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# Which Chair you will choose ?

## Question

Prof. Sushi Rama gives out only one A+ each year in her course on Logic. This year, you and two other students got 100 on all the exams, and you REALLY want that A+. The professor says that she will give the 3 of you a test to determine the winner.

There are 3 chairs in her office. After lunch each of you will sit in one of the chairs, and then she will paste two stamps on your forehead. She shows you the 4 white and 4 black stamps that she has. You will be able to see the stamps on the other two students' foreheads, but not your own, or the two leftover stamps. Then she will ask the student in the first chair to tell what color stamps are on his or her forehead. If the student cannot logically deduce the colors, she will move on to the second chair, then the third chair. If that does not decide the issue, she will continue around the circle of chairs until one of you gives the correct response, with correct reasoning, based on the stamps that are visible and the other students' answers.

After lunch you are the first to arrive at the professor's office. Which chair do you choose and why?

Edited by ujjagrawal

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With this being a Logic class, perhaps we don't need to do any math at all...

Assuming that the teacher wants to be fair about the test, then the only fair way she could go about assigning the stamps is to put one of each color on each students head, that way each student will have the same information as the others. Therefore, the first chair is the choice to make so you can answer first, logical fairness dictates that your head must have a stamp of each color.

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With this being a Logic class, perhaps we don't need to do any math at all...

Assuming that the teacher wants to be fair about the test, then the only fair way she could go about assigning the stamps is to put one of each color on each students head, that way each student will have the same information as the others. Therefore, the first chair is the choice to make so you can answer first, logical fairness dictates that your head must have a stamp of each color.

The difficulty with this approach is that if the students do as the OP requires, the student who made your additional assumption of fairness would pick chair #2.

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I think though that for fairness the teacher would randomly assign the stamps. If she chose the order of stamps, she would be favouring a chair and therefore a student. For example, if she chose 1 black one and 1 white one for each student, then she would be favouring the student in the second chair.

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The difficulty with this approach is that if the students do as the OP requires, the student who made your additional assumption of fairness would pick chair #2.

But the student in the first chair gets to guess first, no? So, if I walk in with the fairness assumption, then I don't really even need to see what the other two students have on their heads in order to determine what mine are.

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But the student in the first chair gets to guess first, no? So, if I walk in with the fairness assumption, then I don't really even need to see what the other two students have on their heads in order to determine what mine are.

First, If the students do as the OP requires, they are not entitled to guess.

If your student who wants the first chair reasons, before stamps have been placed, that he will receive mixed color stamps, then his answer is not based on what he sees nor on the responses of the other students.

Furthermore his reasoning may be incorrect.

He reasons: this is a logic class, and the teacher is fair. But is it a test of logic not to present a rich set of possibilities from which, based on reasoning, a student determines a most favorable chair? And if you say that's what the student who reasons that fairness demands symmetry [and there is only one symmetric case] has done, then: [1] it's not much of a logic test, and [2] it is the teacher, not the students nor their logic, for if these premises are valid they all would want to have chair 1, who is determining the "winner", by allowing one student choose it. Fairness would not permit that.

Only if it is the case that it requires analysis of all possible distributions of stamps to disclose a favorable chair would it [1] be a fair test of logic and [2] be fair to let a student have the choice of where to sit.

Finally, the OP does not state the teacher is fair.

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