## 79. The Second Court Report

- Well, I've been to the most interesting court cases! - said the White Knight after Alice solved the previous problem. - What cases were heard there!
- Please tell me about another trial, - Alice asked.

The task that followed was the most interesting of all that Alice had ever heard.

- Alright, - agreed the White Knight. One of the most interesting trials was last month. There were also three defendants, and only one of them was guilty. The first defendant was the first to speak, then the second defendant spoke, and after him, the third.
- And what did they say? - asked Alice.
- I can't quite remember, - replied the White Knight, but I do remember that each of the defendants accused one of the other two. But who accused whom, I can't recall. Can you find out who is guilty?
- Of course not! - said Alice. - Since you've told me practically nothing! Maybe you can remember at least who of the accused was lying and who was telling the truth?

It's very interesting that you asked that, the White Knight perked up. - Two weeks ago, I told the White King about this trial, who couldn't make it to the court, and the White King asked me the same question. When I told him who of the three defendants was lying and who was telling the truth, he managed to deduce who was guilty through reasoning.

- So who was lying and who was telling the truth? - asked Alice.
- Alas! I don't remember that now! - replied the White Knight.
- What's the point of tackling a hopeless task? - said Alice sadly.
- How glad I am that you said that! - rejoiced the White Knight. - The same thing happened to me last week. I met Shaltay-Boltay and told him about this trial. I told Shaltay-Boltay that a week before meeting him, I had met the White King and told him about the trial, and that the White King, upon learning who of the defendants was lying and who was telling the truth, managed to identify the guilty party. Shaltay-Boltay also
asked me who of the defendants was lying and who was telling the truth, but by that time I had already forgotten and couldn't answer him. Then Shaltay-Boltay said:
- I think I shouldn't even try to solve such a task. It's a hopeless case!
- And Shaltay-Boltay didn't solve the problem? - asked Alice.
- Well, how could he! He solved it! Shaltay-Boltay asked me another question, and as soon as I answered, he was able to solve the problem.
- And what was that question? - asked Alice, burning with impatience.
- Unfortunately, I've forgotten it, - replied the White Knight.
- It's not very easy to get the necessary information out of you, - complained Alice. - Do you remember anything about this question?
- Of course! - replied the White Knight. - I remember clearly that Shaltay-Boltay either asked me if any two statements in a row were true, or if any two statements in a row were false, but I don't remember which of these two questions interested him. I don't remember what I answered him.

Who of the three defendants is guilty?

![](https://cdn.mathpix.com/cropped/2024_05_21_5afa4e43618f5800e92dg-108.jpg?height=909&width=917&top_left_y=1590&top_left_x=612)
79. The second court report. What could the White King have learned from the White Knight that allowed him to identify the guilty party? If the White Knight had told the White King that all three defendants had given false testimony, then the White King would not have been able to find the guilty party. Indeed, $A$ could have been guilty and accused $B$, while $B$ and $C$ could have accused each other (in which case all three would have been lying); or $B$ could have been guilty and accused $C$, while $A$ and $C$ could have accused each other (in which case all three would have been lying again); or $C$ could have been guilty and accused $A$, while $A$ and $B$ could have accused each other. Therefore, the White King could not have learned from the White Knight that all three defendants lied in court.

Could the White King have identified the guilty party if the White Knight told him that exactly two of the defendants lied in court and pointed out who lied? No. Suppose, for example, that the White Knight told the White King: “$A$ spoke the truth, $B$ and $C$ lied.” Then whoever $A$ accused must have been guilty (since $A$ spoke the truth), and $B$ and $C$ both lied and accused $A$ (or perhaps $B$ accused $C$, and $C$ accused $A$). On the other hand, $A$ could have accused $C$, and defendants $B$ and $C$ could have accused $A$, in which case $C$ would have been guilty. Thus, if $A$ was the only defendant who told the truth in court, then neither $B$ nor $C$ could have been guilty. Similarly, if $B$ was the only defendant who told the truth in court, then neither $A$ nor $C$ could have been guilty, and if $C$ was the only defendant who told the truth in court, then neither $A$ nor $B$ could have been guilty. Therefore, if the White Knight told the White King that only one of the defendants (either $A$, $B$, or $C$) told the truth in court, then the White King would not have been able to identify the guilty party. This means that the White Knight did not tell the White King that only one of the defendants (either $A$, $B$, or $C$) told the truth in court.

Could the White Knight have told the White King that all three defendants told the truth in court? No, this is impossible, as the guilty party was certainly lying (since they accused one of the other two defendants, who were innocent).

The only remaining possibility is that exactly one defendant lied in court. But if exactly one defendant lied, then that one must be the guilty party, since if an innocent party had lied, there would have been two false testimonies: one innocent and one guilty. Therefore, the White King could have heard one of three versions from the White Knight.

Version 1. $A$ lied, $B$ told the truth, $C$ told the truth.

Version 2. $A$ told the truth, $B$ lied, $C$ told the truth.

Version 3. $A$ told the truth, $B$ told the truth, $C$ lied.

Now we understand how the White King identified the guilty party, but how can we determine which of the three versions the White Knight reported to the White King? As is known, Shal-tay-Bol-tay either asked the White Knight whether the testimonies of two defendants in a row were false, or whether the testimonies of two defendants in a row were true. The first question would not have made sense (since out of three testimonies, only one is false), as the White Knight would have answered negatively, and this would not have allowed Shal-tay-Bol-tay to determine which of the three versions the White Knight reported to the White King. Therefore, Shal-tay-Bol-tay asked whether the testimonies of two of the defendants in a row were true. If in response to his question, the White King said "yes," then Shal-tay-Bol-tay would have excluded version 2, but still would not have been able to determine the guilty party. But since Shal-tay-Bol-tay determined who was guilty, this means that in response to his question, the White Knight answered "no." Such an answer allowed Shal-tay-Bol-tay to understand that version 2 is the only possible one. Therefore, the guilty party is defendant $\boldsymbol{C}$.