On an island there are 2023 people standing in a line, each of whom is either a knave or a knight: knights always tell the truth, while knaves always lie. If i is odd, the person in position i-th exclaims: "There are at least i knaves"; if i is even, the person in position i-th exclaims: "There are exactly i knaves". How many are the knaves?
Solution
Solution:
The answer is 1348. If the knaves are odd in number, say 2m+1, then the people who tell the truth are all and only those in odd positions less than or equal to 2m+1, which are m+1. There would then be 2m+1 knaves and m+1 knights, for a total of 3m+2 people, but this is impossible since the equation 2023=3m+2 leads to m=32021, which is not an integer. The knaves are therefore even in number, say 2m: the knights will be all and only the people in odd positions less than or equal to 2m, plus the person in position 2m, hence in total m+1. Thus 3m+1=2023 and therefore the knaves are 2m=1348. One verifies that the configuration works.
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Source: MathNet,
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