50. Another Such Interesting Case

- And here's another such interesting case, the King continued. - Regarding the three accused $A, B$, and $C$, the court once again knows that one of them is a knight, another is a liar, and the third is a spy. Opening the trial, the judge stated to the accused:
- I will now ask you a series of questions. You must answer each question briefly - 'yes' or 'no'. As soon as it becomes clear to me who among you is the spy, I will immediately expose him, and the trial will end. As soon as it becomes clear to me that someone is definitely not the spy, I will immediately release the innocent party without waiting for the trial to conclude.

Then the judge asked the defendant $A$:

$3-3187$

- Are you the spy?

$A$ answered, after which the judge asked the defendant $B$:

- Did $A$ tell the truth?

$B$ answered. The judge thought for a moment and asked the defendant $C$:

- Are you the spy?

$C$ answered, and the judge immediately exposed the spy. Who is the spy?

- Just a moment! - Alice exclaimed. - You didn't tell me what the accused answered to the judge.
- You're right, - the King agreed, - but this problem is special: it can be solved even without knowing who of the accused answered 'yes' and who 'no'.

So, who is the spy?
50. An equally interesting case. Since the judge asked defendants $A$ and $B$ the same questions as in the previous problem, we can use the familiar table.

Consider the moment in the trial when the judge asked defendant $C$ whether he was a spy. At this moment, the judge could not assert that any of the three defendants was definitely not a spy, since otherwise the judge would have had to release an innocent person from custody. Thus, cases 1 and 2 are ruled out, as in each of them the judge would have known that $C$ was either a knight or a liar, and would have released $C$. Therefore, we are left to consider cases 3 and 4.

How could the judge have reasoned after hearing the answer from defendant $C$? In case 3, the judge knows that $C$ is either a spy or a knight. If $C$ answered "no" to the judge's question, the judge would not have learned anything new and could not have incriminated anyone. But if $C$ answered "yes," the judge could confidently assert that $C$ was a spy, as a knight could not have said that he was a spy. Thus, in case 3, $C$ was incriminated as a spy.

In case 4, the judge knows that $C$ is either a spy or a liar. If $C$ answered "yes," the judge could not assert that $C$ was a spy (since a liar and a spy could both answer "yes"). But if $C$ answered "no," the judge would have established that $C$ was a spy, since a liar cannot tell the truth and admit that he is not a spy. Thus, in case 4, $C$ would also have been incriminated as a spy.

It is interesting to note that neither you, dear reader, nor I can say which of the two cases (3 or 4) actually occurred, nor can we know what $C$ answered ("yes" or "no") to the judge. We only know that the judge was able to determine who among the accused was the spy, so either everything happened as in case 3, and $C$ answered "yes," or everything happened as in case 4, and $C$ answered "no." In either case, $C$ was incriminated as a spy, so we can confidently say that $C$ is a spy.