Alice is lost in the woods and can't remember what day of the week she is. There he meets the white rabbit that lies on Mondays, Tuesdays and Wednesdays and with the invisible cat that lies on Thursdays, Fridays and Saturdays. The rest of the days of the week both always tell the truth.
Alicia asks both of them What day is it today? to which the rabbit responds: "yesterday I had to lie", while the cat says: "I also had to lie yesterday".
What weekday is today?
The only days the rabbit can say “Yesterday I lied” are Mondays (where he lies) and Thursdays (when he tells the truth). For any other day of the week, there are two consecutive days (today and yesterday) on which either lies both days or tells the truth both days.
If he had to lie both days (today and yesterday), he would be telling the truth, which is not possible and if he had to tell the truth both days he would be lying which is not possible either.
For a similar reasoning we see that the cat can only pronounce the phrase “Yesterday I had to lie” on a Thursday or Sunday so that the only day on which both could answer this is the Thursday.