (TPP,EOP,COP), and it has 12 possible values. To
show there cannot be more, you must show, given two configurations
with the same Parity Vector, that one can be transformed into the
other. The first paragraph of the note from Davis to DDYER, dated
Aug. 4, indicates how the corners can be permuted and reoriented. We
need to be a little more careful about the interaction between the
processes that rearrange corners and edges. Suppose we are
considering going from a configuration A to a configuration B, and
consider the intermediate stage C at which we have permuted and
reoriented the corners to agree with the target configuration B.
First we observe that the EPP of C must agree with that of B, whether
or not A and B have the same CPP. Thus there is an even permutation
of the edges of C that will put them in the desired target positions
of B. This permutation can be generated by swapping appropriate pairs
of edge cubie pairs. Consider the intermediate stage D achieved after
all edge cubies are in their desired positions. Now the EOPs of D and
B agree, so the number of edge cubies with wrong orientation must be
even. This can be corrected by flipping an appropriate set of edge
pairs in place.
The Extended Problem -
If the orientations of face cubies are to be considered then we
Face Orientation Parity (FOP = 0 or 1).
It is the number of quarter turns modulo 2 required to get all the
face cubies back to starting position. Adding this fourth component
to the Parity Vector yields 24 equivalence classes. Existence of
appropriate transformations for moving about within equivalence
classes was indicated by ALAN and CMB in their notes of Jul. 15.
Complementary Configurations -
A configuration may be said to be complementary if every color exposed
on a face of the cube either agrees with that of the face cubie on
that cube face or comes from the parallel opposing face. There seems
to be a fair amount of interest in such positions. It is easy to see
for such configurations that COP and EOP are 0. Thus the
configruation is reachable if and only if its TPP is 0. This can be
determined easily by counting "wrong" cubie faces - ie., edge or
corner cubie faces with the color complementary to that of the face
cubie on the same cube face. TPP is 0 if and only if this number is a
multiple of four. (It is always even.)
If you restrict yourself to 180 degree twists, you can generate only
complementary configurations. Thus the more interesting complementary
configurations tend to be those that require quarter turns to achieve.
A REDUCED PROBLEM ?
Suppose you lock up four of the axles and consider turning just two
adjacent faces. This Two-Faced puzzle deals with only 15 of the
cubies. I got the bright idea of playing with a logically jammed cube
after I had already developed a crude methodology for working cubes.
I figured that with a simpler problem I might gain some new insights.
Actually, I got confused. In retrospect, this is not so surprising.
I think that most of us would grant that if you can work Two-Faced
cubes, you can certainly work the whole thing. The point is that you
tend to encounter many of the same sorts of problems, but you have
fewer degrees of freedom for dealing with them.
The argument already given shows that the Two-Faced cube must have at
least 12 equivalence classes. It is not clear to me that there are
not more in this case. I have not discovered all the tools to show
that two members of the same equivalence class (in the regular sense)
can be transformed into one another. Can anyone out there resolve