65. Three Students and the Colored Caps Three students — A, B, and C — are seated in a straight line, all facing the blackboard at the front of the room. A sits closest to the board and cannot see anyone. B sits behind A and can see A. C sits behind B and can see both B and A. The instructor has five caps: two red and three blue. Without the students seeing which cap is chosen for whom, the instructor places one cap on each student’s head and then returns to the front of the room. One at a time, starting with C and moving forward, the instructor asks each student, “Without touching your cap or turning around, can you tell me the colour of your own cap?” The replies are: C: “I can’t tell.” B: “I can’t tell, either.” A: “Now I know the colour of my cap.” What colour is A’s cap, and how can A be sure? Added 19 June 2009 Show Hint Show Solution Hint: Ask what C would have seen if both A and B had been wearing red caps. Solution: The colour of A’s cap is blue. Reasoning step by step: If A and B had both been wearing red caps, C would have seen two red caps. With only two red caps in total, C would then know his own cap must be blue and could answer immediately. Since C couldn’t answer, A and B cannot both be red. B hears C’s statement and looks at A. If B sees a red cap on A and knows C did not answer, B can reason as follows: “If my cap were red as well, C would have seen the two red caps and would have known his cap is blue. Because C did not know, my cap cannot be red.” Therefore, had B seen A wearing red, B would have been able to deduce that his own cap is blue. But B still could not answer, so A cannot be wearing red. Since A is not wearing red, A must be wearing blue. Thus A confidently answers “blue.” (A blue, B red, C blue is one distribution consistent with all statements, and it is the only possible distribution given the deductions.)
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