Plot hole: Can't fault this massive plot hole to the adaptation, but to the source material; the culprit (forgetting the stupidity of writing an incriminating letter detailing the plan to murder someone, and put it in a desk he shares with her) since there are people outside the room that are about to enter, tears the letter in 3 neat vertical strips, rolls them, puts them in the vase on the mantlepiece, and then opens the side door to slip away...instead of simply pocketing the letter and going through that same door. Nobody was going to search him or anything and could have burned it, torn it into confetti, anything, later. It takes way way longer to do what he did, which needed him to stay there in the room increasing the chances of being found out. And of course he and his accomplice do not retrieve the letter after.
Factual error: Attention to detail in props is always extremely high in this series, and tubular flashlights have been in circulation since the beginning of the century. However the one that the supposed burglar is holding as they make their way through the top floor of the villa looks perfectly modern and unlike any model compatible with the 30s. (00:10:20)
Continuity mistake: Poirot is holding up a letter in an envelope. He turns it to show the front to Lady Mayfield. The flap of the envelope appears to be tucked in. The shot cuts to a close-up, but now the back of the envelope is facing Lady Mayfield and the flap is sealed. He then turns the envelope to show her the front. (00:44:20)
Plot hole: Since the Countess was the only person to be at every dinner when the jewelry thief was in action, it takes a very special kind of idiot to not identify her as the culprit. Japp here did not need at all Poirot's acumen, but simple due diligence cross-checking the guest lists, something there is absolutely no reason he wouldn't do, and yet she is never treated as a special suspect. Also, Japp's job is described as being in jeopardy after the first 3 thefts, a 4th happens, he does not quite solve it (but retrieves at least the necklace), but he's off the hook despite the thief being unidentified, at large and with still the jewelry stolen from the first 3.
Continuity mistake: When Poirot pulls out of the drill hidden in the handle, he holds it triumphantly in his right hand. In the next shot he mounts it on the handle starting the motion with the drill already in the left hand and the rest of the gimlet in the right. (00:40:00)
Plot hole: Nobody hears the sound of a gun being fired past a door they were waiting almost in front of, and the police cannot tell apart a shot fired point blank by one fired 20 feet away and probably at a very sharp angle. Moreover, the bleeding should be all over his face, since leaning the way it is shown in this adaptation is most likely to lead the victim to fall over, and even leave bloodstains out of the window and on the ground below, which someone would have noticed in the crowded factory.
Factual error: Throughout the episode there are sections of dialogue in Greek, which are deliberately left untranslated. The subtitles frequently transcribe the Greek inaccurately. For example, the Greek girl who assists Poirot and Lyall in finding out about the poison says "Elate!" to them, which means "Come!" said to more than one person, but this is transcribed in the subtitles as "Ella!", the form used to a single person. Later on, the girl's grandmother says "Ohi enas Anglos anthras. Mia Anglitha yineka. San esena" ("Not an Englishman. An Englishwoman. Like you"), but this is given in the subtitles as "Ohi enas Anglos anthras, che yenika. Son ethena", which doesn't make sense. (00:40:45)
Factual error: At the end of the report on the findings of the crime scene, the "Italian" inspector tells to his subordinates what literally would translate as "Removes this meddler from here, no more access to the prisoner, that you understand?", which is just wrong in accent, cadence and construct. (00:36:30)
Deliberate mistake: The day of his scheduled departure, Poirot overhears conversations (mainly the one from the window of his room) he couldn't possibly be hearing given the long distance outdoors and the tone of voice. His reactions are shown as if he could hear and not simply see and infer the meaning. (00:23:00)