What is the largest natural n, such that for any scramble that can be solved using an n-move or less solution, that solution is the only optimal one?

[edward_9x]

EDIT: I'm gonna modify my question a bit: The solution doesn't have to be the ONLY optimal one, it just has to be optimal.

In other words, say I had an 8 move solution for a scramble (n=8). Can I be certain that this solution is optimal?
Can n be found using only a simple tree search starting from a solved cube?

 

[edward_9x]

Going through all possible 2-movers, obviously n>=2.
I also know for a fact that it can't be >=9, because the T right matching COLL case with the edges solved can be solved not only in 9 moves, but also 8.

 

[xyzzy]

U U = U2
U D = D U
U2 D2 L2 R2 = L2 R2 U2 D2
R U R' F' U2 = U2 F' L' U L
L R U2 L' R' U = U' L R U2 L' R'
R2 U2 R2 U2 R2 U2 = Rw2 U2 R2 U2 Rw2 U2 = Rw2 Uw2 R2 U2 Rw2 Uw2 = R2 Uw2 R2 U2 R2 Uw2 = etc.

 

[edward_9x]

U U = U2
U D = D U
U2 D2 L2 R2 = L2 R2 U2 D2
R U R' F' U2 = U2 F' L' U L
L R U2 L' R' U = U' L R U2 L' R'
R2 U2 R2 U2 R2 U2 = Rw2 U2 R2 U2 Rw2 U2 = Rw2 Uw2 R2 U2 Rw2 Uw2 = R2 Uw2 R2 U2 R2 Uw2 = etc.

So n is 1 got it

 

[edward_9x]

But what if the solution itself can be different? What is the largest n such that, for any scramble that can be solved in n moves, any optimal solution for that scramble is at most n moves?

 

[bcube]

So n is 1 got it

I might be stating the obvious, but it depends on the turn metric and scramble generator being used.

Assuming you mean 3x3x3, HTM and no scrambling conditions, it is 1 (U U = U2). For HTM and no two consecutive same moves (such as U2 U) it is 2 (U2 D2 U2 = D2).

I am wondering what it is for HTM and no two consecutive moves on the same axis (such as U D is not allowed in the scrambling sequence).

 

[Spencer131]

if you allow slice moves, there are 5 move scrambles that can be solved in 4 moves

M2 U2 M U2 M = F2 M' F2 M

 

[xyzzy]

L U2 D2 L2 R2 = L' R2 U2 D2

 

[Christopher Mowla]

Although I may sound like a bot by saying this, you may enjoy Pseudo 3x3 shorter than 3x3?.

 

[edward_9x]

L U2 D2 L2 R2 = L' R2 U2 D2

We're down to n=4 now. There are really only 3 options left:
n=2
n=3
n=4

 

[ichcubegerne]

The shortest non-trivial algs, that do nothing on the cube, are 10 moves long, hence the answer is 4.
(Non trivial means, that the alg isnt just something like R R' or a commuting variant of algs like R2 L2 U2 D2 R2 L2 U2 D2)
If you are searching for literally any sequence, then it immediatly becomes 2

 

[edward_9x]

Now, to make it more interesting: What if no double moves or consecutive moves of the same type (R R', R R) are allowed?