2) In a conversation between two mathematicians, the first says to the second: "I lied yesterday." The other replies: "I also lied yesterday." Knowing that one of them lies on Mondays, Tuesdays, and Wednesdays (and only on these days), while the other lies on Thursdays, Fridays, and Saturdays (and only on these days), on which day of the week did the conversation take place?

$\begin{array}{llll}(\mathbf{A}) \text { Monday } & (\mathbf{B}) \text { Thursday } & (\mathbf{C}) \text { Sunday } & (\mathbf{D}) \text { such a conversation cannot }\end{array}$ have taken place (E) it is not possible to determine the day uniquely.
2. The answer is $\mathbf{( B )}$.

Let A be the mathematician who lies on Monday, Tuesday, and Wednesday, and B the one who lies on Thursday, Friday, and Saturday. The conversation could not have taken place on Monday because B, who does not lie on Monday or Sunday, would have lied. For the same reason, the conversation could not have taken place on Tuesday or Wednesday. Similarly, we can exclude Friday, Saturday, and Sunday because A, who does not lie on these days, would not be telling the truth. The only possible day is Thursday, when the conversation could indeed have taken place: A is telling the truth (he lied the previous day) and B is lying (he did not lie the previous day), which is consistent with the problem's assumptions.

[Problem proposed by F. Mugelli]