Beaver and Smithy are giants who speak only the truth on some days and only lie on other days. Beaver speaks the truth only on weekends, while Smithy speaks the truth on Mondays, Fridays, and Sundays, and lies on the other days.

One day, Beaver said: "We both lied yesterday."

However, Smithy disagreed: "At least one of us spoke the truth yesterday."

On which day of the week could the giants have such a conversation?

(M. Volfová and V. Žádník)

Hint. Use a clear table to help you see when each giant lies and when they do not.
Information from the assignment for clarity is entered into the table:

|  | Beaver | Koloděj |
| :---: | :---: | :---: |
| Monday | - | + |
| Tuesday | - | - |
| Wednesday | - | - |
| Thursday | - | - |
| Friday | - | + |
| Saturday | + | - |
| Sunday | + | + |

Beaver can claim "we both lied yesterday" either on a day when he is telling the truth and at the same time both giants lied the previous day (which cannot happen), or on a day when he is lying and at the same time at least one of the giants spoke the truth the previous day (i.e., on Monday or Tuesday). Therefore, Beaver can say "we both lied yesterday" only on Monday or Tuesday.

Koloděj can claim "at least one of us spoke the truth yesterday" either on a day when he is telling the truth and at the same time at least one of the giants spoke the truth the previous day (i.e., on Monday or Sunday), or on a day when he is lying and at the same time neither of them spoke the truth the previous day (i.e., on Wednesday or Thursday). Therefore, Koloděj can disagree with "at least one of us spoke the truth yesterday" only on Monday, Wednesday, Thursday, or Sunday.

The only day when the giants can have the mentioned conversation is therefore Monday.