The Guild of Parity Milliners is a cult where members wear blue hats and red hats, and adhere to the code that no two people who know each other can wear the same colored hat.

M, a member of this cult, invites T, who is not part of the cult, to join the cult.

a) Assume T does not know anyone other than M in the cult. T wants to wear only a red hat. Prove that after T joins, there is a way for the group to maintain its code, by possibly changing hats for members.

b) Assume T knows two people in the cult, M and S (and she has no preference of hat color). But M does not have even an indirect connection to S--- i.e., M doesn't know someone who knows someone who knows someone ... who knows S. Prove again that after T joins, there is a way for the group to maintains its code. (T has no hat color preference (unlike part (a)).)

You can assume the club has enough hats of either color. You can assume that "knowing" is symmetric--- if A knows B, then B knows A too.