   Date: Sun, 07 Dec 80 07:24:00 -0500 (EST)   From: David C. Plummer <DCP@MIT-MC >
~~~ ~~~ Subject: maximally distant state, setting the record straight
```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).     