I know that in a 3x3x3 this should never happen (unsolvable state), but is the same true in a 4x4x4?
(the rest of the cube is solved)
The following answer seems to say that everything about the 3x3x3 is true about NxNxN, except that the orientation isn’t fixed.
Why is a single-corner twist not a valid position on a Rubik's cube?
Does that mean that my orientation is wrong?
The cube came solved, and since this isn’t a speed cube, I don’t see how I could have accidentally twisted a corner.
It is impossible.
If you ignore all pieces but the corners, it is equivalent to a 2x2x2 cube. And it is impossible to turn one corner of a 2x2x2.
This is valid for any size, i.e. every NxNxN cube.
Credit goes to william122 who gave this answer first.
Yes. It is impossible, because a 4x4 can be reduced to a 3x3, via reduction, and a corner would be exactly the same as a 3x3 corner.
It is impossible, but I'll try to explain a way to help understand.
Along each of the three axes of rotation there are four layers (I'm not sure if there's a more technical term) which can be freely rotated around their respective axis. Imagine shifting the centre gap so that the second layer becomes thicker and the third layer becomes thinner, like this:
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
| | | | | | | | | | | | | | |
|___|___|___|___| |___|___|___|___| |___|___|___|___|
| | | | | | | | | | | | | | |
|___|___|___|___| -> | | | | | -> | | | | |
| | | | | |___|___|___|___| | | | | |
|___|___|___|___| |___|___|___|___| |___|___|___|___|
| | | | | | | | | | | | | | |
|___|___|___|___| |___|___|___|___| |___|___|___|___|
Imagine doing this 'all the way' so that layer 3 becomes impossibly thin. Now it is functionally identical to a 3 x 3 cube, and we have not removed any functionality from the initial 4 x 4 cube.
This demonstrates that the middle layers do not affect the corners at all.
Hope this helps.
