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 all of the people in the club were knights, then they would be lying. If they were all liars, then they would be telling the truth. Therefore, there is at least one knight and at least one liar. Let the last knight be the nth person. They say, "At least n of us are liars." Since they are a knight, there are at least n liars. However, all of the people before them are knights. Therefore, the n liars must be after the nth person. Since they are the last knight, there are 2n people in all. There are n knights and n liars. Therefore, there are an even number of people in the club. Since 2 is the only even prime, there were 2 people that day. There was a knight and a liar.

Posted by Math Man on 2012-02-26 12:04:29