Continuity mistake: When the mysterious Aussie lawyer gives Miss Dunn his card, he hands it to her with the fingertips pinching the paper in the top corner, while in the reverse shot he's holding it by the bottom corner. Also, just a moment later she gives it back to him using her right hand. Unfortunately, the card was in her left hand. (00:30:20)
Other mistake: Miss Lemon reminds Poirot of the dentist appointment he has at 11. Poirot then receives a phone call from Japp. It turns out during the investigation that the police broke into the apartment at 10:45. Considering that at least a preliminary examination needed to be made and at the time of the phone call Japp was in his office and Poirot was still comfortably having his tisane, everything leads to believe that the dentist appointment should have been later in the day (or the events at the Mews sooner). (00:03:30)
Revealing mistake: Towards the end of the preliminary murder scene inspection, look at the victim in the last close-up, following the part when Poirot picks up paper from the basket. Specifically, the gun hand: the pinky finger of the supposedly dead person, after being too long in the unnatural position, trembles a little. (00:07:30)
Continuity mistake: During the conversation with Laverton-West at the pool, in the close-ups of Poirot and Japp there are always two gentlemen chatting poolside near the bench where the politician dropped the towel. In wider shots, that's not always the case - plus, one of them walks away without a towel but he's back wearing it. (00:27:50 - 00:28:50)
Factual error: As Japp and his moustached pal visit the nightclub in search of Major Eustace, the song "Hindustan" is executed. While it is a song perfectly believable for the year (stated as being 1935) since it was written in 1918 and a huge hit, the version played here is quite a bit different, sounding a lot more like the upbeat version heard first in the Rosemary Clooney-Bing Crosby album "Fancy Meeting You Here", which had other classics reworked for the duet formula with added lyrics - exactly like in the back-and-forth heard here between the singer and the band. Said album came out in 1958, though. (00:30:20)
Character mistake: When the ever gallant Poirot meets Mrs. Waverly, David Suchet hesitates for a second awkwardly as he mistakenly went for a handshake when Julia Chambers offered him a limp downturned hand instead. Not quite the kind of social faux pas Poirot would be caught into. (00:11:55)
Plot hole: Forgiving the implausibility that nobody checks their own watch at the time of the kidnapping, the clock in the living room strikes 12 supposedly ten minutes early: everyone blindly storms from the room when the chimes have not yet ended, and reaches the supposed perpetrator who is by the side entrance. They talk to him briefly and the clocktower signals now the real noon. It's hard to see how 10 full minutes could have passed, and even harder when everybody runs back to the house, to find the clock there signaling 12:11: it took them a mere minute to get there, what took them so long the first time? (00:30:00 - 00:31:30)
Plot hole: When he is taken away from the mansion, Johnnie is hammering with his fists on the car windows, as if he was genuinely in distress and resisting the kidnapping. But as we know, that's not what is happening, he knows his captor well and came willingly. (00:31:25)
Plot hole: In the original story, it is made clear that Mrs. Waverly is the one in the family with the wealth, but it does not go as far as having everyone, guests included, eat plain rice and boiled potatoes, and have the mansion dilapidated with restoration work left unfinished for years. This creates several plot holes: nobody would be interested in asking lavish amounts of ransom money to people who live an impoverished life, and most importantly, the husband would never be able to justify being able to have enough money to suddenly resume construction work on the house (in the original, he simply wanted to use it as spending money).
Continuity mistake: When the milkman places the third bottle on Mr. Gascoigne's steps, they are placed in the sun. When the neighbour approaches, the whole road shows no particular sunlight. Which is fine, as a sudden change of that kind is ordinary. When the other neighbour pops by the corner and inquires, there's a close-up of the bottles, and again there's visible and directional light, pointed differently than the first instance. (00:08:45)
Factual error: Great accuracy went into the dates of this episode - Hastings is following the outcome of the so called "Verity's Match", the 2nd test of the 1934 Ashes series, Australia vs England. The events then should happen between the 22nd and the 25th of June, 1934, compatible with the murder happening on the 16th, and being discovered 3 days after. This however puts it a year before "Murder in the Mews", previous episode where Poirot's dentist was referenced, and that happened in 1935 as stated in the letter to the Chinese laundry.
Plot hole: In the denouement, Poirot says explicitly that the culprit sent the letter to the victim - but the letter in question, in this dramatization was stated earlier (in a change from Agatha Christie's original) to be an invitation to the art gallery, and the culprit is not the manager/art gallery owner! In the actual short story the letter was a personal message of entirely different nature, written and authored by the culprit.