Kill the vampire
There is a vampire who lives in a castle with four underground vaults arranged in a line. A vampire hunter is on the prowl and to avoid her, the vampire sleeps in a different vault every day. But he is bound by a magic spell: he can only choose a vault that is directly adjacent to the vault he slept in the previous day. The vampire hunter can go to one vault every day. If she finds the vampire, she'll kill him. If not, she'll have to wait another day.
Is there a sequence of vaults the hunter can choose to guarantee she'll eventually find the vampire? Can you find a strategy for the general problem with n rooms?