There's a little play by Samuel Beckett called 'Quad' that consists of nothing else than 4 characters walking on and off stage, one at a time. No two actors leave/enter simultaneously, so only one such change happens at any one time. In the course of the play, every possible combination of actors is on stage (16 distinct configurations).
Here is the play in binary, with each actor represented by their own digit (1 means onstage, 0 means offstage): 0000 0001 0011 0010 0110 0111 0101 0100 1100 1101 1111 1110 1010 1011 1001 1000
There is no reason why the 16 values need to appear in this order. For each number there are four others that could be visited, e.g. 1011 -> 0011, 1111, 1001, or 1010
(1) If we always begin with an empty stage, how many different 'paths' can be taken through the 16 configurations? (2) Is it possible to start such a path and end up at a 'dead end', e.g., a situation where you attain 1011 but all of 0011, 1111, 1001, and 1010 have already been visited? (3) How can this be generalized for n actors?
I know this is a solved problem, and probably pretty trivial, but I don't know where to begin researching it.