On the island of Chenonc'è there are 2009 inhabitants, divided into three clans: the knaves who always lie, the knights who never lie, the pages who lie one day and not the next, independently of one another. One day I ask each of the inhabitants how many knaves are on the island. The first says: "there is at least 1 knave"; the second says: "there are at least 2 knaves";... the 2009th says: "there are at least 2009 knaves". I write down in a list the sequence of the 2009 answers, in the order in which they were spoken. The next day I question all the inhabitants in the same way (not necessarily in the same order), and I obtain a list of answers identical to that of the previous day. Knowing that there is only one knight on the island, how many pages are there?
Pick one
Solution
Solution:
The answer is (D). Let be the number of knaves and the number of pages. In each of the two lists there will be exactly true statements (the first ) and exactly false statements (the remaining ones). Therefore between the two lists there are false statements. Now each knave has given 2 false statements, while each page has given only one, since on one of the two days he must have told the truth. Therefore the number of false statements must equal the number of pages plus twice the number of knaves, that is
and since (there is only one knight) one can obtain from which .
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