Recitation: Practice with Logic

Knights and Knaves

You are on an island populated entirely by an interesting kind of people - each citizen on the island is either a knight or a knave:

What can you say about each individual in the following situations?

  1. You meet two individuals. The first says “We are both knaves.” The second says nothing.
  2. You meet two individuals. The first says “We are both the same kind.” The second says “We are each a different kind.”
  3. You meet three individuals. The first says “The second is a knave.” The second says “The third is not a knave.” The third says “I am a knight or the first is a knight.”

There is actually another kind of citizen on this island, the noble:

What can you say about each individual in the following situations? If there isn’t a unique answer, try to list all of the possibilities.

  1. You meet three individuals. You know one is a knight, one is a knave, and one is a noble. The first says “The third is a knave.” The second says “The first is a knight.” The third says “I am a noble.”
  2. You meet three individuals. The first says “I am a knave and only a knave would say we are all knaves.” The second says “We are all knaves.” The third says “I am a knight.”

Hat-wearing prisoners

Four prisoners are given the opportunity to end their sentence early if they can solve a puzzle. Their jailer tells them that he has two white hats and two black hats. The jailer seats three of the prisoners in a line, the first facing a wall, the second facing the first, and the third facing the second (he can see the first also). The fourth prisoner is put in a separate room. The jailer then gives each of the prisoners a hat and tells them that if any of them can say the color of their hat with absolute certainty then they can all go free. How can the prisoners walk free?