13. The Knave of Hearts tells only the truth on Mondays, Tuesdays, Wednesdays and Thursdays. He tells only lies on all the other days. The Knave of Diamonds tells only the truth on Fridays, Saturdays, Sundays and Mondays. He tells only lies on all the other days. On one day last week, they both said, "Yesterday I told lies."
On which day of the week was that?
A Sunday
B Monday
C Tuesday
D Thursday
E Friday
Solution
E

A day on which one of the Knaves says "Yesterday I told lies." is either a day when he is telling the truth but the previous day he was lying, or else a day on which he is telling lies and the previous day he was telling the truth.

So for the Knave of Hearts that day would have to be a Monday or a Friday. Likewise, for the Knave of Diamonds that day would have to be a Friday or a Tuesday.

Therefore a day on which they both said "Yesterday I told lies." is a Friday.