The Guild of Parity Milliners is a cult where members wear blue hats and red hat

Question

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.

Maricha, a member of this cult, invites Thelma, who is not part of the cult, to join the cult.

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

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

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.

Require proper proof

Explanation / Answer

It would appear I am not smart enough to work at Google. Because I couldn't see it. I had considered the prisoners splitting off in to pairs and agreeing that the back one would call out the colour the front one was wearing. Giving us an expected survival rate of 75%. But we can do much better...

The person at the back, lets call him person number 1, will agree that if he sees an odd number of red hats he calls out red, if he sees an even number of red hats he calls out blue. (Obviously any similar scheme would work, but we will use this one.) Unfortunately this means he has a 50% chance of survival but it guarantees everyone else's.

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