A quicker way to reverse NISS a solved cube

[campos20]

TL,DR;
sequence, SLICE, (sequence)'

If you know advanced techniques for Fewest Moves, you probably know reverse NISS. Reverse nissing a solved cube can be useful for finding slices insertions and maybe reduce an achieved solution by a few moves.

Here is an example

Scramble #3 from week 143 at the Fewest Moves Facebook group.

Scramble: R' U' F R2 L' U2 F2 D L F2 U2 B D2 L2 D' B2 U2 D' R' B R' U' F
Solution: F2 R F2 U2 B2 L R' D2 B2 L' B2 R D2 L F2 U' B2 U L' U2 L2 B' L R D2

It is a 25 moves solution.
Now, suppose you wanna try to reduce it. Here is a spot (the best spot for clarity)

F2 R F2 U2 B2 L R' D2 B2 L' B2 R D2 L F2 U' B2 U L' U2 L2 B' # L R D2
# M

The next step would be picking places to apply revese NISS, let's say at ##
F2 R F2 U2 B2 L R' ## D2 B2 L' B2 R D2 L F2 U' B2 U L' U2 L2 B' # L R D2

Usually, revese NISS would be like
## D2 B2 L' B2 R D2 L F2 U' B2 U L' U2 L2 B' M L R D2 [start at where you wanna check, don't forget the slice]
R' U' F R2 L' U2 F2 D L F2 U2 B D2 L2 D' B2 U2 D' R' B R' U' F
F2 R F2 U2 B2 L R' [the solution with the slice up to ##]

What I found is that we can actually start at ##, move to # (call this a sequence), apply the slice and invert the sequence
## D2 B2 L' B2 R D2 L F2 U' B2 U L' U2 L2 B' # [start at where you wanna check and move to #]
M [apply the slice]
B L2' U2' L U' B2' U F2' L' D2' R' B2' L B2' D2' [invert the first sequence]

The usual approach requires 50 moves



The new approach, which produces the same result, requires 32 moves




This also work backwards. Here is the final solution I found

Scramble: R' U' F R2 L' U2 F2 D L F2 U2 B D2 L2 D' B2 U2 D' R' B R' U' F

F2 R F2 U2 B2 L R' # D2 B2 L' B2 R D2 L F2 U' B2 U L' U2 L2 B' L R D2
# M

Now, we wanna check at ##
F2 R F2 U2 B2 L R' # D2 B2 L' B2 R D2 ## L F2 U' B2 U L' U2 L2 B' L R D2

Usual approach
## L F2 U' B2 U L' U2 L2 B' L R D2 [start at where we wanna check]
R' U' F R2 L' U2 F2 D L F2 U2 B D2 L2 D' B2 U2 D' R' B R' U' F [scramble]
F2 R F2 U2 B2 L R' M D2 B2 L' B2 R D2 [the solution up to untill we wanna check, with the slice]



New approach
## D2' R' B2' L B2' D2' # [start at where we wanna check, move to #. Since it's backwards, we are using inverse]
M [slice]
D2 B2 L' B2 R D2 [sequence again]



As we can see, we can insert M' at ## to solve the cube back again and reduce 1 move. It's reverse NISS with 14 moves instead of 50.

How I found this

When we already have a solution, it is just like the inverse scramble. In reverse NISS, instead of applying the scramble, you could invert our solution. Writing it down and cancelling stuff up, one can found why this works.

So, to reverse NISS at some spots with a solution for slices, we can perform
Sequence, Slice, (Sequence)'