### Brain Teasers

# Numbers on Heads

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Queenie, Rachel, and Sara, all perfect logicians, were seated around a table and blindfolded. Then they each got a number taped to their forehead. The three women knew that each number was either 1, 2, or 3, and that the three numbers added up to either 6 or 7. After the blindfolds were removed, each woman was asked whether she knew the number on her head, and if so, what it was. Queenie was asked first, then Rachel, then Sara, and then it repeated. The asker only stopped asking when only one woman did not know her number. If all three women had a 2 on their head, who was the last to not know her number?

### Hint

The following equations might come in handy:2+2+2=6

2+2+3=7

1+2+3=6

### Answer

Sara is the last to not know her number.When each woman got their blindfolds removed, they each saw two 2's, adding up to 4. They then deduced that their own number was either a 2 (which added up to 6) or a 3 (which added up to 7). Because Queenie didn't know her number, she said, "I don't know."

Rachel thought, "If I have a 2, Queenie would not have known if she had a 2 or a 3. If I have a 3, this plus Sara's 2 would have added up to 5, and Queenie would not have known if her number was a 1 or a 2." So Rachel said, "I don't know."

Sara thought, "If I have a 2, neither Queenie nor Rachel would know if they have a 2 or a 3. If I have a 3, Queenie would not have known if her number was a 1 or a 2, and Rachel would have thought, 'If I have a 1, Queenie would not have known if she had a 2 or a 3, and if I have a 2, Queenie would not have known if she had a 1 or a 2.'" So Sara said, "I don't know."

Queenie thought, "If I have a 3, Rachel would have thought, 'If I have a 1, Queenie would have seen my 1 and Sara's 2, adding up to 3, and known her number was a 3. If I have a 2, Queenie would not have known if her number was a 2 or a 3,' and then said, 'I have a 2.' But she didn't know." So she said, "I have a 2."

Rachel thought, "If I have a 3, Sara would have thought, 'If I have a 1, Rachel would have known her number was a 3. If I have a 2, Rachel would not have known if her number was a 2 or a 3,' and then said, 'I have a 2 on my head.' But she didn't know." So she said, "I have a 2 on my head."

Sara thought, "If I have a 2, Queenie and Rachel would both have known the second time around that they each had a 2. If I had a 3, Queenie and Rachel would both have known the first time around that she had a 1 or a 2, and Queenie would have thought, 'If I have a 1, Sara would have seen my 1 and Rachel's 2 and known she had a 3,' and Rachel would have thought the same, except with her own 1 and Queenie's 2." So Sara still doesn't know.

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