The final mystery in The Eight Strokes of the Clock is At the Sign of Mercury. Lupin's lady friend gave him this "impossible mystery" as a joke challenge, at the start of the book. The challenge was to find a good luck necklace that was stolen from her, seven years ago.
Lupin says he has the solution. In order to get the necklace back, she must go through a series of very specific instructions. (He will later admit some of the instructions were unnecessary. He included them to make it more mysterious/adventurous.) The general idea is that she wears a certain outfit and goes to an antique store dedicated to the god Mercury. She has to order the man inside to give her the necklace, and she can't ever show weakness.
The man completely freaks out, and he pretends to faint. She stays firm. But when he pretends to kill himself, she cries in fear and hits the gun away. That breaks her cover, so to speak, and the man realizes this is all some crazy prank.
Lupin is forced to step in and explain things. The man and his wife worked at the manor, when the necklace was stolen. Both of them are Corsicans, and that instantly made Lupin suspicious, because all Corsicans are very superstitious. He figured that they stole the necklace to get the good luck, especially since they're running a store dedicated to the Roman god of luck.
Lupin secretly searched the entire store, looking for the necklace. He didn't find it, but he did find the man's diary, saying that he would give the necklace back if an avenging angel from Heaven, wearing a specific outfit, demanded he hand it over. Hence, the scheme of having Lupin's lady friend wear a specific outfit to get the necklace back. The book does a good job of justifying it, but I still think that this man had a very specific set of circumstances for handing over the necklace, just so we could have a mystery.
Lupin pressures the man, and he has a breakdown, but he realizes that Lupin has no idea where the necklace is. He gives him a statuette of Mercury, thanks him, then kicks him out.
In the end, Lupin realizes the necklace must be inside the statuette of Mercury, by the shop's sign. He couldn't find it inside the shop, because it was outside. The giveaway is the man associates the necklace and the statuette with each other. Also, the statue is off balanced. Without a counterweight inside, it would fall over.
I did not like this mystery much. The detailed series of instructions was good, but things went downhill once the invisible narrator started insulting Lupin's lady friend for ruining everything, by being weak and emotional. Um, she tried to stop a man from committing suicide. I'm on her side, on this one. Besides, it's a little odd for this guy to have "fake suicide attempt" on his list of things to try, to confirm someone is really an avenging angel from Heaven. Also, it's discriminatory to have "all Corsicans are very superstitious" as the mystery solution.
Overall, I'd say that this is a good mystery book, although decidedly old-fashioned today, especially in how Lupin treats women. LeBlanc is good at coming up with interesting ideas and situations for mysteries. I don't always like how he executes the stories, though, and we have stories like this one and The Tell-Tale Film, where the premise was better than the actual story.