There are two types of aproaches to what can be meant by total
(or maximal) randomizing. One is to consider states of the cube that
are maximally far away from being solved (that is the minimal number of
quarter twists needed to solve from a given state is maximallized.
the other, which may be more useful for making the cube more difficult
to solve, is based on maximallizing the amount of time it would take
a person (or a program, for that matter) to solve the cube. Of course,
since this amount of time is both dependent on the state of the cube, and
how the person (or program) goes about solving it. Thus the set of most
randomized positions is dependent on who is solving the cube.
the advantage of the first definition is that it seems to be of more
theorectical significance (the number of quarter turns needed to
solve from this position gives you the diameter of the state graph).
The second approach seems more kludgy since it much less well defined
since the amount of time it takes to solve the cube if a function of
many variable besides simply the initial state.
This distinction is much like differing strategies for writing chess
playing programs, you could write it assuming the oponent to be a
"perfect" player (as if a complete look ahead to the leaves of the tree
appraoach were being used by the oponent) or by considering how the
oponent is likely to play (have the program try to confuse the oponent
by taking advantage of something that the oponent wont recognise).
the second method thus will take into account the knowledge-base
available to the solver (for example trying to trick the solver into
thinking that the cube is in one of the easy to solve classes but really
isnt, thus leading the solver down a blind alley)
This gives me an idea for a game where oponents are each trying to bring
a randomized cube into two different final states (each of which
is hopefully equally far away from the initial state, just to make the
game fair) by alternately taking turns making quarter turns. of course
one must make up some rules so that that the game terminates (ei going
around cycles in the state graph an indefinite number of times is
disallowed), and these rules might not be very obvious,although
they would prob be similar to some of the stalemates rules of chess.