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Hat Exchange (Posted on 2005-02-24) Difficulty: 3 of 5
At a party with N people, each person is identified by a unique number between 1 and N. Each person is also wearing a hat with their number on it. They decide to play a game. Each person passes his or her hat to the person with the next highest number (so person 2 gets hat 1, person 3 gets hat 2, and person 1 gets hat N). The game proceeds from there in rounds; on each round, each person may choose to either keep the hat they currently have, or swap hats with exactly one other person.

What is the minimum number of rounds, as a function of N, that it would take for every person to get their own hat back? Note: each person can see their own hat at any time.

See The Solution Submitted by Avin    
Rating: 3.0000 (5 votes)

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Some Thoughts Maximum | Comment 2 of 8 |
As in each round, by doing swaps, half the "wrong-hat" people can manage to get their hats back, a maximum is the base 2 logarithm of N, rounded up.
  Posted by Federico Kereki on 2005-02-24 15:34:05
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