Queen Shayra does not have the most impressive following, but she intends to expand her influence. If she could settle down at the center of an N Dimensional 3-Cube and build a palace there, she would attract all sorts of interesting folk to her realm. Of course, before she may build a palace, she needs to make sure that the N Dimensional 3-Cube is fit for colonization (no rats, etc).
For this purpose she will send her Brave Knight on a journey over all 1-Cubes of a realm, except for the center, which is easy enough to deal with during the building process as it is only one cube. Brave Knight moves from cube to cube in chess knight jumps (two over in one direction, one in another), and he may not pass through a cube that he has been to already because Shayra is an optimization enthusiast.
For which $N$ is there a $3\times\dots\times3$ cube with such a path? Whenever for some $N$ such a path exists, construct it.