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?

There are two ways to look at the statements (as far as the alternating statements of a knave). Either each set of statements in a line counts as a single statement which are both true or false, and the next group of statements is the opposite (if, of course, that person is a knave); or the first statement is true, the second is false, etc, regardless of the lines in the problem.

First assume that they count together. That means that both sentences in any one line have the same value (true/false), and if we know one, we know the other.
Based on my original assumption that B cannot be first (or there will be no full solution), B must be a liar or a knave. His first set of statements must be false, since he says, "I was first," and we are assuming that can't be the case.
Also, the first part of that statement ("No you didn't") must be false, so Ajax was first.
The first part of Cicero's first statement ("You were not," referring to B being first) must be true. Therefore, his second statement ("I arrived first") must also be true, but we already have that Ajax was first. Both cases cannot be true, so that must mean that the statements in a line do not count together.

Therefore, each statement counts independently, but we know that every other statement by any of the people has the same value.
Assuming still that B is not first, his second statement must be false. He is not a knight; we know that his other two statements are the same, but could both be true (if B is a knave) or false (if he is a liar). There is no way of figuring this out from his statements alone (as those are the only ones he made).
If Balthazar's second statement is false, then Cicero's first statement, which denies it, must be true. Therefore, he (Cicero) is not a liar, but the next statement ("I arrived first") could be true or false, depending on whether he is a knight or a knave. His third statement, however, must also be true, and Balthazar did not arrive second. Therefore, Balthazar arrived third.
B's last statement, which agrees with Cicero's last statement that is true, must also be true. Therefore, B is a knave, and his first statement ("No you didn't" in response to A's "I arrived first") must be true, as well as the third. Therefore A did not arrive first; he arrived second and Cicero was first.

Therefore, a little more inspection shows that all of Ajax's statements are false; he must be a liar. Also, Cicero's statements are all true; he is a knight.

The men, in the order they arrived, are:
1st: Cicero, knight
2nd: Ajax, liar
3rd: Balthazar, knave

Posted by DJ on 2003-06-25 11:38:36