Red Hat, Blue Hat
Date: 11/16/2002 at 19:24:48 From: Ken Subject: IMP Mathematics My math teacher tells us that he has gotten an answer to this math problem, but he always tells us that an answer is worth nothing unless you can prove it. That is our teacher's problem - he can not prove it, therefore I don't think that the answer is correct. The question comes from the Math Lessons of the Integrated Mathematics Program. There are 3 children, A, B, and C. The children tell their teacher they think the homework they do is just busy work. So the teacher says they will get one chance to get an A for the semester without having to do any more homework. All they have to do is to answer one of his math problems right. The problem is that he has 5 hats, 3 red and 2 blue. The children are told to stand in line with their eyes blindfolded. The teacher puts one hat on each of their heads and then discards the remaining 2 hats so they cannot be seen. Then the first child (child A) is told he can look at the other two children and, judging by the color of their hats, he can guess the color of the hat he wears. He can either guess or pass. (If they pass, there will be no effect on their homework, if they guess right there will be no more homework, and if they guess wrong they have to do all their homework plus grade papers after school. So the children will only guess if they are 100% positive they are correct.) The guess and pass process is carried through with the rest of the children. The main problem is that in one particular situation student A passes, Student B passes, and then without even opening his eyes Student C guess correctly what hat is on his head. I dont understand how he knows without even looking... Thank you, Ken
Date: 11/18/2002 at 11:44:27 From: Doctor Roy Subject: Re: IMP Mathematics Hi, Thanks for writing to Dr. Math. Student C must be wearing a red hat. Let's think about it from the beginning. Start with student A. Student A doesn't know which hat he has on. That must mean at least one (so both B and C or just one of B and C) must be wearing a red hat. If both B and C are wearing blue hats, A must have a red hat. So, one or both of B and C is wearing a red hat. But A could be wearing a red or blue hat in this case, so A doesn't know. So, let's go to student B. Student B, being very clever in realizing what student A is thinking, realizes that one or both of himself and C is wearing a red hat. If student B sees a blue hat on student C, then student B must be wearing a red hat. But this is impossible, since student B does not know which hat he has. That means that student B MUST see a red hat on student C. So, that leaves the possibility that student B has a blue or a red hat. That leaves student C. Student C, also being quite clever, realizes the situation, and realizes that he must have a red hat, since neither student A nor student B could deduce the colors of their own hats. That's the solution to the problem. There is no way that student C could have a blue hat and produce the same results. If student B also had a blue hat, student A would know he had a red hat. Or if student B had a red hat, student A would not know his own hat color, but student B would know that since C had a blue hat, B must have a red hat. I hope this helps. - Doctor Roy, The Math Forum http://mathforum.org/dr.math/
Date: 11/28/2002 at 14:51:53 From: Ken Subject: Thank you (IMP Mathamatics) Thank you Doctor Roy, you are a genius. Your explanation was much more clear than that of my teacher. Thank you again... if I ever have another math question I know exactly whom to ask. Thanks Ken
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