On the island, there live two kinds of natives: Truth-tellers, who always speak the truth, and Liars, who always lie. When a stranger met three natives, Alan, Bruno, and Ctibor, he asked them which group they belonged to.

Alan stated: "Bruno is a Liar."

Bruno said: "Alan and Ctibor are either both Liars or both Truth-tellers."

Ctibor did not speak.

Could the stranger determine with certainty whether any of these natives was a Truth-teller or a Liar? 

(M. Volfová)
First, let's assume that Alan is a Truth-teller. In such a case, his statement would be true, and Bruno would be a Liar. Bruno's statement would therefore not be true, which would mean that Alan and Ctibor belong to different groups. Since Alan is a Truth-teller, Ctibor would be a Liar.

Now, let's assume that Alan is a Liar. In such a case, his statement would not be true, and Bruno would be a Truth-teller. Bruno's statement would therefore be true, which would mean that Alan and Ctibor belong to the same group. Since Alan is a Liar, Ctibor would also be a Liar.

$S$ can be determined with certainty that Ctibor is a Liar.

Another solution. Consider all possible cases, where for each of the three natives $(A, B, C)$, we consider each of the two cases $(\mathrm{P}, \mathrm{L})$. In total, we get eight possibilities, which we will compare one by one with the statements of Alan and Bruno. Any contradiction with one of these statements is marked in the last row of the table:

| A | P | P | P | P | L | L | L | L |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| B | P | P | L | L | P | P | L | L |
| C | P | L | P | L | P | L | P | L |
| contradiction with | A | A B | B |  | B |  | A | A B |

With certainty, it can be determined that Ctibor is a Liar.

Evaluation. 2 points for the correct conclusion; 4 points for the quality of the commentary.