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Metapuzzle (Posted on 2012-09-13) Difficulty: 5 of 5
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?

No Solution Yet Submitted by Math Man    
Rating: 5.0000 (1 votes)

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Solution Solution and Proof (sort of) Comment 4 of 4 |

If we don't assign the claims to a specific letter (A, B or C), we get 11 permutations of what they might have said about one another - since we're not assigning to a particular letter, once we have the permutation LIAR, LIAR, KNAVE, (for example) we don't need the permutations LIAR, KNAVE, LIAR or KNAVE, LIAR, LIAR, as the starting point is largely arbitrary.

Of these 11 permutations, 4 would either be impossible answers or impossible to draw any inferences.

Of the remaining 7 permutations, only 4 had unique solutions that would have allowed the logician to conclusively solve the puzzle:
(note the permutation on the left is what the characters are claiming about the next person, not what they actually are. To the right is what they are)

LIAR, KNIGHT, KNAVE      [knight, liar knave]
LIAR, KNAVE, KNIGHT      [liar, knight knave]
LIAR, KNIGHT, KNIGHT     [knight, liar, knave]
KNAVE, KNAVE, KNIGHT   [liar, knight, knave]

When the first friend finds out what A said of B, he can't discern ANY information.

If A had called B a Liar, then the friend would have known that C was the Knave. Since he didn't have the answer, A must have either called B a Knight or a Knave.

Now we can assign the possible permutations to actual characters (once again what the characters said is on the left and what they are is on the right)…

A             B             C

Knave, Liar, Knight          [Knave, Knight, Liar]

Knave, Knight, Liar         [Knight, Knave, Liar]

Knave, Knight, Knave     [Knight, Knave, Liar]

Knave, Knave, Knight     [Liar, Knight, Knave]

Knight, Knave, Liar           [Liar, Knave, Knight]

Knight, Liar, Knave           [Knave, Liar, Knight]

Knight, Knight, Liar          [Liar, Knave, Knight]

Knight, Liar, Knight          [Knave, Knight, Liar]

Knight, Knave, Knave     [Knave, Liar, Knight]

 

So when our logician tells friend number 2 that his first friend couldn't discern any of the types from A's claim about B, he knows that one of the above combinations is correct.

Even with this extra information, B can't ascertain anything when he hears B's claim about C, so B can't have claimed C was a Liar, or he'd have known that A was a knave, and he can't have claimed C was a knight or he'd have known that B was knave. So B must have claimed that C was a Knave, leaving friend number 3 with the following options:

A             B             C

Knave, Knave, Knight     [Liar, Knight, Knave]

Knight, Knave, Liar           [Liar, Knave, Knight]

Knight, Knave, Knave     [Knave, Liar, Knight]

Friend number 3 goes through the same process. (If he knew about friend two, he would know who everyone was… but he doesn't). Since friend 3 also can't figure out any types, we know that C didn't call A a Knight (otherwise he'd know B was a Knight) and we know he didn't call him a Liar (or he'd know that B was a knave)… No, C obviously called A a Knave, once again leaving 3 possibilities

A             B             C

Knave, Knight, Knave     [Knight, Knave, Liar]

Knight, Liar, Knave           [Knave, Liar, Knight]

Knight, Knave, Knave     [Knave, Liar, Knight]

 

We can immediately see that only one combination is common to both lists and this is the answer.

A = Knave

B =Liar

C = Knight

A, (the Knave) lies and claims that B (the Liar) is a Knight. B (the liar) lies and calls C (the Knight) a Knave. C (the Knight) tells the truth, that A is a Knave.

Edited on April 12, 2013, 8:13 am
  Posted by scott on 2013-03-22 12:05:31

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