perplexus.info :: Liars and Knights : The way to Normalville

The way to Normalville (Posted on 2012-10-05)

In Normalville, every inhabitant is either a knight, a liar, or a normal. Knights always tell the truth. Liars always lie. Normals tell the truth and lie completely at random. You are going to Normalville when you see a fork in the road. There are two ways to go. One of them leads to Normalville. You see three inhabitants by the fork, A, B, and C. You know that only one of them is a normal, but you are not sure who it is. In two "yes" or "no" questions, how do you find the way to Normalville? (Each question can only be addressed to one person, but it can be a different person for each question.)

Submitted by Math Man

Rating: 3.7500 (4 votes)

Solution: (Hide)

Ask A, "Are you a knight if and only if B is a normal?" Suppose A says, "Yes." If A is a knight, then A is a knight if and only if B is a normal, so B is a normal. Since only one of them is a normal, C is not a normal. If A is a liar, then A is not a knight if and only if B is a normal. Since A is not a knight, B is a normal. Therefore, C is not a normal. If A is a normal, then C cannot be a normal since there is only one normal. Therefore, if A says, "Yes," then C is not a normal.

Suppose A says, "No." If A is a knight, then A is not a knight if and only if B is a normal, so B is not a normal. If A is a liar, then A is a knight if and only if B is a normal. Since A is not a knight, B is not a normal. If A is a normal, then B is not a normal because there is only one normal. Therefore, if A says, "No," then B is not a normal.

Now, you know one of them that is not a normal (B or C). Therefore, that person is either a knight or a liar. Ask this person, "Are you a knight if and only if the left road leads to Normalville?" Suppose they say, "Yes." If they are a knight, then they are a knight if and only if the left road leads to Normalville, so the left road leads to Normalville. If they are a liar, then they are not a knight if and only if the left road leads to Normalville. They are not a knight, so the left road leads to Normalville. Therefore, if they say, "Yes," then take the left road.

Suppose they say, "No." If they are a knight, then they are not a knight if and only if the left road leads to Normalville, so the right road leads to Normalville. If they are a liar, then they are a knight if and only if the left road leads to Normalville. They are not a knight, so the right road leads to Normalville. Therefore, if they say, "No," then take the right road. Now, you know the way to Normalville.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
w a i t ......	Ady TZIDON	2012-10-10 11:23:22
re(2): First question only	broll	2012-10-08 01:33:12
re(5): no cigar	Math Man	2012-10-07 13:20:51
re: First question only	Ady TZIDON	2012-10-07 10:30:36
re(4): no cigar	Steve Herman	2012-10-07 09:37:26
Thoughts on the second question.	broll	2012-10-07 06:49:10
First question only	broll	2012-10-07 06:21:14
re(4): no cigar	Ady TZIDON	2012-10-06 16:21:29
re(3): no cigar	Math Man	2012-10-06 14:58:00
re(3): no cigar	Ady TZIDON	2012-10-06 13:26:59
re(2): no cigar	Steve Herman	2012-10-06 10:10:52
re: no cigar	Math Man	2012-10-06 09:15:20
no cigar	Ady TZIDON	2012-10-06 05:12:36

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