The best known bounds put the maximum number of qtw at between 21
and 104. At most about 1 percent of the cube's positions are
within 18 qtw of solved.
At each qtw, the number of positions increases by a factor of
almost sqrt(12+5*sqrt(6))+sqrt(6)+2 ~ 9.374. [I say ``almost''
because this rate of increase does not take into account the
identities of length 12 qtw or greater. Eventually, long identies
reduce this factor to zero. But between six and seven qtw, the
factor is about 9.356]. If we want to show a 2n qtw process
optimal, we need only scan the table of n-1 qtw processes, so the
increase is more like a factor of 3 per qtw.
Finally, I would like to call attention to the fact that showing an
18 qtw process to be optimal requires about 150 megabytes and two
days on a PDP-20 for each such proof. Finding someone to lend you
a big machine and a big disk for two days is not something you can
take for granted. And if you want to do it on your Apple, be
prepared to sit around swapping 2400 floppies in and out for the
next decade or so, as that two day figure is mostly disk latency
rather than CPU time.