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.
Plot hole: When Poirot and Hastings break into the house, they search the entire house for the box, without success. However, later on, when they are in the kitchen, the housekeeper comes down the stairs - having been lying in wait for them. How did Poirot not run into her when he was initially searching upstairs, particularly when he was making such a thorough search?
The Adventure of the Cheap Flat - S2-E7
Plot hole: The professional hitman from the US does not have a gun of his own, and does not bother in the long time elapsed between the apartment's hit and the nightclub (it could even be hours) to check if the gun he just grabbed is loaded or not. Who in the world would not take a split second of his time to peek at the cylinder and see how many bullets does he have? An assassin planning to kill with that gun, even.
The Disappearance of Mr. Davenheim - S2-E5
Plot hole: The killer has a perfect (in movie logic at least) plan but he amplifies its risks to 1000000% getting caught on purpose with an incriminating item that links him to his secret identity and that is amazingly likely to have him questioned and face the wife of the deceased (who recognizes him in 0.5 seconds) and other witnesses. Even without Poirot it is impossible to imagine a trial happening without him having to appear in some capacity. And in this TV dramatization, the motive of Lowen to hate Davenheim is obvious, but not the other way around, so it appears even more baffling that he'd utterly ruin his perfect escape just to mess with a rival apparently he had a upper hand on. Moreover, the dramatization makes his plan even more absurd, as instead of pawning the valuable ring like in the original and being only marginally involved, here he has it days after the murder, and robs the Inspector chief! Any bobby would do. Finally, here he also wears prosthetics, again drawing attention.
The Disappearance of Mr. Davenheim - S2-E5
Plot hole: The safe is behind a painting that acts like a real door. When Mrs Davenheim spots it, it's wide open. But Japp questioned her in the same room and the painting was in place back then. It really makes no sense that with a story depicted as being so big (it makes the first page of the national newspapers) and with an investigation in place, nobody in a week has checked the safe at all. (00:27:50)
Plot hole: The whole premise of the plot is that nobody is staying at End House but Nick (and the servants); she even says explicitly that there is just one room that is usable and everyone else stays at the Majestic. During the party though, Freddie tells Maggie to go get her coat which is "in my room." They are at End House, not the hotel, how would Maggie have a room there? (00:35:00)
Plot hole: The killer's plan needs to have Maggie at End House - but it's achieved through Poirot's proposal to have Nick call her to act as added protection. But this makes no sense; Maggie's presence would have not protected Nick in any of the alleged attempts to her life; the frame would have crushed her too, she would have died in the car, she couldn't have prevented the rock from falling, and Nick was shot while the most famous detective was with her, even! A random little girl from Yorkshire would have not mattered, so it makes no sense that Poirot endangered someone for no gain.
