Problem 5. Six gnomes are sitting around a round table. It is known that exactly two gnomes always tell the truth, and they sit next to each other. Additionally, exactly two gnomes always lie, and they also sit next to each other. The remaining two gnomes can either lie or tell the truth, and they do not sit next to each other. A treasure hunter walks around the table and asks the gnomes where they have hidden the gold.

- The first gnome said it is in the cave.
- The second said - at the bottom of the lake.
- The third said - in the castle.
- The fourth said - in the enchanted forest.
- The fifth said - at the bottom of the lake.

Where did the gnomes hide the gold? Justify your answer.
Answer: in the cave.

Solution. We have two gnomes who always tell the truth. Let's call them truthful gnomes (denoted by the letter T in the diagram).

There are two gnomes who always lie. Let's call them liars (denoted by the letter L in the diagram).

And there are two gnomes who can both lie and tell the truth. Let's call them ordinary gnomes (denoted by the letter O in the diagram).

From the problem statement, it follows that the arrangement of the gnomes is as follows:

![](https://cdn.mathpix.com/cropped/2024_05_06_83a6c5e4a5c18bfc804cg-3.jpg?height=456&width=519&top_left_y=1145&top_left_x=467)

Since the two truthful gnomes sit next to each other, there should be two identical answers in a row. But this is not the case, so the truthful gnomes must be either the first and sixth gnomes, or the fifth and sixth gnomes.

However, opposite each truthful gnome sits a liar, so the fifth gnome cannot be truthful (since the second gnome gave the same answer as him). Therefore, the first and sixth gnomes are truthful. This means the gold is hidden in the cave.

Criteria

16. The correct answer is provided.

2 b. The arrangement of truthful, ordinary, and lying gnomes is indicated, along with the correct answer.

4 6. The correct answer and justification are provided.