   Date: Thu, 31 Jul 80 14:28:38 -0400 (EDT)   From: Allan C. Wechsler <ACW@MIT-AI >   Subject: [no subject]

Yeah, we have a prejudice against regarding
180-degree twists as atomic. I understand
your feeling that a 180-degree twist is
intuitively a single operation.

Many of the cube-hackers at MIT became
interested in the mathematical aspects of
the cube, and the preference for counting
quarter twists arose from this (admittedly
rather Spartan) mathematical viewpoint.

When the cube first appeared, the mathematicians
among us instantly exclaimed, with great
delight, "Wow, here we have a group, whose
elements are possible manipulations of the
cube, and whose binary operation consists
of following one manipulation with another."
We immediately got interested in group-
theory questions like, "What is the order
of this group?" "Does it have any interesting
subgroups?" and, in general "What kind of
object is this group? Does understanding
it help us solve the cube better?"

There are several common ways of representing
groups. One is as a subgroup of a permutation
group. This doesn't really help in the case of
the Hungarian Cube, because it is too close
to what the cube really is: few new facts or
insights are revealed. Another way is with
generators and relations. This means, to list
a few basic group elements from which the whole
group may be derived by multiplying them together.

We soon figured out (along with hundreds of other
mathematically-inclined cube-hackers) that the
whole group of possible manipulations could be
generated from six elements: the quarter-twists
of each of the six faces. This observation
later turned out to be crucial in calculating
the order (number of possible states) of the group.

Hence our predilection for counting quarter-turns.
The half-turns were already accounted for, and
we thought of them as two juxtaposed quarter-turns.
I guess some of us believe that the mathematical
structure of the cube group is built on quarter-
turns. Those whose delight in the cube is not
mathematical will not agree: after all,
a half-twist is as easy as a quarter-twist
to perform. But you will miss things like
the fact that many useful manipulations
are 8, 12, or 24 quarter-turns long. If you
count half-turns, you get a whole spectrum
of random move counts, thus missing some
fundamental (and as yet little-understood)
kinship between these manipulations.

Of course, if you are not interested in such
things, any measure of complexity (why not
count equator twists? why not penalize for
counter-clockwise twists, since they are
marginally harder for right-handed people to
do?) will suffice.

---Wechsler     