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Mutually friendly (Posted on 2002-07-01) Difficulty: 2 of 5
Prove that any group of six people contains either 3 mutual friends or 3 mutual strangers.

(For the purpose of this problem any pair of people must be either friends or strangers.)

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

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Solution Mutual friends Comment 11 of 11 |
Take any person in the group. Call them Joe. They are either friends with at least 3 people or strangers with at least 3 people. Suppose they are friends with at least 3 people. If any two of the three are friends, then those two and Joe are 3 mutual friends. If they are all strangers, then they are 3 mutual strangers. Either way, there is a group of mutual friends or strangers. A similar argument proves that if the first person is strangers with at least 3 people, then there is a group of mutual friends or strangers.

  Posted by Math Man on 2011-04-25 20:08:22
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