david1971xx: A king wants his daughter to marry the smartest of three extremely intelligent young princes.
The princes are gathered into a room and seated, facing one another, and are shown two black hats and three white hats. They are blindfolded, and a hat is placed on each prince's head, with the remaining hats hidden in a different room.
The king tells them that the first prince to deduce the color of his hat without removing it or looking at it will marry his daughter. To prevail, a prince must also explain his reasoning clearly. A wrong answer or a poor or inadequate explanation will mean death. The blindfolds are then removed.
You are one of the princes. You see two white hats on the heads of the other two princes. After some time passes, you realize that the other princes are unable to make a deduction. You rise and state your hat color, and your reasoning, and you win the princess. What color was your hat? How did you know?
david1971xx: There are only three logically possible cases to consider:
A. A prince (any prince) sees two black hats B. A prince sees one black hat and one white hat C. A prince sees two white hats.
If Chauncey (a random prince,not necessarily you) saw two black hats, he would immediately rise and state "My hat is white! I see two black hats and the only remaining color is white!" So this rules out Case A.
So obviously, no prince has seen two black hats. Can any prince be seeing one black hat and one white hat? That is, is Case B possible? Consider this case exists (Case B) as a hypothesis.
If Chauncey were to see Alphonse with a white hat and Balthazar with a black hat, he should ask himself, "Can I myself be wearing a black hat?" So Chauncey imagines, in this hypothetical world where he sees one of each and for the sake of the argument, that his hat is black and then asks the consequences.
Chauncey realizes that if he saw one black and one white, and if in that case he himself were indeed wearing a black hat, then Alphonse would see two black hats and Alphonse, therefore, would have jumped up immediately and said "Mine's White!".
Therefore no prince can be seeing a black hat and a white hat. Therefore every prince is seeing the same thing -- two white hats. Therefore, your hat is white and you get the princess.
david1971xx: Well under the rules as stated, a correct color without an adequate explanation = death but we can make an exception in your case. Did your head explode when you figured it out?
The princes are gathered into a room and seated, facing one another, and are shown two black hats and three white hats. They are blindfolded, and a hat is placed on each prince's head, with the remaining hats hidden in a different room.
The king tells them that the first prince to deduce the color of his hat without removing it or looking at it will marry his daughter. To prevail, a prince must also explain his reasoning clearly. A wrong answer or a poor or inadequate explanation will mean death. The blindfolds are then removed.
You are one of the princes. You see two white hats on the heads of the other two princes. After some time passes, you realize that the other princes are unable to make a deduction. You rise and state your hat color, and your reasoning, and you win the princess. What color was your hat? How did you know?
A. A prince (any prince) sees two black hats
B. A prince sees one black hat and one white hat
C. A prince sees two white hats.
If Chauncey (a random prince,not necessarily you) saw two black hats, he would immediately rise and state "My hat is white! I see two black hats and the only remaining color is white!" So this rules out Case A.
So obviously, no prince has seen two black hats. Can any prince be seeing one black hat and one white hat? That is, is Case B possible? Consider this case exists (Case B) as a hypothesis.
If Chauncey were to see Alphonse with a white hat and Balthazar with a black hat, he should ask himself, "Can I myself be wearing a black hat?" So Chauncey imagines, in this hypothetical world where he sees one of each and for the sake of the argument, that his hat is black and then asks the consequences.
Chauncey realizes that if he saw one black and one white, and if in that case he himself were indeed wearing a black hat, then Alphonse would see two black hats and Alphonse, therefore, would have jumped up immediately and said "Mine's White!".
Therefore no prince can be seeing a black hat and a white hat. Therefore every prince is seeing the same thing -- two white hats. Therefore, your hat is white and you get the princess.