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 The Prime Club (Posted on 2004-02-03)
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 5 of 10 |
The queue cannot be all knights, for the first would be lying.

The queue cannot be all liars, for the first would be telling the truth.

So the queue must have some knights and some liars. The knights must be at the beginning of the queue, and the liars at the end.

If there were more knights than liars, the last knight would be lying.

If there were more liars than knights, the first liar would be telling the truth.

Thus, there must be the same number of knights and liars, and since the total number must be a prime number, there were TWO people in all, a knight and a liar.
 Posted by e.g. on 2004-02-03 11:48:56

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