3. On the Island of Logic, there live 40 knights (who always tell the truth), 25 liars (who always lie), and several sophists. A sophist can only utter phrases that a knight or a liar could not say in their place. For example, standing next to a liar, a sophist can say, "We are both liars" (because if he were a knight, this statement would be false, and if he were a liar, it would be true). One day, a sophist made two statements about the island's inhabitants:
1. "There are exactly ... liars on the island."
2. "There are no more sophists on the island than liars."
Reconstruct the first statement of the sophist. How many sophists are on the island?
Answer. 1) There are exactly 26 liars on the island; 2) 27 sophists.

Solution.

1) If a sophist were a liar, he would have to tell the truth, and then he would say that there are $25+1=26$ liars. If he were a knight, there would be 25 liars, and he would lie.
2) Let C be the total number of sophists on the island. If a sophist were a liar, he would have to tell the truth, and then he would say that $\mathrm{C}-1 \leq 25+1$, or $\mathrm{C} \leq 27$. If a sophist were a knight, he would have to lie, and then it would be true that $\mathrm{C}-1>25$, or $\mathrm{C}>26$. The only natural number that satisfies both inequalities is 27.

## Grading Criteria.

Correctly stating the claim about liars without justification: 1 point.

Correctly stating the number of sophists without justification: 2 points.

For justifying each of the claims, an additional 2 points are added.