Plot hole: The hole in the hat was pre-existing and the bullet was fired in advance and dropped. Poirot notices immediately the hole when Nick puts the hat down but somehow missed it while she was wearing it as she supported him. He also does not notice that the bullet has not been fired at that time - which would have been still hot if just fired, not at all if dropped. Most importantly, in the spot where they were sitting, it's not even possible to think where a shooter could have ever been hidden; nobody wonders where the mysterious shooter could have been or cares to investigate the circumstance despite knowing the very moment the supposed shot happened. The fact is basically ignored throughout the rest of the episode.
Character mistake: In the hotel lobby, Poirot, indignant of Hastings' attitude towards the murder case, says "After three attempts had been made at the life of M. Nick, Poirot gave his word he would protect her." But he did that after four attempts; Nick already avoided 3 "mysterious accidents" (painting, car, boulder) and Poirot witnessed the 4th, the bullet that pulled him into the whole case to begin with. (01:02:20)
Continuity mistake: When Mr. Vyse is beginning to talk about the new will, in the wide shot encompassing the whole table, the open folder is raised, as the left page is sitting on top of another object. In the close-up of the letter, the folder is perfectly level with the table. (00:34:45)
Factual error: Nick says that she made the will 6 months before the events of the episode, just before her appendix surgery. The surgery happened on February 27th, so it would be the end of August now. But at the beginning of the first episode Hastings was reading a (movie prop) copy of the Daily Express which had on the cover "Perry makes a new record - Champion two years in succession", complete with a picture where he is unmistakably with his rival Gottfried Von Cramm. The photo (and the mention of the 2 years record) is about the Wimbledon final who took place on 5 July 1935. That's almost 2 months earlier than it should be. (00:04:45)
Continuity mistake: Once Ghost Nick enters the room and Poirot turns on the lights, the Crofts stare at her, and you can see in frame the right arm of "Mne LeMon", sitting by Milly. Next shot though, Miss Lemon is standing on her feet. (00:40:10)
Continuity mistake: Commander Challenger in front of his boat laughs with Hastings "I hope my alibi is in order." As he says that, Hastings nods. During this close-up, his hat is casting a completely different shadow, covering his whole shoulder instead of just the back of the neck as in the rest of the scene. (01:03:35)
Continuity mistake: Jim Lazarus pulls over by the protagonists to complain about being unable to see Nick. During the scene, Paul Geoffrey's different facial expressions are shown in the side mirror of the car, but each time we cut back to him, he has the same stoic face listening to Poirot. (00:47:00)
Continuity mistake: Poirot rudely awakens Hastings with his three questions. You get a good view of Hastings' wristwatch on the bedstand. When at the end of the Q&A Hastings turns his attention to the watch, it is in a completely different position. (00:41:25 - 00:42:35)
Continuity mistake: Poirot and Hastings are talking with Freddie, by her invitation. She says "You'll be giving me a diploma next" and three extras walk by her chair. In the following shot she completes the action of standing up, but the extras are still there, behind their original positions. (00:44:15)
Continuity mistake: When Freddie holds the Mauser 1910 gun, Alison Sterling holds it in a very characteristic way (thumb and and pinky finger stick out) in the front views. It is held in a much more orthodox way in the views from behind. (00:43:50)
Continuity mistake: Starting the seance, Poirot ask everyone to hold their hands. Lazarus barely stifles a laughter. Between that shot and the previous, his hand goes from above the table to underneath it. (01:38:25)
Continuity mistake: Miss Lemon's ice cream quite literally melts between shots (and the actress ignores the dripping on her fingers). (00:51:15)
Continuity mistake: When Poirot shows Freddie the bullet, in the close-up his hand is right by the sugar bowl, but he was shown putting the cigarette holder at greater distance, closer to Freddie herself. (00:25:05)
Audio problem: Nick gets supposedly grazed by a bullet fired while the plane's noise has already diminished considerably, making quite unlikely that any gunshot could be drowned out by it. (00:06:50)
Continuity mistake: Poirot shows Hastings Nicki's hat; he holds it in a different way (using the thumb to pinch, or with the light brim tucked in between index and middle) between shots. (00:08:55)
Continuity mistake: Poirot and Hastings are eating after their first visit to End House. During the entire scene, natural light changes in intensity depending on the angle of filming. (00:20:15 - 00:21:10)
Continuity mistake: During the sentence when Hastings tells Poirot he's got "No imagination at all", the lady sitting behind the Belgian faces the other way between shots. (00:02:25)
Continuity mistake: When Japp talks to Poirot and Hastings about the gallery troubles, the peppermint rock candy he is waving is wrapped differently between shots. (01:13:40)
Continuity mistake: Poirot meets Hastings about to go play golf with a Stiffie. Hastings holds his hat by the brim, which turns into the crown in the reverse shot. (01:02:05)
Factual error: Jim Lazarus drives a Vauxhall DX (14/6 Light Six) Stratford Tourer, a 1936 model. This episode takes place in the summer of 1935. (00:46:20)
