8.5. On an island, there live 33 knights, as well as liars and fantasists. Each resident of this island was asked in turn: “How many of you are knights?”. Ten different answers were received, each of which was given by more than one resident. Knights always tell the truth, liars always give a wrong number that has not yet been mentioned, and fantasists always give a number that is one more than the previous answer. Was the number 40 necessarily mentioned?
Answer: Yes.

Solution. Let's call a number $m$ "initial" if it was named, but the previous number $m-1$ was not named. According to the problem, the fantasizers did not name initial numbers. If an initial number was named by a liar, then neither another liar nor a knight could name it again. But each number was named at least twice. Therefore, initial numbers could only be named by knights, and 33 is the only initial number. Moreover, every number other than 33 must have been named by at least one fantasizer, since liars cannot name the same number twice. This means: 1) numbers less than 33 were not named (otherwise, the smallest of them would be initial); 2) all ten named numbers are consecutive (but not necessarily in ascending order).

Therefore, all natural numbers from 33 to 42 inclusive were named, and 40 is among them.

The situation described in the problem is possible, for example: 39L; 40F; 33K; 34L; 36L; 35L; 36F; 37F; 38L; 41L; 42F; 33K; 34F; 35F; 36F; 37F; 38F; 39F; 40F; 41F; 42F; 33K; ...

It is not required for participants to provide such an example.

Grading criteria.

“+" A complete and well-reasoned solution is provided

“士” A generally correct solution is provided, but it contains minor gaps or inaccuracies

"Ғ" The correct answer is provided, but the solution contains both correct and incorrect statements (for example, it is claimed that the numbers were necessarily named in ascending order)

"Ғ" It is only justified that numbers less than 33 could not be named, but there is no further progress or there are errors in further reasoning

“F” Based on one or several examples, it is claimed that the numbers from 33 to 42 inclusive were named, but this is not proven

“-" Only the answer (“Yes” or “No”) is provided

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