Three women were seated around a table. After blindfolded, a numbered disc was pasted on each of their foreheads. The women were truthfully told "Each of you has either a 1, a 2 or a 3 on your forehead, and the sum of your numbers is either 6 or 7".

After the blindfolds were removed, each woman in turn was asked to name the number on her forehead without seeing it. The question was repeated until only one woman failed to name the number on her forehead.

When it was logically possible to name the number on her forehead, each woman did so; when it was not logically possible to name the number on her forehead, she would said "I don´t know my number", and waited until the question was repeated to her next time around.

Each woman had a 2 on her forehead. Which woman failed to name her number?

Woman 1: Sees (2) (2) and knows that she can have either a (2) or (3) and answers that she doesn’t know

Woman 2: Sees (2) (2) and knows that she can have either a (2) or (3), a knows that if  W1 had seen a (3), she (W1) would have been left with either a (1) or (2), and if she had seen a (2), she would have been left with (2) or (3), compounding the fact that she doesn’t know either

Woman 3: Sees (2) (2) and knows that she can have either a (2) or (3), a knows that if  W1 or W2 had seen a (3), they would have been left with either a (1) or (2), and if they had seen a (2), they would have been left with (2) or (3), compounding the fact that she, also, doesn’t know either

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Because all three women know that they can have only a (2) or (3), but also know that with either option, the other women still have two choices, therefore none of the women ever “know” what number they have…

Posted by Nick on 2005-12-16 14:31:01