One way they solve a crime is to look at what was necessary to commit the crime and what was extra. If it was necessary to kill someone, from a criminal’s point of view, then what were the extra things that were done to the victim - such as wrapping the head of the victim or stealing her shoes (two examples from real life cases they solved). Those actions can reveal the killer.
The Vidocq Society member who was interviewed also said that, "In terms of interviewing an anger-retaliatory type, one of the things that must be understood by the detectives is, contrary to popular opinion, the perpetrator does not feel any guilt for committing the killing. When they leave the killing site, they many times have a sense of well-being and a sense of relief and a sense of charm because they've just had 50 pounds of emotional baggage taken off their shoulders. Police don't expect that."
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