MathDB

There are

12

mathematicians living in a village, each of whom belongs to the

\sqrt2

-clan or belong to the

\pi

-clan. Moreover every mathematician's birthday is in a different month and every mathematician has an odd number of friends among them the mathematicians. We agree that if mathematician

A

is a friend of mathematician

B

, then so is

B

is a friend of

A

. On his birthday, every mathematician looks at which clan the majority of his friends belong to, and decides to join that clan until his next birthday. Prove that the mathematicians no longer change clans after a certain point.

Problem Statement