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The Prime Club (Posted on 2004-02-03) Difficulty: 2 of 5
There is a nightclub in Truth town called the Truth Club which is made up entirely of knights and liars.

Sometimes they start singing a song. One person sings "At least one of us is a liar", the next person sings "At least two of us are liars", continuing on like this such that each person says one more person than the last person; each person singing exactly one line. (If there were 10 people in the club, the only person who hadn't sung a line would sing the last line, "at least 10 of us are liars" and then the song would be done.)

One day when you know there was a prime number of people in the club, you hear the start of the song "At least...", but don't hear the middle; all you know is that they sang the song through completely. Even though you only hear those two words at the start of the song, you can tell how many people are in the club. How many people were there?

See The Solution Submitted by RoyCook    
Rating: 3.7500 (4 votes)

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Solution | Comment 3 of 10 |
There are two people in the "knight-club".


If there are N people in the knight-club, then the last one, who sings "At least N of us are liars" must be lying; if he were a knight, he'd be lying, since then there could be at most (N-1) liars in the knight-club. But then the first person, who sang "At least one of us is a liar" must be a knight; he is telling the truth. There is at least one liar and one knight in the knight-club. So N >= 2.

But if N > 2, and N is a prime number, you get paradoxes:

e.g. if N=3 (prime number)

knight: "At least one of us is a liar."
2nd person: "At least two of us are liars."
liar: "At least three of us are liars."

Is the 2nd person a liar? Then there are two liars in the knight-club, and he is telling the truth. Is he a knight? Then only one liar is present,and he is lying.

If N=4 (non-prime number):

knight: At least one of us is a liar.
knight: At least two of us are liars.
liar: At least three of us are liars.
liar: At least four of us are liars.

That is fine. There is no paradox if N is not a prime number.
Edited on February 3, 2004, 9:49 am
  Posted by Penny on 2004-02-03 09:34:24
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