In Y-town all crossroads are Y-shaped, and there are no dead-end roads. Is it true that if you start from any point in the city and start walking along the roads, turning alternatingly left and right at each crossroad, eventually you will arrive at the same spot?

Yes, it is true. If you start walking forward, eventually you will end up in a loop. It is easy to see that your entire path, including the starting spot, must belong to this loop. Therefore, eventually you will end up in the starting spot again.

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  1. I am picturing a grid of hexagons. With a Y at each vertex. L-R-L-R will just wander in a zigzag. Am I picturing the pattern wrong?

    1. HI Daniel. Your honeycomb pattern indeed satisfies the condition. If you start from any place and keep switching between LEFT and RIGHT at the intersections, you will see that you will end up in your starting spot. This is true for all kinds of patterns with Y-intersections only.