Two (one knight and one liar). The key is to realize that it has to be an even number of people, or else the song is paradoxical. If an even number 2m are singing, then one can easily verify that the first m are Knights and the second m are liars. If there is an odd number 2n+1, however, then consider the n+1th singer. He sings "At least n+1 of us are liars". If he is telling the truth, then so was everyone before him (since they all sang "At least x of us are liars" for some x less than n), so there are n+1 knights, which means there could be at most n liars. Thus, he lied, resulting in a contradiction. Similarly, if he were lying, then everyone after him would also be lying, but this would mean that there were n+1 liars, resulting in his having told the truth. Again, contradiction. Once we know that it has to be an even number of singers, the behaviour of the bouncers guarantees that it must be two singers, the only even prime. |