[Spoilers of very good stories]
The Hydra is one of those monsters that people tend to still know: the many-headed beast, whose heads grew back double when amputated. The Hydra of legend also had incredibly poisonous blood and a single immortal head. Hercules did not take on the Hydra alone, he went in a-chopping while his nephew used a firebrand to sear the bloody stumps, thereby preventing the heads from re-growing. When he finally found the immortal head, he tore it off, and buried it so that while alive, it could not regrow. On re-reading these old stories, I'm impressed by how much each story has a component of outsmarting (which we'll revisit for the Augean Stables) rather than just pure strength.
Hercule Poirot's Hydra is a rather more slippery beast: Rumor. Dr. Oldfield, a middle-aged country doctor, has discovered just that with the death of his wife. Now his practice is falling off, people give him stares on the street, and he's even received nasty letters. Poirot eventually gets out of him the particulars: people believe that he poisoned his wife with arsenic in order to marry his pretty young assistant.
Poirot visits the town and interviews a few people, including a servant (dismissed immediately after the wife's death) and a nurse. He tells people in "strict confidence" that he is there to investigate Mrs. Oldfield's death, so naturally word gets around town much faster than otherwise. The assistant, Jean, is horrified by the idea of exhuming the body for an official investigation. Nurse Harrison, on the other hand, comes around to the idea, and recounts an overhead conversation that seems quite damning to Jean and Dr. Oldfield. Poirot calls up the Home Office, who exhume and autopsy the body, and discover lethal levels of arsenic.
Someone comes forward with further information: another suspect behaved quite suspiciously in the days leading up to Mrs. Oldfield's death! Poirot encourages her, but when she comes forward with news of a compact filled with arsenic, she's gone too far. She's been followed in secret by Poirot's valet Georges, who saw her purchase the compact and hide it. She is the one who poisoned Mrs. Oldfield, under the delusion that Dr. Oldfield was in love with her and would marry her. She started the vicious rumors, and then seized on the chance to do away with her rival when Poirot arrived -- oh, but she overstepped.
This story is even shorter than the first, between 5,000 and 6,000 words by my estimation. The cast of main characters is kept small, and I noticed that the character names are very different so that it's easy to keep track. Which is helpful, because the story is entirely populated with stock characters and stock motivations. That doesn't make it a bad story, but it's hard for a character to catch in one's mind without anything interesting to grab hold of. Ergo, differentiating the characters as much as possible through names and speaking styles.
"There's no smoke without fire" is the most important phrase of the story, almost a theme. It's a horrible phrase in terms of spreading gossip, circular logic that lets a rumor perpetuate itself. But instead of repudiating it, Poirot takes it as true in a way that Dr. Oldfield refused, and that was enough of a clue for him to find a way forward.
I had been thinking of this as a very fair puzzle, but in that case it's hard to account for the compact. I spent the morning thinking about it, and I decided that a mystery in fact has two basic parts: figuring out whodunnit, and figuring out how to prove it. I believe that the Fairness Doctrine applies only to the first part by necessity.
My thinking is as follows: the reader has only access to a certain subset of the clues available to the detective, and has the inability to act in the detective's space. The first part of the story, then, is a matter of presenting all the information needed to come to the correct conclusion. This is do-able, like writing a logic puzzle. With care, you can ensure that the reader has all the information that the detective uses to solve the crime (or more!) and thereby make the puzzle fair.
The second part, however, is where real art comes in. In a good story, the reader can become convinced of the guilt of one of the suspects -- but a real mystery can remain about how on earth to prove it. And this second part, it seems to me, is often where mystery stories fall down. This is where the detective tends to do things behind the scenes, where coincidences tend to most help the detective, and where suspects do dumb things and fall into traps. And frequently the reader doesn't have a fair chance to evaluate just how artificial it all is, because he or she does not know who the detective is targeting (even if it is known exactly what the detective is doing!).
Proof, the essential element of this second part, requires certain things: a solid understanding of the setting and the people, and the ability to act in this space. In order to feel fair, the reader needs as much of the first as possible in order the judge the outcome -- but the reader will never have the second. There will always be more than one way to do it, as Mr. Wall would say. One person would have Georges follow the suspect, another person might lay a careful logical trap and prove a contradiction, another might reconstruct the crime and prove that only one person had means, motive, and opportunity.
I'm unsure about how well Christie manages the second part in The Lernean Hydra. She loves these "induce the criminal to incriminate herself" endings, but they're so hard to get right, and the reader is left with a sense of a house of cards, that by making one sensible move, the criminal could have evaded detection. Remember too that often in this step the detective is proving his case not only to the characters but also to the reader! Poirot's actions in this story just seem reckless on first examination. If the murderer had just remained calm, then Dr. Oldfield would likely have been hanged. Of course, Poirot relies always on "the psychology": he tailored his plan according to the sort of person the criminal was. (And had this plan failed, we assume, he would have had another up his sleeve.) This is not the sort of plan he would have used for a stolid and cautious individual who doesn't care about blaming someone else for the crime.
From a craft point of view, this suggests an easy way to expand a short story into a novel: making plans fail and then recovering from them, or simply taking more time with it. Stretch it out, throw in lots of red herrings to make it look like someone else might have done it, and build up the tension a bit more. The best example of this is Christie's The Mysterious Affair at Styles (public domain in the US), where it was perfectly clear who did it and why, but proving it was a terrible ordeal that involved shifting attention from the real criminal.
So, what did I learn from the Lernean Hydra? Again, a simple plot is necessary, something that won't take too long to explain, giving you room to elaborate and confuse the issue. Second, that as long as you're careful and spend a lot of thought on it, you can save a lot of space in the proving step with a trap rather than a careful (and potentially boring) logical analysis. If you writer the story knowing how the criminal will get caught, then you can probably do a better job of setting the stage early on. Third, Christie's use of stock characters is interesting to me. The country doctor, the invalid wife, the inveterate gossip -- varying too much from the stereotypes would attract too much attention. It might have made the doctor more interesting as a person to have a passion for juggling, but it would be distracting and Chekhov would definitely disapprove.