7.3. Each of the thirteen dwarfs is either a knight, who always tells the truth, or a liar, who always lies. One day, all the dwarfs in turn made the statement: “Among the statements made previously, there are exactly two more false ones than true ones.” How many knights could there have been among the dwarfs?
Answer: 6

Solution. The first two statements are obviously false, as there were fewer than two statements made before each of them. The third statement is true, as there were 2 false statements and zero true statements made before it. The fourth statement is false, as it adds one true statement to the two false ones, while the fifth statement is true again.

Reasoning similarly, we find that all gnomes making an even-numbered statement were lying, while those making an odd-numbered statement were telling the truth. Thus, the knights are the gnomes who spoke under the numbers: $3,5,7,9,11$ and 13.

Evaluation criteria.

"+" A complete and well-reasoned solution with the correct answer is provided

“士” A generally correct reasoning with minor gaps or inaccuracies and the correct answer is provided

"干" Only the numbers of the gnomes who told the truth are correctly indicated, but any explanations are missing

“-" Only the answer is provided

“-" An incorrect solution is provided or it is missing