I'll keep notes here about some of those classic puzzles which
require you to move rings and ropes around to get something free.
Note: mathematical these can be modeled as follows. Hold one
piece steady and parameterize the motion of each other piece in terms
of center of mass and orientation (6 variables per piece). The puzzles
simply require finding a path between two points in 6N-dimensional space.
I suppose there is an aesthetic mathematical question even after it
is proved that the points are topologically connected: we might for
example try to find the shortest path or the path which stays furthest
from the boundary. (Not entirely sure what a good metric in on this space...)
A five-ring puzzle