When Mr. Mathwiz visited the IoK&L* he was surrounded by 20 or 21 male inhabitants, who formed a perfect circle and then each said a single word (either "Knight" or "Liar") about the man on his right.
The wiz then evaluated correctly the exact numbers of liars and the truth-tellers.
What were those numbers?
*IoK&L, is an imaginary island, inhabitated by Liars and by Truth-Tellers a.k.a. Knights.
If all of the inhabitants were of the opposite type, then they would have said the same thing. Therefore, whatever they said, there are two solutions with opposite types for each person. Make the number of people x. If one solution has k knights, then there are x-k liars, so the other solution has x-k knights and k liars. For Mr. Mathwiz to know the number of knights and liars, x-k and k must be the same number. Therefore, x=2k, so there are an even number of inhabitants. That means that there are 20 inhabitants, so x=20 and k=10. Therefore, there are 10 knights and 10 liars.
Edited on June 4, 2011, 2:20 pm
Posted by Math Man
on 2011-06-04 14:18:18