Black and White Hats
David, Enrique, and Gema are logic experts and know the following is true:
- Each is wearing a hat.
- The hat is either a black hat or a white hat.
- They cannot all wear a white hat.
- They cannot see their own hats.
- Gema is wearing a blindfold and cannot see anyone’s hat.
On a warm day in early spring, David says, “I don’t know what color hat I am wearing.”
After hearing this, Enrique says, “I don’t know what color hat I am wearing.”
After hearing both of these statements, the one wearing the blindfold, Gema, knows what color she is wearing.
What color is Gema wearing?
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I recommend drawing a picture to solve/follow this solution; it gets pretty tricky.
We know the three (we'll call them D, E, and G) cannot all be wearing white hats. Since D knows this, if E and G had white hats, he would instantly know he was wearing black. Therefore, E and G are not wearing white hats. There are now three scenarios (in terms of EG): BW, WB, and BB. We must prove that either this scenario is or is not BW in order to get our answer.
Assume that the scenario is BW. There are two more sub-scenarios, if you will: BBW and WBW. WBW is clearly impossible because E would know his hat color: black. BBW is a bit more tricky; but it is possible.
If the scenario is BBW, then that means D would've seen E with black and G with white. Since E knows that G is white, he can deduce that he, himself, is wearing black.
Therefore, since all of the scenarios with G wearing a white hat are impossible, Gema must be wearing a black hat.