Three young men named Ernesto, Fontleroy, and Gildenstern arrived singly at an inn and awaited the innkeeper. When she arrived at the front desk, all three asked for the best room. The innkeeper explained that, since it was not possible for them all to have the best room, the man who had arrived first could have a spacious room overlooking the village square, the second to arrive could have a small room with a partial view of the garden, and the third would have to settle for a drafty loft by the back alley, but it was the last room she had to offer. The following conversation ensued:
Ernesto: I am a knight.
Fontleroy: While I am only a knave.
Gildenstern: I agree with you there, Fontleroy.
Ernesto: Gildenstern is a knight.
Fontleroy: No, he is a liar.
Gildenstern: Then let me say: I did not arrive first.
Ernesto: Fontleroy is the liar.
Fontleroy: Following Gildenstern's lead, let me say: I did not arrive first.
Gildenstern: Ernesto is a knave.
Ernesto: If that is so, then the most honest of us did not arrive last.
Fontleroy: Ernesto, you are a knight.
Gildenstern: Ha!
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 man, and what room did the innkeeper assign to each?
Before beginning, not the following:
1) Each person makes four comments, which will be labeled as such: Ernesto's first comment is (E1), Gildenstern's last is (G4), etc.
2) For each person, their first and third comments must both be true or both be false; there cannot be a situation where one is true and the other a lie. Similarly with their second and fourth comments.
Thus:
a) We know that Fontleroy is not a knight, because a knight would never call himself a knave. Thus, F1 - and by extension F3 - are lies.
b) If Ernesto was a knight, then Gildenstern would also be a knight (via E2), but then Ernesto would have to be a knave (via G3). Thus, Ernesto cannot be a knight.
c) Since Ernesto is not a knight, F4 is a lie, and thus so is F2. Therefore, Fontleroy is not a knight, and Gildenstern is not a liar.
d) Also since Ernesto is not a knight, E1 is false and therefore so is E3. Thus, Fontleroy is not a liar - and since he's not a knight either (see c above), he must be a knave. Since F2 and F4 are lies, F1 and F3 must be the truth.
e) Since Fontleroy is a knave, G1 must be true and therefore G3 as well. Thus, Ernesto must be a knave (via G3), and since E1 and E3 are false, E2 and E4 must be true. Gildenstern is therefore a knight (via E2), and G2 and G4 are true.
From the above we know that: G2, F3 and E4 are all true. Thus, Gildenstern did not arrive first, nor did Fontleroy, and Gildenstern (the only knight and therefore most honest) did not arrive last. This means that only Ernesto could have come first, and the only place in which Gildenstern could have arrived is second. Finally, Fontleroy came last. If the innkeeper awards rooms to those individuals based on the order they arrived (rather than their honesty), the Ernesto gets the best room, the Gildenstern, and finally Fontleroy.
Solution
