While @2012rcampion is technically correct I will add what I think is a more naively implementable answer that uses both more face turns and more sequences (although still not all that easy to perform):
$2$ sequences totalling $14$ face turns
Firstly one $8$ move sequence to permute three corners for which we ignore the effect on edges, such as:
U' R U' F' R2 F U2 F'
Secondly one $6$ move sequence to permute three edges which does not affect the corners whatsoever, such as:
R2 D' U F2 U' D
Now we can:
Apply the first move in different cube orientations until all the corners are placed correctly (Note: most could probably place the first four corners without learning any algorithms)
Apply the second move sequence in different orientations, possibly prefixing and postfixing with any necessary turns to align the three you wish to permute and undo that change (that is conjugation), until the cube is solved.
Assuming we cannot perform any conjugations:
3 sequences totalling $23$ face turns
Do the same as before for the corners
For the edges also learn a $9$ move sequence to permute three edges on a side rather than in a slice, such as:
F2 D L' R F2 L R' D F2
(Note this is really the result of a conjugation: we move the three into a slice, permute with a reorientation of the $6$ move sequence from before, and then move them back out of the slice, it is also set up in such a way as to cancel a face turn).
If one needs to permute, for example, three edges neither in a single face nor in a single slice one can permute either one unsolved and two solved or two unsolved and one solved to get to a set of states one can directly can solve).
Update: ais523 asked "How does this handle orienting the edges and corners once they're placed?"
This is, indeed, not immediately obvious from the above - the answer is also not all that enlightening on its own - "We can do two things: 1. turn the cube; 2. move those already placed edges".
As such I will run through an example of a typical position one would normally think of as "there are two edges that need orienting" on the last layer.
You can see it in this jsfiddle (Using the work of Lars Petrus's Roofpig. Each time the cube is reoriented it is being set up for one or two sequential edge-permutations using the conjugation-included form, F2 D L' R F2 L R' D F2
. I have left in two D F2 F2 D
sequences that would, of course, just become D2
for clarity.
Here is a rundown of that:
First one may setup the "scramble" like this: B' L' B2 U2 F2 R' F2 U2 B'
(or if you prefer less faces L' R F R' L D2 L' R F R'
). This should leave you with U
and F
fully solved, all the corners in the correct locations, while the remaining three edges on D
are incorrect, where one would identify the ones adjoining B
and L
as "flipped".
Now the (uber-convoluted!) solve using only (1) cube-orientations (x, y, z are turns of the whole cube clockwise if one were to look at R, U, or F respectively) and (2) the edge-permutation above. Steps 1-6 permute edges in such a way that we are left with two faces each requiring only edge permutations, steps 7-10 then solves those two faces:
orient the cube such that R->U
& D->F
: y x
apply face 3-edge: F2 D L' R F2 L R' D F2
orient the cube such that R->F
& D->U
: x2 y
apply face 3-edge twice (or reflect the sequence & perform it once): F2 D L' R F2 L R' (D F2 F2 D) L' R F2 L R' D F2
orient the cube such that R->F
& D->R
: y z'
apply face 3-edge: F2 D L' R F2 L R' D F2
orient the cube such that R->U
& D->R
: z'
apply face 3-edge twice (or reflect the sequence & perform it once): F2 D L' R F2 L R' (D F2 F2 D) L' R F2 L R' D F2
orient the cube such that R->F
& D->U
: x2 y
apply face 3-edge: F2 D L' R F2 L R' D F2