In 1973, Richard Michael Wilson showed a theorem from which there follows:
Theorem: [R.M. Wilson 1973]: Assume our underlying graph G (also called cage graph) is connected, not a cycle, and does not have a cut vertex.
In particular there follows that, under the assumptions on G, in the bipartite case 50% of the initial states can be transformed into the target state, and that in the non-bipartite case every such puzzle can be solved.