Home > Logic > Liars and Knights
| Another Race (Posted on 2003-12-15) |
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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?
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Submitted by Gamer
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Rating: 3.3636 (11 votes)
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Solution:
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(Hide)
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First of all, every other statement someone makes is logically equivalent. Also, since Knights, Knaves, and Liars are never mentioned by anyone, you can break the 6 groups of 6 statements into 12 logically equivalent statmenets:
About the 1, 3, 5 positions in the race:
1: A B C D E F
3: D F E F A B
5: B C A A F F
About the 2, 4, 6 positions in the race:
2: C D D E B C
4: E E F B D A
6: F A B C C D
If two people say the same thing, it means everything else they say is logically equivalent. For example, C and D both said that A came in fifth place, but differed in opinion about who came in first and third place. Since both of them couldn't have come in first place, anything in the same column with the A in fifth place (1: C D, 3:E F, 5: A A, in this example) is eliminated. When this is applied to all of them, the following grid can be constructed:
_A_B_C_D_E_F
1|? X X X X ?
2|? X X X X ?
3|X ? X ? X X
4|X X O X X X
5|X ? X ? X X
6|X X X X O X
F thinks E was in fifth place. Since E is in sixth place, his statements about first and third place must be wrong as well.
_A_B_C_D_E_F
1|O X X X X X
2|X X X X X O
3|X X X O X X
4|X X O X X X
5|X O X X X X
6|X X X X O X
Filling in the only remaining choices gives the solution grid. So the order is A F D C B E; A is a knave and the others are liars. |
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