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?
Call the women A, B, and C. The last woman to fail to know her number is C.
On the first round, all three women say, "I don't know my number." All three can say only that they are a 2 or a 3.
On the second round, A knows she is a 2.
If A were a 3, then C would have known she was a 2 on the first round, but C didn't know her number: C would have seen a 3 and a 2 and said, "I'm a 1 or a 2, but I can't be a 1 because then A would have seen a 1 (me) and a 2 (B) and would have known she was a 3 on the first round. She didn't know her number, so I must be a 2." That didn't happen, so A can't be a 3. A knows she is a 2.
B uses the same logic as A: "If I'm a 3, then C would have known she was a 2 on the first round (since I didn't see a 2 (A) and a 1 (C) and say I was a three). She didn't know her number, so I can't be a 3, and I know I'm a 2.
That leaves C as the last person not to name her number.