8. "Did you answer the previous questions honestly?"

40 gnomes answered "yes" to the first question, 50 to the second, 70 to the third, and 100 to the fourth. How many honest gnomes are there in the underground kingdom?
Answer: 40 honest gnomes.

## Solution.

On the 4th question, both an honest and a liar will answer "yes," so there are 100 gnomes in the underground kingdom.

An honest gnome will answer "yes" to one of the first three questions and "no" to two. A liar, on the other hand, will answer "yes" to two of the first three questions and "no" to one.

The answer can be obtained either by an equation or by reasoning.

Method 1. Let $x$ be the number of honest gnomes. Then, the total number of "yes" answers to the first three questions will be $x + 2 \cdot (100 - x) = 200 - x$, i.e., $200 - x = 40 + 50 + 70 = 160$, from which $x = 40$.

Method 2. In total, there were $40 + 50 + 70 = 160$ "yes" answers to the first three questions. If all the gnomes were telling the truth, there would be 100 "yes" answers to the first three questions. Since each liar gives one more "yes" answer, there were $160 - 100 = 60$ liars. Therefore, there are 40 honest gnomes.

## Grading Criteria.

- Correct solution - 7 points.
- Reasoning to find the number of liars but not the number of honest gnomes - 5 points.
- Correct answer without justification - 2 points.