"The day must be a Thursday!"

Little story behind the title! 

Lion lies on Mondays, Tuesdays, and Wednesdays and The Unicorn, on the other hand, lies on Thursdays, Fridays, and Saturdays.


That is on Sundays both must be telling the truth. 

Suppose Lion and Unicorn made those statements today.

Lion - “Yesterday I was lying,”  

Unicorn - “So was I,”  (“Yesterday I was too lying,” ) 

If it was Sunday today, then Lion's statement would have been lie as lion tells truth on Saturdays. But as per data, both must be telling the truth on Sundays. So it can't be Sunday today.

For rest of all days, one must be telling the truth and other must be lying.

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CASE 1 : Lion is lying and Unicorn is truth teller.

For Unicorn's statement - “Yesterday I too was lying,” to be true it must be Sunday today. But on Sunday, lion also speaks truth. And lion's statement can't be true on Sunday as concluded earlier. 

Hence, today must be the case below.

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CASE 2 : Lion is truth teller and Unicorn is lying.

Again for Lion's statement - “Yesterday I was lying,” to be true it must be Thursday today. 

And if today is Thursday Unicorn is lying with it's statement - "“Yesterday I too was lying,” as it was Wednesday yesterday where Unicorn always tells truth on Wednesday. 

Hence, today, on Thursday, Unicorn must be lying with his statement while Lion is telling the truth. Both are as per behaving the given data.