I have just moved to an island on which every intersection is the meeting of three two-way streets. On a lark I decide to go running, turning right at the first intersection, left at the next, and alternating in this way to decide my route. Prove that eventually I’ll return home.
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On any given street my state can be described by the street name, the direction I’m traveling on it, and the direction of my last turn (left or right). That means that each step determines that one that precedes it: If I’m running west on Primrose Lane having just turned right from Willow Avenue, then on the preceding step I was running on Willow Avenue toward Primrose Lane having just turned left, and so on backward. Since the island contains a finite number of such triples, I must eventually hit a triple for the second time. In most cases this implies a contradiction: Primrose Lane can’t be the first repeated triple, because I could only reach that triple via Willow Avenue, which would make Willow the first repeated triple. The only triple that doesn’t produce such a contradiction is the very first one … where my house is.
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