It has been said that the two main motives for murder as sex and money, especially money. The problem this produces is that it tends to make it relatively obvious who the murderer is—whoever gets the money or gets the lover.
One of the traditional solutions to this problem is for the murderer to invent a clever way to kill the victim without being there. Another is for them to fake an alibi, or to fake the time of death so that they can establish an alibi.
The final of the classical solutions is for the murderer to have an accomplice. This produces the problem of motive for the accomplice.
Probably the simplest solution is economic—for the accomplice to share in the proceeds through the primary murderer’s discretion afterwards. Other motives involve love, a shared desire for revenge, and being tricked into it without understanding what was going on (then either not being aware of the significance of the help or being afraid to speak up).
An interesting alternative was used in the Death in Paradise episode, A Dash of Sunshine (Series 2, Episode 6). Here, the murder was perpetrated entirely by an accomplice, in exchange for the murderer having murdered the accomplice’s victim in another country, several months before. That is, each murder was committed by a person with no connection to the victim, as a quid pro quo. This was a very interesting setup, as it led to some unusual qualities in the murder.
Ordinarily, a murderer who has no connection to the victim is impossible to catch. In these modern days of DNA evidence, that may be somewhat less true, if the murderer has been foolish enough to use an online DNA service, or if a close relative has. Even there, though, that isn’t tremendously useful if there was no other evidence linking the murderer to the crime. Moreover, it’s not overly likely to be present.
There was a time, in the dawn of forensic science, where fingerprints were new technology and criminals did not know to wear gloves. That time did not last long. Similarly, there was a time when criminals did not know to avoid letting their DNA get onto a crime scene, but they do know it now. Admittedly, avoid fingerprints is easier than avoiding leaving any DNA, but at the same time, it’s not that hard. Gloves will do it for anything that one touches, and beside that, some simple precautions like wearing pants and a long-sleeved shirt, and ensuring that one’s hair doesn’t shed (a hat, a hair net, etc) will generally suffice. It’s not like the police are going to swab every inch of the floor in the hope of catching a stray skin cell. Tests cost money.
DNA evidence is much more of a problem for the mystery writer in crimes of passion, where the murderer would not have thought to take precautions beforehand. On the other hand, it is far more likely in crimes of passion that here will be an innocent excuse for why the murderer’s DNA is present at the crime scene. If a man lives in the same mansion as the victim, it is of no significance that his DNA was found in the same room as the victim. It would almost be odd to not have found it.
All that allowed for, it is still nearly impossible to catch a murderer with no connection to his victim. This makes the quid-pro-quo murderer very difficult to catch—unless the murderer makes a mistake and leaves physical evidence, the only way to catch him is through his connection to his partner—the one who benefits.
Interestingly, in the Death in Paradise episode, the killer made mistakes and left forensic evidence and this is how the detectives caught the pair. Perhaps the biggest of these mistakes was killing the victim in his own rental house. This was especially odd as the other murderer had made the reservations at the rental house, meaning that they cooperated in the terrible decision to proceed with a link from the murderer to the victim. It was explained, and I think reasonably, that the murderers were careless because they were arrogant. They assumed that across the ocean, in a country with a minimal police force, a staged robbery would be accepted as such and murder would not be suspected. It was still a terrible idea, and one that they didn’t make with the first murder, back in England.
I can’t think of any examples of the other way around—of finding the connection between the murderers then finding the connection between the killer and the victim. Given how many murder mysteries have been written in the last 100+ years, it probably has been written, of course, but it would be interesting to write the second.