Dark. Wet. Cold. Your vision is foggy and blurred. You look left, right, and all of it is sand. You are on a beach, stranded from your friends and family. But this is not like a game of Survivor, you are alone. Or are you? Suddenly a thought occurs to you. There are people on this island. They could save you! However, there is a twist. On this island, all the inhabitants are either knights, who always tell the truth, or knaves, who always lie.
You see two men approaching you. Somehow, you know that one is a knight and one is a knave. However, which is which? You desperately desire to ask them how to get off the island and return home safely. But who can you trust? You could question them for hours. Nevertheless, would that really get you any closer to the truth?
What is one question you could ask each man to determine who is a knight and who is a knave?
Let’s investigate this a bit.
What if you asked the question “are you a knight?”
The knight would respond with “yes” since he always tells the truth. The knave would respond with “yes” since he always lies. Therefore, there is no way to tell them apart, since they both responded the same. Drat! That is not helpful at all!
What if you asked the question “are you a knave?”
The knight would respond with “no” since he always tells the truth. The knave would respond with “no” since he always lies. Again, there is no way to tell them apart. Phooey! That is not helpful either!
We need to get to the bottom of this problem and our questions so far are getting us nowhere fast. Now take a second, before you read on, and think about some possible questions that might give you the information you need to determine each man’s true identity.
Done? Well here is the line of reasoning. Ask each man a question about the other man. This concept may be a little bit difficult to grasp at first, but you must grasp it, your life depends on it!
What if you asked the two strangers the following “Is the other man a knight?”
If you asked the knight, he would respond with “no” since he always tells the truth and the other man is NOT a knight. If you asked the knave, he would respond with “no” since he always lies and the other man IS a knight and he must lie.
Again, since they both answered the same way, there is no way of determining who is who. We need to go deeper. What if we asked each man a question about a question about the other man?
“If I asked the other man, ‘is the other guy a knight’, what would he say?”
Now let’s investigate their responses. Suppose I ask the knight this question. We know from before that when we ask the knave “is the other man a knight” he responds with “no.” Hence, the knight will convey this fact honestly and tell us “no.”
Suppose I ask the knave this question. We know from before that when we ask the knight “is the other man a knight” he responds with “no.” However, the knave is crafty. He will not transmit this fact to us honestly. Instead, the knave will lie to us and respond with “yes.”
Huzzah! We have a difference. Therefore, if we ask the knight this question, he will respond with a “no” and if we ask the knave, he will respond with a “yes”.
Now we know who we can trust to get us off this terrible island! In the words of a wise hologram, that, detective, is the right question.