[saiya-jin]

I do similar mental exercise with higher dimensions, but for me with easier situation - escaping from certain area without crossing boundaries.

1 dimensional - an endless line with 2 spots marking boundaries to some subpart X. If you're inside, you need to enter 2nd dimension to escape that subpart X without crossing boundaries (and ie come back to original dimension).

2 dimensional - circle, you're inside, you need to enter 3rd dimension to escape without touching the circle.

3 dimensional - sphere, entering 4th dimension to escape without touching the sphere.

I can't go with my simpleton mind much further but it should scale indefinitely :)