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In a room are 13 men, who are either Knights (truth tellers) or Knaves (lie tellers).

the first man looks around and says "The number of knights in this room is no greater than one"

the second man looks around and says "The number of knights in this room is no greater than two"

the third man looks around and says "The number of knights in this room is no greater than three"

and so on.

how many knights exactly are in the room?

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There are 7 knights. The 7th knight is the first correct, and from 7-13 are 7 knights.

Every knight added eliminates a knave while increasing the threshold for truthiness. So, since one is increasing at the same rate that the other is decreasing, the result will end up near the middle.

If there are 6 knights, that would work down to knight 8, which would be untrue. If there are 8 knights, that would work down to knight 6, which is also untrue. 7 knights works down to knight 7.

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Man 13 must be telling the truth... there are not more than 13 knights so he is a knight.

If he is the only knight, man 1 must be a knight which is a paradox. Man 1 is a knave.

As we know now that there is no more than 12 knights, Man 12 is a knight.

This can continue until we find that men "8-13" are knights while men "1-6" are knaves.

As this means there are at least 6 knight and 6 knaves, man 7 says "there are no more than 7 knights". He is correct so is a knight too.

Ergo: "7-13" are knights while men "1-6" are knaves

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