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(part a and part b are different questions)
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.
Expert Answer
Answer:
person 1 –
- Follows the only rule ‘if the first person sees an odd numeral of red hat he calls absent red, if he see an even number of crimson hats he call out blue.’ he call out blue since he sees an even number of crimson hats in addition to dies.
Person 2
- Know that including him there is an even number of crimson hats. He looks forward and can also observe an even number of red hat. This means he is tiring a blue hat.
- Had he been exhausting a red hat after that the person behind him would have see 3, an odd numeral, which he knows not to be the case.
Person 3
- Know that to start readily available be an even number of red hat. He knows so as to none of them contain gone, that is to articulate nobody previous than perhaps the first being has declared themselves in the direction of be wearing a red hat, so counting himself readily available be an even number of crimson hats.
- He looks presumptuous and can observe one red hat, an out of the ordinary number. That this has distorted between him and the persons in front of him income he is exhausting a red hat.
Person 4
- Know that to start readily available were an uneven number of red hats. He knows one has gone. So as well as himself there is an odd figure of red hats. He looks arrogant and can zero (an even number,) of red hats. That this has changed between counting and excluding him means he is wearing a red hat.
Person 5
- Knows that in the direction of start there was an even numeral of red hats. He knows that two have gone. So including himself readily available is an even number of red hats.
- He looks forward and can zero (an even number,) of red hats. That this has not changed sandwiched between including an exclusive of him means he is wearing a azure hat.
