Welcome to the Island of Knights, Knaves, and Sir Mix-a-lot! There are a few things you should know about this island:
- Knights always tell the truth
- Knaves always lie
- There is exactly one Knight named “Sir Mix-a-lot”. Like every other knight, he cannot lie.
Today, we are on the island of sleepwalking knights and knaves. At night, half of the population sleepwalks around the island. When a resident sleepwalks, they are living in a dream world where everything is the opposite of reality. In other words, every belief held by a sleepwalking resident will be wrong. For example, if you ask an awake knight if he is a knight, he will say “yes”. If you ask a sleepwalking knight if he is a knight, he will say “no”! He is sleepwalking, so he believes he is a knave. Since he is actually a knight, he answers all questions truthfully, so he will tell you that he is actually a knave!
Let’s say you run into a resident of this island during the night. Last time, you wanted to know if someone was a knight or a knave. Now you want to know if this other resident is sleepwalking or not. Can you figure it out in one question?
One possibility is “Are you a knight?” An awake knight will tell the truth and say “yes.” An awake knave will lie and also say “yes.” Easy enough so far, and we now know that no awake resident can say “no.” But what do sleepwalkers say?
A sleepwalking knight will believe himself to be a knave. When asked if he is a knight, he will tell you the truth about that belief and say “no.” A sleepwalking knave will believe himself to be a knight. He will then lie to you about that belief and will also say “no.” So a “yes” answer means that person is awake, and a “no” answer means that person is sleepwalking. We don’t know if we are talking to a knight or a knave, but we do know if that person is sleepwalking or not.