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.
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)
Factual error: It is well established that the episode takes place in 1935 (Nairobi Daily Press dated Saturday July 27 1935, the poster in town advertises the meeting for "Today, Wednesday September 4th", day of the week consistent with the year), but Poirot and Hastings are stopped on their way to the train station by a Wolseley Series II - 14/56, a model that entered production in mid 1936. (00:17:00)
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.
Factual error: At the party, Marcus Hardman tells Bernard that the Countess recently arrived from Russia, and she describes herself as being in exile. Which made sense in the source material, set right after the Russian Revolution, but less sense in this adaptation, set in the mid 1930s. If she stayed in Russia that long, she would have spent 15-20 years with zero privileges from her rank at that point, and nothing from her old wealth, seized by the communist government.
Other mistake: Investigating Poirot's investigation, the clumsy constable crushes a tiny egg under his foot. Poirot crawls to him and points with his cane at the other eggs in the bush. But he points practically at his shoe, and the eggs are in a different spot to the left. (00:20:10)
Plot hole: The murderer is in possession of chloroform from the doctor's cabinet, but there's no realistic timeframe when they could have done it, nor any explanation why the doctor notices it completely at random when Poirot visits, just approaching the medicine cabinet, something he routinely opens and looks through during the day.
Continuity mistake: The Countess arrives at the hotel at the beginning of the episode. In the overhead shot she reaches the top of the stairs with nobody close to her, just the porters behind her, but in the next shot there are other people in the lobby, who at the next cut are even closer to her to fit in frame. (00:02:10)
Plot hole: The murderer planned everything making it seems accidental. They perfectly managed to fool the coroner and need only to be alone at the mansion to get rid of the murder weapon. So what they do is...they go through great risks to make everyone know someone tried to murder the wife too, making it obvious that there is a killer at large, which only makes the investigation more stringent and most importantly, keeps the police on the premises for surveillance preventing them from disposing of the weapon (like almost everything in this episode's plot it is not a flaw of the original story, but of the dramatization, who added practically every minor character).
Continuity mistake: At the end of the 'lovely day' with the Countess, Poirot stops a cab and opens the door for her. Just before the cut, notice a woman with a white dress and a brown hat crossing the street in front of the taxi. The extra crosses the street again in the same direction when he says "Oui" in the following shot. (00:28:50)
Continuity mistake: During the exhibition of the much acclaimed contralto, at one point it is shown that Lady Runcorn is going upstairs and is looking at the smoking person. Back to the host who is on his own at his table. At the first cut, suddenly there are more people standing in attendance in the background (pretty much everyone was seated earlier) and his glass has also refilled. (00:10:15)
Continuity mistake: During the party, the colonel expresses to Poirot his disappointment for the jazz record being played, music "from the inferior races." He puts a tart in his mouth, closing his lips past the piece of fruit. Cut to the wider shot, and he is still holding a tart that sports the strawberry he just chewed on. (00:19:00)
Factual error: Poirot is visiting an art exhibition with the Countess, and expresses his admiration for a painting by Marc Chagall. Amazingly enough, that painting is "Les Plumes en Fleurs", something Chagall will create in 1943, years after the time when this pre-WW2 episode takes place. (00:24:50)
Continuity mistake: Poirot enters Japp's office when the inspector is dealing with the tribulations of modern machinery. As Poirot sits down, you can notice the crumpled paper by the phone being different in the reverse shot, when you can also see Japp's hand being at the corner of the typewriter. But he brings it up from his lap in the next shot. (00:38:15)
Continuity mistake: In the flashback during Poirot's narration, the scene with Colonel Curtisstalking to the victim at the military club happens differently from the first time around, with Clayton switching hand position after a different line of dialogue, starting off in discontinuity with the previous shot (originally the continuity was flawless). (00:44:40)
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)
Factual error: Hastings and Miss Lemon decide to investigate on their own. In the outside view of the first suspect they go question, a large contrail is visible on the left of his building (contrails were not a complete impossibility in the 30s, but it's rather odd to randomly see one in an establishing shot for this timeframe). (00:22:40)
Continuity mistake: Poirot and Hastings are in front of the corpse at Marsdon Manor. When Miss Rawlinson makes her comment about him taking advice, she is almost with her back against Hastings and in a dramatic light totally different from what shown in establishing shot. (00:17:45)
Revealing mistake: In the reveal at the end of the episode we see a prolonged close-up of the newspaper "Nairobi Daily Press." Not only the fateful article linked to the murder is written in a font different and larger from the one used in the rest of the first page, but you can also see where the fake part abruptly ends, resuming mid-sentence the article of a real (or realistic) newspaper used as base for the prop, back to the right font and consistent style. (00:50:30)
Continuity mistake: The morning after his arrival at the village, Poirot sits with Hastings for breakfast (prime minister Baldwin is mentioned on the radio). Hastings is holding the newspaper with his left hand and the right is on his knee. New shot and both hands are on the table. It keeps happening during the scene. (00:14:25)