## 85. And what would you say?

- I remember one trial in great detail, - the White Knight told Alice. - I remember clearly that there were three defendants and that only one of them was guilty. I remember distinctly that the first defendant accused the second, and the second confessed to being guilty. As for the third defendant, he either confessed to being guilty or accused the first defendant, but I can't tell you which.
- Some time ago I told this problem to Humpty Dumpty, - the White Knight continued, - and he asked me how many of the three statements were true. I don't remember what I answered him, but after I answered, Humpty Dumpty was able to solve the problem.

Who among the defendants is guilty?
85. And what would you say? Suppose the White Knight told Humpty Dumpty that all three defendants were lying. Then Humpty Dumpty could not have preferred one of the two options: either $C$ is guilty and accused $A$, or $A$ is guilty and accused $C$ (since in both cases all three defendants were lying).

Humpty Dumpty could not learn from the White Knight that all three defendants were telling the truth, because all three could not have been telling the truth (since both $A$ and $B$ accused $B$, while $C$ accused someone else).

If the White Knight had told Humpty Dumpty that exactly two of the defendants were lying, then he would have known that the defendants $A$ and $B$ were lying (because if either of them had told the truth, the other would also have told the truth), and $C$ was telling the truth. But then either $C$ accused himself and was guilty, or $C$ accused $A$ and $A$ was guilty, and it would have been impossible to determine which of these two scenarios was true. Therefore, in this case, Humpty Dumpty could not have determined which of the defendants was guilty.

Humpty Dumpty could have determined the guilty party only in one case: if the White Knight told him that exactly two statements were true. This would mean that $A$ and $B$ both told the truth (since their statements are consistent, if one were false, the other would also be false, but then we would have two false statements), and $C$ lied. Since $A$ and $B$ both told the truth and accused $B$, defendant $B$ must be guilty.