perplexus.info :: Logic : Three women, three numbers

Three women, three numbers (Posted on 2005-12-15)

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?

See The Solution Submitted by pcbouhid

Rating: 3.3750 (8 votes)

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An Answer

| Comment 2 of 28 |

There are the following possible permutations which yield a total of 6 or 7

{{1,2,3},{1,3,2},{1,3,3},{2,1,3},{2,2,2},{2,2,3},{2,3,1},{2,3,2},{3,1,2},{3,1,3},{3,2,1},{3,2,2},{3,3,1}}

If w1 sees {3,3} then she must be 1;

If w1 sees {2,1} then she must be 3;

When w1 answers negatively (since {2,2} is ambiguous), these are excluded and the remaining possibles are

{{1,2,3},{1,3,2},{2,1,3},{2,2,2},{2,2,3},{2,3,1},{2,3,2},{3,1,3},{3,2,2},{3,3,1}}

If w2 sees {3,3} then she must be 1;

If w2 sees {2,1} then she must be 3;

When w2 answers negatively (since {2,2} is ambiguous), these are excluded and the remaining possibles are

{{1,2,3},{2,1,3},{2,2,2},{2,2,3},{2,3,2},{3,2,2},{3,3,1}}

If w3 sees {3,3} then she must be 1;

If w3 sees {2,1} then she must be 3;

When w3 answers negatively (since {2,2} is ambiguous), these are excluded and the remaining possibles are

{{2,2,2},{2,2,3},{2,3,2},{3,2,2}}

Now if w1 sees a 3, then she must be a 2.

When w1 answers negatively (since {2,2} is ambiguous), both w2 and w3 knows that they have a 2.

Posted by goFish on 2005-12-15 10:39:43

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