The last day the King could kill the condemned man is Saturday. The King promised not to let him know he is going to be shot next day. However, if he is going to be killed in Saturday, when he is still alive on Friday evening, he knows he is going to be killed tomorrow (since Saturday is the last day to execute him). So he is not going to be killed on Saturday (because he knows). According to this, he cannot be killed on Friday also (because the King cannot kill him on Saturday, on Thursday he knows he is going to be killed on Friday and so on). Thus, the King couldn't kill him at all.
However in reality, the King killed him at dawn on Tuesday and he didn't lie.
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