That's why I'm not clear on the goal, is it:
Find an algorithm, whose performance is inconsequential,
that can determine the minimal number of moves to sort
Or is it something like:
Find an optimal algorithm for sorting the tapes such that
the only atomic operations move(m,n) and
examine(n) -- which tells you what tape is
in slot n.
If any amount of preprocessing and analysis is allowed,
then any number of hueristics could be useful for find
a path from the starting order to sorted order.
If nothing else, you can do a breadth first search
of all the possible permutations of moves untill you
achieve the desired ordering.