From:

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