Suppose you live in a finite city with no roads going in or out. Every road intersection is a T-junction. You leave your house then proceed to go left, right, left, right, and so on indefinitely; do you ever get back home?
My Attempt . . .
I think the Pigeonhole Principle is relevant here. There's a potentially infinite route but only finitely many roads, meaning you're bound to use at least one road at least twice, either in the same direction or in the opposite direction, and in both cases, either left or right as your next turn.
Where do I go from here?
Please help :)
Consider this process as a sequence $S_n$ of "states", where the state is given by a directed road and the direction (left or right) of the next turn. This process is deterministic both in the future and past directions. Since there are only finitely many states, eventually you come to a state that has occurred before, say $S_n = S_k$, $k < n$. But then if $k > 1$, $S_{n-1} = S_{k-1}$. Conclude that the first state to occur more than once is the initial state, i.e. you do come back home.