British puzzle creator Hubert Phillips invented this genre of puzzle in the 1930s.
I have slightly modified a puzzle attributed to H.P. (a.k.a. Caliban).
One of five people is a murderer.
Here are their statements:
Bernie: Becky didn’t do it.
Benny: Betty did it.
Bella: Becky is innocent.
Betty: Becky did it.
Becky: I did not do it.
Can you identify the killer?
Given k people, including the killer, lied – provide your answers and reasoning for all possible values of k.
3 people (Bernie, Bella, Becky) say Becky did not do it and 1 (Betty) says that she did.
1 or 3 of these 4 people are lying. So we cannot have 0 or 5 liars.
Other possible k:
k = 1. Betty must be lying, so she is the killer.
k = 2. Betty and Benny must be lying, so the killer is not Betty (or Becky). Since the killer must be a liar, it is Benny.
k = 3. Betty and Benny must be telling the truth, but this is a contradiction, so k cannot = 3.
k = 4. Betty must be telling the truth. So Becky is the killer. No contradiction.
In summary, the only possibilities are
k = 1. Betty is the killer.
k = 2. Benny is the killer
k = 4. Becky is the killer.