B3. One of the four gnomes, Anne, Bert, Chris, and Dirk, has stolen gold from the king. Each of the gnomes, who know each other inside and out, makes two statements. If a gnome is a liar, at least one of those two statements is a lie. If a gnome is not a liar, both statements are true.

Anne says: "Bert is a liar." and "Chris or Dirk did it."

Bert says: "Chris is a liar." and "Dirk or Anne did it."

Chris says: "Dirk is a liar." and "Anne or Bert did it."

Dirk says: "Anne is a liar." and "Bert or Chris did it."

How many of these eight statements are true?
B3. 5

First, consider the case where Anne stole from the king. Then the last two statements of Bert and Chris are true, and those of Anne and Dirk are false. Therefore, Anne and Dirk are liars. Since both statements of Chris are true, Chris is not a liar. From this, we conclude that Bert is a liar, because his first statement is a lie. Now that we know whether each person is a liar, we simply count that out of the eight statements, five are true.

You can apply similar reasoning for the cases where Bert, Chris, and Dirk stole from the king. Since the problem is symmetric (cyclically swapping the names 'Anne', 'Bert', 'Chris', and 'Dirk' does not change the problem), we find the same result each time: five of the statements are true.