Let's look at different examples, with different underlying graphs (cage graphs).
Every animal has to eventually transfer to its cage. Therefore, if the cage graph is not connected, if there are two vertices without a walk between them, then the corresponding puzzle might be insolvable (depending on the initial and target state, of course).
More examples of cage graphs which may have no solution are shown below:
| If the graph has a so-called "cut vertex"--- a vertex x where there are two other vertices y and z such that every walk from y to z must meet x--- then the animals cannot be moved beyond this cut vertex. Try it out in the applet below. | For cycles like the one shown below, the cyclic ordering of the animals along the cycle remains unchanged. So again, often such puzzles cannot be solved. |