Three people, named Adam, Beth, and Carl, are wearing either a red or blue hat.
They’re standing in a straight line, facing forward. Carl can see both Adam and Beth, Beth can see only Adam, and Adam cannot see anyone.
They know there are two hats of one colour and one hat of the other colour, but they don’t know which colour they are wearing.
Carl is asked whether he knows the colour of his own hat. Carl answers, “No, I don’t know.”
Beth, who can see only Adam, is asked the same question. She also replies, “No, I don’t know.”
Finally, Adam, who cannot see anyone, is asked the same question.
He confidently answers, “Yes, I know the colour of my hat.”
What colour is Adam’s hat, and how can he be so sure?
Adam can confidently determine the colour of his hat based on the responses from Carl and Beth.
When Carl sees Adam and Beth, he has two possibilities:
- Both hats are the same colour (either both red or both blue).
- Their hats are different colours (one red and one blue).
If Carl saw two hats of the same colour (two reds or two blues), he could have deduced that his hat must be of the opposite colour since there are only two hats of one colour.
However, Carl says he doesn’t know the colour of his hat, which means he sees one red hat and one blue hat on Adam and Beth.
Now, Beth knows that Carl sees one red and one blue hat. If Beth sees a red hat on Adam, she would deduce that her hat is blue, as Carl would have known his hat colour if he saw two red hats.
Similarly, if Beth sees a blue hat on Adam, she would deduce that her hat is red, as Carl would have known his hat colour if he saw two blue hats.
However, Beth also says she doesn’t know the colour of her hat, which means she cannot deduce her hat colour based on Adam’s hat.
Now, Adam knows that both Carl and Beth see one red and one blue hat, and Beth cannot deduce her hat colour based on his hat.
This means that Adam’s hat must be the same colour as Beth’s, as that is the only scenario in which Beth would be unable to determine her own hat colour.
Since Beth is wearing a blue hat (as observed by Carl), Adam must also be wearing a blue hat.