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Who is a Liar, Who Tells the Truth?

Date: 9/4/96 at 19:7:34
From: Anonymous
Subject: Who is a Liar, who Tells the Truth

The inhabitants of a certain island have a strange characteristic: 
those who are swimmers always tell the truth, and those who are non-
swimmers always lie.  

You visit the island and meet a group of three native inhabitants. 
You ask the first person, "Are you a swimmer?" The answer is 
unintelligible. The second person immediately says, "That was 'yes.' 
I am a swimmer, too." The third person says, "They are both lying." 

Is each of these three people a swimmer or a nonswimmer?  

My answer: The first is a nonswimmer, the second is also a nonswimmer, 
the third one is a swimmer.

Date: 9/4/96 at 21:54:41
From: Doctor Robert
Subject: Re: Who is a Liar, who Tells the Truth

Put yourself in the place of the second of the three natives.  If the 
first person is a swimmer he will say "I am a swimmer."  If the 
first person is a non-swimmer, he will also say "I am a swimmer" 
(because he always lies).  So, the thing that the second person hears 
is "I am a swimmer."  

When the second person says that the first person says he is a 
swimmer, the second person is telling the truth and is therefore a 

Now, the third person says that the second person is lying - which is 
wrong.  So that makes the third person a non-swimmer.  Unfortunately, 
we do not know whether the first person is a swimmer or non-swimmer.  
Using S for swimmer and N for non-swimmer, the possibilities are

   SSN   or NSN  
[See also the Classic Problem "Liars and Truthtellers" in the
Dr. Math FAQ:   .]

-Doctor Robert,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Logic

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