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?

If a liar starts the singing he would make a true statement.
Therefore, the first person to sing must be a knight and there must be at least one liar present.
There being at least one knight present, the last person to sing must be liar.
If the second to sing is a liar, he makes a true statement unless he is also the last.
If the second to sing is a knight, then we need at least one more liar.
If the third is a liar, any subsequent liars would render him honest. There would now be four in the group.
If knights continue to sing, an equal number of liars are needed to keep all behaving according to type. This produces an even number.
Therefore, there are two at the club. A knight starts singing and a liar finishes it.
Edited on February 3, 2004, 9:29 am

Posted by retiarius on 2004-02-03 09:23:58