Three young men named Ajax, Balthazar, and Cicero arrived singly at an inn and awaited the innkeeper. When she arrived at the front desk, the innkeeper explained that no rooms were available, but as the rains were especially harsh that season she was willing to put up the three men as best she could. The man who arrived first could sleep in a spare bunk in the stableboy’s room, the second to arrive could sleep in the stable, and the third would have to bunk in the pighouse, which at least was warm and dry. The following argument ensued:

Ajax: I arrived first.
Balthazar: No you didn't! I was first.
Cicero: You were not! I arrived first.
Ajax: That's a lie! I arrived first, as I said before.
Cicero: Well, Balthazar did not arrive second.
Balthzar: Agreed.

The innkeeper knew that everyone in these parts was either a knight who always told the truth, a liar who never told the truth, or a knave whose statements strictly alternated between truth and untruth. Using deductive reasoning, what is the disposition of each young man, and what berth did the innkeeper assign to each?

I broke the statements down by person, to make it easier to find a knave. Then, since there were only 6 combinations possible, I then wrote down each combination and the truthfulness for each statement.

Statements:
A:A was 1st. C was not 1st. A was 1st.
B:A was not 1st. B was 1st. B was not 2nd.
C:B was not 1st. C was 1st. B was not 2nd.

Order: A,B,C
A:T,T,T
B:F,F,F
C:T,F,F - not valid

Order: A,C,B
A:T,T,T
B:F,F,T - not valid
C:T,F,T

Order: B,A,C
A:F,T,F
B:T,T,T
C:F,F,T - not valid

Order: B,C,A
A:F,T,F
B:T,T,T
C:F,F,T - not valid

Order: C,A,B
A:F,F,F
B:T,F,T
C:T,T,T

Order: C,B,A
A:F,F,F
B:T,F,F - not valid
C:T,T,F

There is only one valid combination (C,A,B). So Ajax is a liar is the stable, Balthazar is a knave in the pighouse, and Cicero is a knight in the stableboy’s room.

Posted by Ender on 2003-06-25 11:07:33