While Gwen and Sybil are talking in the back of the auditorium at the flower show Mary is seen in the background talking to another lady. Right after that she is in front of the crowd listening to her grandmother give the Grantham Cup to Mr. Mosley. Then right after that she goes up to Matthew in the back of the auditorium.
Matthew is angry at Mary after dinner and leaves abruptly. Mary see's him leave and follows immediately behind him. It takes her a mere 9 seconds to reach the front hall, and Matthew has already donned his large coat, his hat, and has left through the front door. He is 20 feet away walking home when Mary see's him through the door glass. He did all that in the few seconds that she was behind in reaching the front hall. Impossible.
Cora asks Mrs. patmore to make Apple Charlotte for a dinner for Anthony Strallan and says it's a new desert. However Apple Charlotte has been around since the late 1700s and is supposed to be named after Queen Charlotte the wife of King George III.
After breakfast, Robert Crawley tells Mr. Carson, "I'll be in the library, please lady me know when her Ladyship comes down." Yet a few minutes later, Robert walks outside and Cora (her "Ladyship") is sitting on a bench.
Edith Crawley addresses her letter to "His Excellency the Turkish Embassador". At the time depicted the country was known as the Ottoman Empire. It only became the Republic of Turkey in 1922.
When O'Brien and Thomas are having the discussion about who told about Lady Mary and Mr Pamuke she says that Evelyn Napier could not have been the one to do it because he wasn't in on it. However he knew something went on between the two and later tells Mary herself that he knew all about it.
Daisy says "I hate this room" when making the fire in Lady Mary's room. She has a flashback to the scene where Lady Mary and Lady Grantham and Anna are carrying My Pamuk in the way of explaining why she hates the room. But they weren't carrying the dead Mr Pamuk near Lady Mary's room when Daisy saw them, they were carrying him into his guest room which was on the other end of the gallery.