Trianglia is a jacked-up island where no road has a dead end, and all the crossroads are "Y" shaped.
The young prince of Trianglia mounts his horse, and is about to go on a quest to explore the land of Trianglia. He gets to the road by his palace, when the mother queen comes out and shouts:
"But Charles, how will you find your way back?".
"Don't worry Elizabeth", the prince replies, "I will turn right in every second crossroad to which I arrive, and left otherwise. Thus I shall surely return to the palace sooner or later."
Is the prince right?
Clarifications: I'm assuming that the roads are arbitrarily crooked, but that they invariably form three-legged junctions. I'm even willing to admit non-planar graphs (maybe one road crosses over a bridge over another road).
Charles will always eventually get back to his staring place. Proof is as follows:
Model the situation with a state machine. At any instant, Charles is on a particular road, headed in a particular direction, and planning to turn either left or right at the next intersection. Thus, there are (number of roads) times 4 states. (The first times 2 is for which direction on the road, and the second times 2 is for which direction the next turn will take.)
Charles's next state is a function of nothing but his current state. Since there are a finite number of states, Charles will eventually enter some state for the second time.
However, the state transition function is fully invertable: Charles's previous state can be uniquely determined from his present state.
So we know that every state has a successor, there are a finite number of states, and no two distinct states have the same successor. The only way this can be is if each state is part of a simple loop.
Therefore, Charles will always return to his starting point.
Posted by Jim Lyon
on 2002-09-11 19:52:53