While reading Problem Solving Strategies - Crossing the River with Dogs and Other Mathematical Adventures by Ken Johnson and Ted Herr, I came across a problem I was not sure how to solve.
One of the five members of a company's board of directors was suspected of sleeping during a board meeting. It was known that only one board member had actually slept, but no one (except the five members) knew who it was. The company vice president questioned the members and they made the following statements:
Davis: The snoozer was either Rawls or Charlton
Rawls: Neither Vongy nor I was asleep.
Charlton: Both Rawls and Davis are lying.
Bobbins: Only one of the Rawls or Davis is telling the truth.
Vongy: Bobbins is a liar.
When the board chairperson (she was not questioned) was consulted, she said that three of the board members always tell the truth and two of them always lie. Who slept in the meeting?
Perhaps I am simply glossing over one facet of the information given that I need to solve it. It seems like a simple problem on its hinge, so I am a bit flustered I am not able to solve it!
If you could give me a run-down of the logic used to solve it, that would be lovely.
Note: this is NOT homework - simply for my own benefit.
If Charlton is telling the truth, that means that both Davis and Rawls are the liars; in particular, Vongy is telling the truth, which would mean that Bobbins is also a liar, contradicting the fact that there are only two liars and there three truth-tellers.
So Charlton is lying. That means that at most one of Davis and Rawls are lying.
If Bobbins is telling the truth, then the second liar must be one of Rawls and Davis, which would again mean that Vongy is telling the truth, making Bobbins a third liar; this is impossible, so Bobbins is lying.
So we now know that the two liars are Charlton and Bobbins, and the remaining three are truth-tellers.
Therefore, the snoozer is either Rawls or Charlton (since Davis is telling the truth), but cannot be Rawls (since Rawls is telling the truth).
So Charlton is the one who fell asleep.
Alternatively: Bobbins and Vongy cannot both be telling the truth, so at least one of them is a liar. In particular, Charlton cannot be telling the truth, since that would give at least three liars. So the two liars are among Charlton, Bobbins, and Vongy, hence both Davis and Rawls are telling the truth; this suffices to establish that Charlton is the one who fell asleep.