Home > Logic > Liars and Knights
| Metapuzzle (Posted on 2012-09-13) |
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A logician met three people, A, B, and C. He knew that one was a knight, one was a liar, and one was a knave, but he did not know which was which. The logician asked A, "What type is B?" A either said that B was a knight, B was a liar, or B was a knave. Then, he asked B, "What type is C?" B either said that C was a knight, C was a liar, or C was a knave. Finally, he asked C, "What type is A?" C either said that A was a knight, A was a liar, or A was a knave. The logician now knew what type each person was.
The next day, the logician met a friend. He told his friend about his conversation with A, B, and C. The friend asked, "What type did A say B was?" The logician told him. The friend was not able to figure out what type any of A, B, and C was.
The day after that, the logician met another friend. He told the second friend the same puzzle he told the first friend. He also said that the first friend asked what type A said B was, but that the first friend could not solve what any of them was. He did not tell the second friend what type A said B was. The second friend asked, "What type did B say C was?" The logician told him. The second friend could not figure out what type either A, B, or C was.
The day after that, the logician met a third friend. He told the third friend the same puzzle he told the other two friends. He talked about the first friend not being able to solve what any of A, B, and C was. However, he did not talk about the second friend. The third friend asked, "What type did C say A was?" The logician told him. The third friend could not figure out what type any of A, B, and C was.
What are A, B, and C, and what did they say?
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Submitted by Math Man
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Rating: 5.0000 (1 votes)
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Solution:
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There are 27 cases for what A, B, and C said.
Case: A said that B is a: B said that C is a: C said that A is a:
1 knight knight knight
2 knight knight liar
3 knight knight knave
4 knight liar knight
5 knight liar liar
6 knight liar knave
7 knight knave knight
8 knight knave liar
9 knight knave knave
10 liar knight knight
11 liar knight liar
12 liar knight knave
13 liar liar knight
14 liar liar liar
15 liar liar knave
16 liar knave knight
17 liar knave liar
18 liar knave knave
19 knave knight knight
20 knave knight liar
21 knave knight knave
22 knave liar knight
23 knave liar liar
24 knave liar knave
25 knave knave knight
26 knave knave liar
27 knave knave knave
Case 1 is impossible. There is exactly one knight, and that knight would not say that somebody else is the knight.
Case 2:Neither A nor B can be the knight, so C is the knight. Then, A is a liar, so B is a knave. Case 4 and Case 10 are similar.
Case 3:Again, C is the knight. That makes A the knave and B the liar. However, B would be telling the truth, so this case is also impossible. Case 7 and Case 19 are similarly impossible.
Case 5:A cannot be a knight. It could be true that A is a liar, B is a knave, and C is a knight, or A is a knave, B is a knight, and C is a liar. The logician would not be able to figure out what A, B, and C were. Case 11 and Case 13 cannot be solved by a similar argument.
Case 6:A cannot be the knight. Suppose B is the knight. Then, C is a liar, so A is a knave. C would be a liar telling the truth. Therefore, C is the knight, so A is the knave and B is the liar. Case 16 and Case 20 can be solved by a similar argument.
Case 8:A cannot be the knight. Suppose B is the knight. Then, C is a knave and A is a liar. A would be a liar telling the truth. Therefore, C is the knight, which means that A is the liar and B is the knave. Case 12 and Case 22 are similar.
Case 9:A cannot be a knight. Suppose B is the knight. Then, C is a knave and A is a liar, but A would be telling the truth. Therefore, C is the knight. That makes A the knave and B the liar. Case 21 and Case 25 are similar.
Case 14:It could be that A is a knight, B is a liar, and C is a knave. It could also be that A is a liar, B is a knave, and C is a knight. It could even be that A is a knave, B is a knight, and C is a liar. This case cannot be solved.
Case 15:It could be that A is a knight, B is a liar, and C is a knave, or A is a knave, B is a liar, and C is a knight. This case cannot be solved. Case 17 and Case 23 are also unsolvable.
Case 18:A could be a liar, B could be a knight, and C could be a knave. Also, A could be a knave, B could be a liar, and C could be a knight. The logician would not be able to solve this case. Case 24 and Case 26 are similarly unsolvable.
Case 27:It could be that A is a knight, B is a knave, and C is a liar. It could also be that A is a liar, B is a knight, and C is a knave. It could also be that A is a knave, B is a liar, and C is a knight. This case is unsolvable.
In summary, here are the cases that the logician could have solved what A, B, and C were.
Case: A said that B is a: B said that C is a: C said that A is a: A: B: C:
2 knight knight liar liar knave knight
4 knight liar knight knave knight liar
6 knight liar knave knave liar knight
8 knight knave liar liar knave knight
9 knight knave knave knave liar knight
10 liar knight knight knight liar knave
12 liar knight knave knight liar knave
16 liar knave knight liar knight knave
20 knave knight liar knight knave liar
21 knave knight knave knight knave liar
22 knave liar knight knave knight liar
25 knave knave knight liar knight knave
If the logician told the first friend that A said that B is a liar, then either Case 10, 12, or 16 must hold. Then, the friend would know that C is a knave. Since the friend did not know what anybody was, A did not say that B is a liar. Now, we have these cases.
Case: A said that B is a: B said that C is a: C said that A is a: A: B: C:
2 knight knight liar liar knave knight
4 knight liar knight knave knight liar
6 knight liar knave knave liar knight
8 knight knave liar liar knave knight
9 knight knave knave knave liar knight
20 knave knight liar knight knave liar
21 knave knight knave knight knave liar
22 knave liar knight knave knight liar
25 knave knave knight liar knight knave
If the second friend knew that B said that C is a knight, then he would know that either Case 2, 20, or 21 holds. Then, B would have to be a knave. If the second friend knew that B said that C is a liar, then either Case 4, 6, or 22 would be true. In all three cases, A is a knave. Since the second friend did not know what any of them was, B said that C is a knave. The second friend knew that either Case 8, 9, or 25 holds.
The third friend knew about the first friend, but not about the second friend, so he had nine cases. If he knew that C said that A is a knight, then either Case 4, 22, or 25 would hold. Then, he would know that B is a knight. If he knew that C said that B is a liar, then either Case 2, 8, or 20 must hold. Then, he would know that B is a knave. Since the third friend did not know what anybody was, C said that A is a knave. The third friend knew that either Case 6, 9, or 21 is true. We also know that either Case 8, 9, or 25 is true. Therefore, Case 9 is the correct one, so A is a knave, B is a liar, and C is a knight.
A:B is a knight.
B:C is a knave.
C:A is a knave.
A is a knave, B is a liar, and C is a knight.
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