Proof that Out of 6 Persons Either 3 are Friends or 3 are Strangers

Did you know that out of 6 people, either 3 are friends or 3 are strangers? Three are friends in the question means mutual friends: A and B are friends, B and C are friends, and A and C are friends.  Three are strangers also mean that out of three persons, none of whom know each other.

In the mathematical proof below, we represent the individuals by vertices and the relationships by edges. As we can see, we can form triangles. In the triangles below, a green edge connecting two vertices means that the persons represented by two vertices are friends and a red edge connecting two vertices means that the persons represented by two vertices are strangers. A triangle formed by edges of the same color represents three mutual friends or three mutual strangers. We highlight this relationship by filling the interior of the triangle with the appropriate color. » Read more

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