While traveling, you come upon a fork in the road. One path is the short route to your destination, the other is the long 'scenic' route. You do not know which route is which, but a man at the intersection does.
(Case 1) He is either a knight (always tells the truth) or a liar (always lies). By asking him one question, can you determine which route is the short route?
(Case 2) He is a knave (strictly alternating true and false statements), but you do not know if his next statement will be true. By asking him one question, can you determine which route is the short route?
How does this differ from case 1?
(Case 3) He could be a knight, a knave, or a liar. With two questions, can you determine which route is shorter?
Is it possible to determine which route is shorter with one question?
1. Are you the type that would say that the left road is the long route?
If the left is long, then a knight would say that and say, "Yes." A liar would say that the left is short, but would lie and say, "Yes." If the left is short, then a knight would tell the truth and say, "No." A liar would say that the left is long, but would lie and say that they would not say that.
2. This is impossible. A knave could lie or tell the truth.
3. a. Are you a knave?
b. Is the left road the long route?
Suppose he says, "Yes" to a. Then, he cannot be a knight. Therefore, he is either a liar or a knave who told the truth and is going to lie to the next question. Then, you know that he will lie to b, so you will know which route is long.
Suppose he says, "No" to a. Then, he cannot be a liar. Therefore, he is either a knight who always tells the truth or a knave who lied and is going to tell the truth. Then, you know that he will tell the truth to b, so you will know which route is long.
It is impossible to find out in one question.

Posted by Math Man
on 20110104 18:21:49 