From:

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Date: 7 DEC 1980 0108-EST From: MJA at MIT-MC (Michael J. Aramini)

well it is possible that to maximally distant states are half twist apart

WRONG! (I assume you meant "two" for "to" and typo'ed). Read

ALAN's previous message. In the half twist metric, there exist

odd distances away, and there exist even distances away. A QTW

takes the cube from odd to even or from even to odd. The maximally

distant state is the state such that the fewest number of QTW

required to solve it is maximized. This must be odd OR even, and

thus, two states that are maximally distant must be both odd or

both even, which means the distance between them is even, or an

EVEN number of QTW. A single QTW is ODD, and thus cannot separate

maximal states.

also if you count half twists as one twist (i dont, but its

still worth thinking about) does that change the set of

maximally distant states?

Maybe it does, maybe it doesn't. It is much harder to tell because

counting half twists has no analog to the QTW odd/even property of

distance, and this is one reason several of us don't count half twists.

For example, (R L R) and (L [RR]) are equivalent manipulations, but in

half twist counting, one is three and the other is two moves. (assume []

means grouping two moves into one.)

also it is possible that there exists states for which all directions

lead closer to home (and twist put the cube in a state closer to home)

but the state is not necessarily maximally distant (to use

a continous analogy, think of think of a hill in a funtion of

two variables, that is not necessarily the maximum value of the

function)

We have been saying this all along! Simple example: (RRLL UUDD FFBB)

is a local max (any twist takes you closer), and it is definitely not

absolute max (abs max must be at least 21 from combinatoric arguments).