FILE No.67Unsolved
A box holds five hats: three black and two white. Three people (Ann, Ben and Cara) each take one hat from the box and put it on without seeing its colour. The remaining two hats stay in the box.
They stand in a line. Ann can see Ben and Cara's hats, Ben can see only Cara's hat, and Cara can see nobody's hat. None of them can see their own.
In turn they are asked whether they know their own hat colour.
Hearing that, Cara says, "I know."
What colour is Cara's hat?
If you get stuck, open the hints one at a time, in order.
What combinations does Ann's "I don't know" rule out?
What did Ben see when he looked at Cara?
Why did Ben say "I don't know"?
What colour is Cara's hat?
Choose the answer closest to the truth.
Black
The answer is: black.
There are only two white hats. If Ann had seen white hats on both Ben and Cara, she would have known her own hat was black. Since she said she didn't know, Ben and Cara cannot both be wearing white.
Ben can see Cara's hat. If Cara were wearing white, Ben would reason: "If my hat were also white, Ann would have seen two white hats and known hers was black. But she didn't know, so my hat must be black." In that case Ben would know his own color. Since he said he didn't know, Cara cannot be wearing white. Therefore Cara's hat is black.
We treat "I don't know" as an empty statement rather than a clue.
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