First, the murderer writes anonymous letters to Hercule Poirot, taunting him with the crimes that he is about to commit. He signs the letters "A.B.C.," which is both a reference to the diabolical plot of killing people one by one as he works his way through the alphabet and also...
First, the murderer writes anonymous letters to Hercule Poirot, taunting him with the crimes that he is about to commit. He signs the letters "A.B.C.," which is both a reference to the diabolical plot of killing people one by one as he works his way through the alphabet and also a way of throwing suspicion on Cust, whose name, Alexander Bonaparte Cust, also can be abbreviated as the initials A.B.C.
Second, Franklin Clarke deliberately intends to throw suspicion on Cust, who is innocent. He "hires" Cust as a travelling salesman of silk stockings. On the day of each murder, a man selling silk stockings was seen in the vicinity.
Third, Cust is an epileptic who suffers bouts of short-term blackouts following a head injury he received during WWI. Cust cannot recall his whereabouts during the last murder, and he was found with blood on his sleeve and a knife.
Fourth, when the police search Cust’s room, they find a box of ABC railway guides that are identical to the guides the killer leaves at each crime scene. They also find a typewriter and paper that is identical to those that the A.B.C. murderer used in his letters to Poirot.
Fifth, Poirot doubts that Cust is guilty, in part because he has a solid alibi for the second murder at Bexhill.
In terms of Cust's possible motives for each murder or how he would profit from these crimes, Franklin does not need to show that Cust has motive or would profit. Franklin has planned the killings to look like the work of a maniacal serial killer to mask the true motive, which is the killing of his brother, Carmichael Clarke, the third victim.
Franklin devises an elaborate scheme to murder a series of people in what looks like a madman’s serial murder plot to kill people in alphabetical order based on their first and last names and location of the crime. Thus, his first victim is Alice Ascher in Andover (AAA), then Betty Barnard at Bexhill (BBB), and then Carmichael Clarke in Churston (CCC). If the murderer were a madman, he needs no motive for the killings, including a profit motivation.
Franklin (not Cust) writes letters to Poirot about the crimes as a way of establishing the pattern, throwing suspicion on Cust and giving himself time to kill his brother. Poirot wonders why A.B.C. would write to him rather than Scotland Yard, but it is because Franklin deliberately sends the “C” letter to the wrong address but one that looks like Poirot's. This causes a delay in delivery of the letter so that Poirot cannot warn Carmichael Clarke in time.