There is a wonderful alogorithm for ANY pattern, which constitutes
an existence proof. Given that you know how to "solve cubes", you
can achieve a given
pattern if it is achievable simply by "solving" to that position,
which may in fact be faster than some set of arcane transforms.
Before the "neat" algorithms for the Cruces Plummeri et Christmani
were discovered, they were achieved by cubemeisters only by
"solving" to that position, lambda-binding the target state
(lessee, this guy wants to go here, etc.) to the desired pattern.
I will burn some neurocomputrons tonight to describe the
algorithms for the Crosses.
Incidentallly, if you try to solve for some pattern and come to
a roadblock of the form "this cant come here because its here"
(we need two of these cubies ( a local jargon
for the little cubes)), or a parity/trinity argument, you have
proven that you can't achieve it.