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Math Help - 12 bastions on a wall of a fortress

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    12 bastions on a wall of a fortress

    There are 12 bastions on a wall of a fortress, and a watchman in each bastion. You don’t know neither the shape of the wall nor the distribution of the bastions on the wall. At midnight, every watchman starts from his bastion in a direction on the wall. Every watchman proceeds at constant speed, so that he would walk around the wall in 1 hour. When two watchmen meet, both turn back. They go through a bastion without any delay. Prove that at noon, every watchman arrives to his original bastion.
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    Quote Originally Posted by james_bond View Post
    There are 12 bastions on a wall of a fortress, and a watchman in each bastion. You don’t know neither the shape of the wall nor the distribution of the bastions on the wall. At midnight, every watchman starts from his bastion in a direction on the wall. Every watchman proceeds at constant speed, so that he would walk around the wall in 1 hour. When two watchmen meet, both turn back. They go through a bastion without any delay. Prove that at noon, every watchman arrives to his original bastion.
    That's a nice one, I liked it! If anyone wants to give it a try, please don't read the following spoiler...

    The nice step is to see that, after one hour, the watchmen are in bastions (but not necessarily their own) and that their directions are the same as the directions of the watchmen who were occupying these bastions at the beginning. This is because when two watchmen meet they go on "like they had exchanged their identities". After one hour an undisturbed watchman goes back to his own bastion. Up to several identity switches, this stays true in the general situation.

    The relative order of the watchmen around the wall doesn't change. That's why after one hour their positions were just "rotated" by some number of bastions. Plus, the directions remain the same, so that after another hour, the same rotation will have been applied to the positions. Finally, repeating 12 times the same rotation (acting on 12 elements) gives the identity back.
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