[1] Start with the edge pairing algorithm at the top of
this page on bigcubes.com
Rw' F R F' Rw
What does this algorithm do? It swaps 3 wing edges and does 3x3x3 scrambling. Despite how it scrambles the cube, we can still use it
without modifying it.
We can use it if we position the 3 wing edges
like so (see the applet, but ignore the algorithm)
The two wing edges in the top layer are already correctly positioned, but the one in the R slice needs to be moved in the bottom face in the back on the left side. Hence,
[2] Add R D
Rw' F R F' Rw
R D
[3] Now, everyone should be familiar with the algorithm
r U2 r', as we use it when we insert a center row into a center.
What most people don't bother to learn about this brief algorithm piece is that it also
1) Swaps the two wing edges in the top front edge
2) Swaps the wing edge in the r slice in the bottom front with the wing edge in the l slice in the top back.
Using this knowledge, the idea is to involve all 3 unsolved wing edges we have in that
formation with r U2 r'. Since r U2 r' involves 4 wing edge pieces and we only have 3 messed up, if we are not careful, we could easily mess up more than 3 edge pieces with r U2 r' (which we don't want to do: we want to keep it 3).
Hence, we need to use r U2 r' to:
1) Solve back one of the 3 wing edges
2) Pull another wing edge, which was originally unseparated from its composite edge, out of place.
[3] We add y2 r U2 r' y2 AND U2 (just try me about the last U2)
Rw' F R F' Rw
R D
y2 r U2 r' y2 U2
But why do this and why use r U2 r'? As we already know, r U2 r' does affect center rows. In fact, it affects 3 center rows as well as the 4 wing edges. By using it, we will create 3 unsolved 1x(n-1) blocks consisting of a wing edge and center row each. This is the main idea for common algorithms I derive in my thread and YouTube videos.
Once we have 3 1x(n-1) blocks, which can be anywhere on the cube, we need to move all of them into either the l or r slices so that we can do a quarter turn to solve back two of the three 1x(n-1) blocks and pull another 1x(n-1) block (all done within the slice we do the quarter turn in) out of place. The net result is two 1x(n-1) blocks will be out of place.
Since two of the three 1x(n-1) blocks are in slice l (on the 7x7x7, both have white and blue wings: the 1x(n-1) in the top layer on the left has a blue-and-white edge with a blue center row and the 1x(n-1) in slice l in the back (touching the previous one just mentioned) has a white center row. The third 1x(n-1) block is in slice r in the top. It has a red-and-green wing edge and a white center row.
[4] Add r2 D2 to put the third 1x(n-1) block next to the other two in slice l (so that all blocks are TOUCHING!).
Rw' F R F' Rw
R D
y2 r U2 r' y2 U2
r2 D2
[5] Do a quarter turn to accomplish the process previously described. In this case, we must do l' as opposed to l.
Rw' F R F' Rw
R D
y2 r U2 r' y2 U2
r2 D2
l'
[6] Reverse r2 D2 with D2 r2
Rw' F R F' Rw
R D
y2 r U2 r' y2 U2
r2 D2
l'
D2 r2
Now notice that we have a single edge parity algorithm, but it also discolors two 1x(n-2) center rows. In order to fix them, we have to add some set up moves on the very outside of the algorithm. The set up moves we need to do are l' U2 l. However, we must first restore the 4 edges in
M first.
[7] We add D' F R F' to bring the original top front composite edge back into its original location so that we have all 4 composite edges in M restored:
Rw' F R F' Rw
R D
y2 r U2 r' y2 U2
r2 D2 l' D2 r2
D' F R F'
[8] Finally, we can add in the set up moves l' U2 l and invert them at the end (inverse of l' U2 l = l' U2 l), so we have:
l' U2 l
Rw' F R F' Rw
R D
y2 r U2 r' y2 U2
r2 D2 l' D2 r2
D' F R F'
l' U2 l
So this is a way to make a single edge parity algorithm an alternate way than using (r U2)5, but not the same way as with commonly practiced algorithms.
Still, I'm glad I had the chance to find it by messing around and not by using a tutorial.
