Home > Logic > Liars and Knights
| No Room at the Inn (Posted on 2003-06-25) |
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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?
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Submitted by Bryan
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Rating: 4.5000 (10 votes)
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Solution:
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(Hide)
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Call the men A, B, and C, and start by considering the first four lines. In line 1, A claims to have arrived first. Lines 2 - 4 consist of two statements apiece, with each man denying the one before him arrived first, and asserting he was first.
- If A is a knight, both of B’s statements in line 2 are false, making him a liar, and C’s statements in line 3 are true then false, making him a knave.
- If A is a knave telling the truth in line 1, then his next statement will be false, B is again a liar and C is a knave. But A’s first statement in line 4 is true, therefore A is not a knave telling the truth in line 1. If A is a knave lying in line 1, then his first statement in line 4 is true, C did not arrive first, so B arrived first, both of B's statements in line 2 are true, making him a knight, and both of C's statements in line 3 are false, making him a liar.
- If A is a liar, then from line 4, C must have arrived first, making him a knight, and B’s statements in line 2 are true then false, making him a knave.
Regardless of which scenario, the order of speakers is knight, liar, knave, knight, liar, knave, etc., and the knave’s last statement is false, meaning his next statement will be true. So when B and C agree in lines 5 and 6, they must be a knight and a knave telling the truth. A, B, C must be liar, knave, knight, in that order. Thus C arrived first (line 3) and B did not arrive second (line 5) so he was last.
Cicero gets the bunk,
Ajax gets the stable, and
Balthazar gets the pighouse.
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