Solving $3\times 3\times 3$ Rubik's cube positions using a $4\times 4\times 4$ cube? Someone hands you a $4\times 4\times 4$ cube for solving. It is scrambled as if it were a $3\times 3\times 3$, i.e. without ever separating the $4$ central pieces of each face (which implies that edge pairs remain together too).
If this restriction is lifted now, is it possible that you might be able to solve this cube in fewer moves than you would a regular $3\times 3\times 3$ by utilizing the extra moves that $4\times 4\times 4$ provides?
Equivalently, are there any valid $3\times 3\times 3$ cube positions that are reachable from solved state in fewer moves on a $4\times 4\times 4$?
 A: The answer is YES.  There are some "Pseudo $3\times 3\times 3$" $4\times 4\times 4$ positions which, although they can be exclusively solved by turning outer layer faces only (e.g., using a $3\times 3\times 3$ solver), if one applies certain $4\times 4\times 4$ move sequences (which consist of both inner and outer slices), the resulting solution will be fewer moves than optimal solutions retrieved from a $3\times 3\times 3$ solver.
This is a "trivial position" found by Stefan Pochmann in the speedsolving.com thread, Pseudo $3\times 3$ shorter than $3\times 3$?.
As you can see, I was a participant in that thread as well, and I found the following more complicated position.  (I was motivated to find what may be considered "non-trivial" positions of this type -- to show that Stefan's wasn't the only $4\times 4\times 4$ position like this.) 

Uw2 2R2 Uw2 Rw2 U2 Rw' D' U' R2 D' U' s2 Rw (15 htm, 13 stm)

versus

z2 x' B2 L U D R2 B e' m' B' U' D' L' s' U2 (17 htm, 14 stm)

(You may verify that 17 htm is move optimal from any move optimal 3x3x3 computer solver you wish.)
In fact, based on the eight short PLL parity fixes that I listed in the last post of that thread (I eventually found four more and put all $12$ in this section of the $4\times 4\times 4$ parity algorithms speedsolving wiki page), we can make dozens of such examples by combining two short PLL parity move sequences together.
I never explored the topic further to know if there is another method for finding such positions; and therefore, I cannot claim that my method is the only method for finding them.
