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.
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?
Suppose A was first. Then, B's first statement is false, so B is either a knave or a liar. Either way, B's third statement is false, so B was second. Then, C was third. That makes C's statements true, false, and false, which is impossible. Therefore, A was not first.
Suppose B arrived first. Then, C's first statement is a lie, so C is either a knave or a liar. If C is a liar, then C's third statement is false. If C is a knave, then his second statement is true and his third statement is false. Either way, C's third statement is false, so B arrived second. That contradicts what we supposed, that B arrived first. Therefore, C was first.
Since C arrived first, all of A's statements are false and all of C's statements are true. Therefore, A is a liar and C is a knight. B's first statement is true, so B's third statement is true. Therefore, B was not second, so he was third. That makes A second. Also, B's second statement is false, so B is a knave.
Posted by Math Man
on 2011-01-30 21:46:10