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?

[Sorry for strange characters from copy & paste in my last post - I'll try and get it right here]

The women are named A, B and C reflecting the order in which they answer. The situation is described (2,2,2) implying that A, B and C are respectively wearing the number 2 on their foreheads. A1 labels A’s 1st answer.

Each woman knows that she is wearing 2 or 3 (i.e. not 1)

Answers:

A1 – DK (don’t know)

B1 - DK

now A knows that she is wearing 2.

Why?

Consider A’s only alternative situation (3,2,2): A's answer, seeing B and C both wearing 2, is DK. However this answer informs B that she (B) is wearing 2. B can't be wearing 3 (because the sum of the numbers would then be 8) and she can't be wearing 1 (because A would have seen B:1 & C:2 and would have known that she was wearing 3).

C1 - DK

Why?

Consider C’s only alternative situation (2,2,3): A’s answer, seeing (B:2, C:3) is DK (she could be wearing 1 or 2). B’s answer is then also DK, for A would have answered the same had B been wearing either 1 or 2.

Thus after hearing A&B both answer DK, and seeing them both wearing 2, C is unable to distinguish between situations (2,2,2) and (2,2,3).

A2: I’m wearing 2 (for explanation see above)

B2: I’m wearing 2

How does B know she’s wearing 2?

Consider B’s only alternative situation (2,3,2): A’s 1^st answer (seeing 3&2) would have been DK. B seeing 2&2 answers DK. B’s answer tells A&C that they’re both wearing 2. (Had either A or C been wearing 1, B would have had to have answered 3.) Thus C’s first answer (C1) would have been that she was wearing 2 and not DK.

The solution: C is the woman last to know her number

Posted by Gadsden on 2005-12-15 19:41:42