# Lost in Gnu York

Wanda the Wanderer is driving the streets of Gnu York. She doesn't know the what the entire map of Gnu York looks like, but she knows this much:

• Gnu York is flat
• Every intersection of streets is four-way, resembling a "+"
• There are no overpasses, tunnels, or dead ends
• There are no streets leading out of Gnu York
• There are only finitely many intersections

Whenever Wanda reaches an intersection, she decides whether to go left, right or straight according to the repeating pattern

Left $\to$ Right $\to$ Straight $\to$ Left $\to$ Right $\to$ Straight $\to\dots$

So, if she turned left before, she will turn right at the next intersection, then go straight at the one after that, then left at the one after that.

At noon, Wanda passes by city hall. Prove that she will eventually pass by city hall again.

Below is a an example of what Gnu York might look like, with $\color{red}{\star}$ being city hall, along with how Wanda's path would start out. However, your proof should work for any road layout which obeys the given rules, not just the one below.

• Do you want to add the condition that there is no way to drive in or out of Gnu York? Otherwise, a trivial map (say, with a single intersection +and the roads extending west to Filladelphia and south to Phlorida) shows she will not pass city hall again. Commented Sep 18, 2015 at 20:43