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 donīt 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?
Let the three ladies be A, B, and C with A going first, B second, and C third. Each of A, B, and C see a total of 4 points and each realize that she must have a 2 or a 3. A knows that B will then see a total of either 4 points (A's 2 and C's 2) or 5 points (A's 3 and C's 2). But if A had a 3, B would see a total of 5 points (A's 3 and C's 2) and would know that she, B, could only have a 1 or a 2. But if she had a 1, A would have seen a total of 3 (A's 1 and C's 2 and know that she, A, could only have a 3. This did not happen, so B can not have a 1. Therefore, if B does not know what number she has on her first turn, A must have a 2. We can employ similar logic for C relative to B. C must have a 2 or 3, but if she had a 3, B would know that she, B, could only have a 1 or 2, but if B had a 1, A would have seen a total of 3 and realize that she, A, must have a 3. Therefore, A can determine her number on the second turn as can B. C does not know and can not determine whether her number is a 2 or a 3.
Gordon S
Solution
