When I went to the race track in Racing Town, a town made up only of Knights which always tell the truth, Knaves which tell truths and lies in an alternating pattern, and Liars which always lie, a race between 6 citizens of that town had just finished.
I went to the 6 citizens and asked each of them the order that all 6 finished. They all gave me different responses, each thinking themselves as winning, displayed here left to right as first to last.
A: A C D E B F
B: B D F E C A
C: C D E F A B
D: D E F B A C
E: E B A D F C
F: F C B A E D
From what they said, I was able to figure out what the correct order was. What is it?
The answer is:
A F D C B E
A is a Knave, and all the others are liars
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Since SOMEONE must have won, only one of them can have gotten the first one correct.
Therefore, there is either zero or one knights. Also, since one told the truth in the first position, they can't ALL be liars.
Case (1): If there is one knight,
then one of the six people have the actual order, and we merely try each in succession.
case (1.1):
if A is the knight, then B told the truth in only the 4th position.
impossible
case (1.2):
if B is the knight, then A told the truth in only the 4th position.
impossible
case (1.3):
if C is the knight, then B told the truth in only the 2nd position.
impossible
case (1.4):
if D is the knight, then C told the truth in only the 5th position.
impossible
case (1.5):
if E is the knight, then D told the truth in only the 6th position.
impossible
case (1.6):
if F is the knight, then A told the truth in only the 2nd position.
impossible
Therefore Case (1) isn't the case. There is no knight, and at least one knave.
Case (2):
(let X be an unknown letter) there are only two possibilities in each case 2.x.x,
because the other possible 4 combinations would let each person lie twice in six times.
Case (2.1) If A is a knave and told the truth in the first position:
A X D X B X
Case (2.1.1)
A E D F B C
impossible 'cause then C told the truth in only 4th position.
Case (2.1.2)
A F D C B E
POSSIBLE - A is a Knave and tells the truth first, the others are all liars.
Case (2.2) If B is a knave and told the truth in the first position:
B X F X C X
Case (2.2.1)
B E F A C D
impossible 'cause then D told the truth in 2nd and 3rd position.
Case (2.2.2)
B A F D C E
impossible 'cause then D told the truth in only 3rd position.
Case (2.3) If C is a knave and told the truth in the first position:
C X E X A X
impossible can't be because D agrees in only one of the positions.
Case (2.4) If D is a knave and told the truth in the first position:
D X F X A X
impossible can't be because C agrees in only one of the positions.
Case (2.5) If E is a knave and told the truth in the first position:
E X A X F X
Case (2.5.1)
E D A C F B
impossible 'cause then C told the truth in only 6th position.
Case (2.5.2)
E C A B F D
impossible 'cause then D told the truth in only the 4th position.
Case (2.6) If F is a knave and told the truth in the first position:
F X B X E X
Case (2.6.1)
F A B D E C
impossible 'cause then D told the truth in only the 6th position.
Case (2.6.2)
F D B C E A
impossible 'cause then B told the truth in 2nd and 6th position.
Edited on December 15, 2003, 9:22 am
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