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Bob's Hat (Posted on 2002-06-13) Difficulty: 4 of 5
Three friends, Alan, Bob and Cecil, sit down at a park bench. Each of them is a Knight or Liar and each knows what the other two are.

Forgetful Bob, well known for his flamboyant hats that are either red, white or blue, asks Alan: "What colour hat am I wearing today?"

Alan replies: "There is a chance Cecil would tell you it's blue. If you saw it, there is a chance you would say it is white."

Bob sighs, "I should have asked Cecil - he always tells me the truth..."

What colour hat is Bob wearing?

See The Solution Submitted by Nick Reed    
Rating: 3.6667 (15 votes)

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Solution Bob's hat Comment 11 of 11 |
Suppose Alan and Bob are both knights. Then, Bob's statement is true, so Cecil is a knight. Since Alan is a knight, Cecil would say that Bob's hat is blue. Therefore, it is blue. However, Bob would not say that his hat is white. That is impossible, so Alan and Bob are not both knights.

Suppose Alan is a knight and Bob is a liar. Then, Bob's statement is false, so Cecil is a liar. Since Alan is a knight, Cecil would say that Bob's hat is blue, so it is not blue. Also, Bob would say that his hat is white, so it is not white. That makes it red. Therefore, if Alan is a knight and Bob is a liar, then Bob's hat is red.

Suppose Alan is a liar and Bob is a knight. Then, Cecil is a knight. Since Alan is a knight, Cecil would not say that Bob's hat is blue. Cecil is a knight, so it is not blue. Also, Bob would not say that his hat is white. Since Bob is a knight, it is not white, so it is red. Therefore, if Alan is a liar and Bob is a knight, then Bob's hat is red.

Suppose Alan and Bob are both liars. Then, Cecil is a liar. Since Alan is a liar, Cecil would not say that Bob's hat is blue. Cecil is a liar, so it is blue. However, Bob would say that his hat is white. That is impossible, so Alan and Bob are not both liars.

Answer:Bob's hat is red.


  Posted by Math Man on 2013-05-01 14:03:11
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