<Blue hats and Red hats> The Guild of Parity Milliners is a cult where members w

Question

<Blue hats and Red hats>

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.

Explanation / Answer

Person 1 - Follows the only rule 'if the first person sees an odd number of red hats he calls out red, if he sees an even number of red hats he calls out blue.' he calls out blue because he sees an even number of red hats and dies. Person 2 - Knows that including himself there are an even number of red hats. He looks forward and can also see an even number of red hats. This means he is wearing a blue hat. Had he been wearing a red hat then the person behind him would have seen 3, an odd number, which he knows not to be the case. Person 3 - Knows that to start there were an even number of red hats. He knows that non of them have gone, that is to say nobody other than possibly the first person has declared themselves to be wearing a red hat, so including himself there are an even number of red hats. He looks forward and can see one red hat, an odd number. That this has changed between him and the people in front of him means he is wearing a red hat. Person 4 - Knows that to start there were an even number of red hats. He knows one has gone. So including himself there is an odd number of red hats. He looks forward and can zero (an even number,) of red hats. That this has changed between including and excluding him means he is wearing a red hat. Person 5 - Knows that to start there were an even number of red hats. He knows that two have gone. So including himself there is an even number of redhats. He looks forward and can zero (an even number,) of red hats. That this has not changed between including and excluding him means he is wearing a blue hat.

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