## 83. Another Case

"I remember it as if it were yesterday," the White Knight began, "I once attended a trial that the Baboon couldn't make it to. As is often the case with us, there were three defendants, and only one of them was guilty. When the first defendant was asked if he pleaded guilty, his answer was brief: either 'yes' or 'no,' though I can't recall which. Then the second defendant was asked if he pleaded guilty. His answer was also one-word: either 'yes' or 'no,' though I also can't remember which. Then the third defendant was asked whether the first defendant was guilty or not, and the third defendant either said that the first defendant was not guilty or claimed that the first defendant was guilty. But what exactly the third defendant said has faded from my memory. Do you have any idea who is guilty?"

"Of course not!" Alice replied.
"I also couldn't determine who was guilty," the White Knight admitted. "But I remember something else: I forgot who among the defendants was telling the truth and who was lying, but I clearly remember that at least one of them was telling the truth and at least one was lying. Can you determine who is guilty now?"
"Of course not!" Alice replied.
"I also couldn't determine who was guilty," said the White Knight, "but if it helps you at all, let me tell you that last week I met the Baboon, who started asking me about the trial. At that time, I still remembered what each defendant had said, and I recounted their statements to the Baboon. I also told the Baboon that at least one of the three defendants was telling the truth and at least one was lying. Upon learning this, the Baboon, through logical reasoning, managed to find the guilty party.

Now Alice (and, therefore, you, the reader) has enough information to solve the problem.

Who is guilty?
83. Another case. There are 8 versions of the testimonies given by the defendants \(\boldsymbol{A}\), \(\boldsymbol{B}\), and \(C\) during the trial. Indeed, \(A\) could give two versions of his testimony, each of which could be combined with two versions of \(B\)'s testimony, resulting in 4 versions of the testimonies of \(A\) and \(B\). (Let's list these versions: 1) \(A\) and \(B\) both admitted guilt; 2) \(A\) admitted guilt, \(B\) claimed innocence; 3) \(A\) claimed innocence, \(B\) admitted guilt; 4) \(A\) and \(B\) both claimed innocence.) Each of the four versions of the testimonies of \(A\) and \(B\) corresponds to two versions of \(C\)'s testimony, so the total number of testimonies of \(A\), \(B\), and \(C\) reaches 8.

In each of the 8 versions of the testimonies, any of the three could be guilty (at least in principle). Therefore, the total number of "scenarios" (by which we mean a set of each of the three defendants' testimonies and their actual guilt or innocence) reaches 24. Of course, if we knew which of the 24 versions corresponds to reality, we would know who was lying and who was telling the truth. Therefore, let's compile a summary table of all 24 scenarios. This table will be useful not only for solving this problem but also for one of the following problems. All necessary explanations are provided after the table.

| Case | What the defendants said | \(A\) guilty | \(B\) guilty | \(C\) guilty |
| :---: | :---: | :---: | :---: | :---: |
| 1 | \(A\): I am innocent | L | T | t |
|  | \(B\): I am innocent | t | L | t |
|  | \(C\): \(A\) is innocent | L | t | t |
| 2 | \(A\): I am innocent | L | t | T |
|  | \(B\): I am innocent | t | L | T |
|  | \(C\): \(A\) is guilty | T | t | T |
| 3 | \(A\): I am innocent | L | t | T |
|  | \(B\): I am guilty | L | t | L |
|  | \(C\): \(A\) is innocent | L | t | t |
| 4 | \(A\): I am innocent | L | T | t |
|  | \(B\): I am guilty | L | T | L |
|  | \(C\): \(A\) is guilty | t | L | L |
| 5 | \(A\): I am guilty | T | L | L |
|  | \(B\): I am innocent | t | L | t |
|  | \(C\): \(A\) is innocent | L | t | t |
| 6 | \(A\): I am guilty | T | L | L |
|  | \(B\): I am innocent | t | L | t |
|  | \(C\): \(A\) is guilty | t | L | L |

![](https://cdn.mathpix.com/cropped/2024_05_21_5afa4e43618f5800e92dg-176.jpg?height=434&width=1340&top_left_y=254&top_left_x=358)

The letters "L" and "T" (from the words "Lies" and "Truth") indicate whether the corresponding defendant is telling the truth (T) or lying (L). In case 5B (at the intersection of row 5 and the column "B guilty"), we see that \(A\) is lying, \(B\) is lying, and \(C\) is telling the truth. (By case 5B, we mean the scenario where \(A\) admitted guilt, \(B\) claimed innocence, \(C\) stated that \(A\) is innocent, and in reality, \(B\) is guilty.

Other examples: in case 8C, all three defendants lied; in case 3B, all three told the truth; in case 4C, defendant \(A\) told the truth, while \(B\) and \(C\) lied.

Barmaglot, after learning what each defendant said and that at least one statement was true and at least one was false, was able to determine who was guilty. What information, which the White Knight could have provided, allowed Barmaglot to determine who was guilty? Suppose Barmaglot learned from the White Knight that \(A\) claimed innocence, \(B\) claimed innocence, and \(C\) claimed \(A\) was innocent (thus we are within case 1). With this information, Barmaglot could have excluded the guilt of defendant \(C\) (since in case 1C, all three defendants lied), but perhaps he could not have determined who was guilty: \(A\) or \(B\) (since in case 1C, at least one statement was true and at least one was false; a similar situation is observed in case 1B). Therefore, the White Knight could not have provided Barmaglot with this information (since Barmaglot determined who was guilty). How would things have been in case 2 (where \(A\) claimed innocence, \(B\) claimed innocence, and \(C\) claimed \(A\) was guilty)? In this case, Barmaglot could not have determined who was guilty (since both case 2A and case 2B could have occurred). We encounter a different situation in case 3, where at least one true and at least one false statement are possible only in sub-case 3C. Therefore, if the White Knight had informed Barmaglot that \(A\) claimed innocence, \(B\) claimed innocence, and \(C\) claimed \(A\) was innocent, Barmaglot, through reasoning, would have concluded that \(C\) was guilty. Therefore, it is not excluded that the White Knight provided Barmaglot with this information. By analyzing the other cases (4, 5, 6, 7, and 8), the reader will find that only in case 6 (in addition to the already known case 3) could Barmaglot have determined who among the three defendants was guilty. As in case 3, the guilty party would be defendant \(C\).

Thus, regardless of whether the testimonies the White Knight told Barmaglot belonged to case 3 or case 6, in both cases, the guilty party (by a fortunate coincidence) would have been defendant \(C\).