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Hats in a Bag
Zack, Susie and Peter are standing in a line. All are facing in the same direction.
Zack is facing the wall. Susie is behind Zack and Peter is behind Susie.
They all know that Mr. Smith has four black and two red hats in a bag.
Mr. Smith asks them to close their eyes and randomly puts one hat on each of them.
He tells them to open their eyes.
He then asks a question," Tell me the color of the hat you are wearing?"
After a few minutes of silence, Zack started jumping in excitement and yelled
that he was wearing a ______ hat. Remember that they can only look in front of them.
Which color was Zack’s hat?
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In solving this problem, we should assume that all participants think logically,
do not make logic mistakes, extract all logical consequences completely and in full,
and acts according to logical situation.
Peter is the last in the line. He sees Zack and Susie, that are before him.
Peter sees their hats. But Peter keep silence.
It means that Peter does not see two red hats before him - otherwise Peter
would recognize/guess that his hat is black.
Since Peter does keep silence, Zack, who is the first in the line and does not see
Susie and Peter, concludes that at least one hat before Peter is black.
OK. But Susie also keeps silence. In her brain, Susie makes the same logic as if
she is Peter. Since Peter keep silence, Susie concludes that
- either her hat is red or Zack's hat is red, but only one of the two (NOT two are red);
- either her hat is black AND Zack's hat is black.
So, now Susie knows (she derived it in her brain) that
- either her hat is red, or Zack's hat is red, (but only one of the two is red);
- either her hat is black AND Zack's hat is black.
So, if Susie sees red hat on Zack, she must conclude logically that her hat is black.
But Suzie keeps silence - so Zack, thinking logically, concludes that Susie sees his,
Zack's, hat black.
Thus Zack concludes that his hat is black.
ANSWER. Zack's hat is black.