I’d spent a considerable amount of time on this problem before I finally gave up and looked at the solution, where I discovered essentially the deductions identical to mine.

In the solution the author starts with an assumption that one of the two opposite possibilities is true, reasons until he encounters a contradiction, and declares that, since we encountered it, the opposite must be true.

But if you explore the opposite assumption it also leads to a contradiction. I suspect that the problem’s conditions are inconsistent and there is no solution, but would still like a confirmation that I didn't miss something obvious.

The preliminary conditions:

There are two types of people on an island: knights and knaves. Knights always tell the truth and knaves always lie. Everyone is mute and uses cards to signal YES or NO. The cards are RED and BLACK, but we don’t know which is which.

The problem:

While Abercrombie was on the island, he attended a curious trial: A valuable diamond had been stolen. A suspect was tried, and three witnesses A, B and C were questioned. The presiding judge was from another land and didn’t know what the colors red and black signified. Since non-inhabitants were present at the trial, the three witnesses were willing to answer questions only in their sign language of red and black.

First, the judge asked A whether the defendant was innocent. A responded by flashing a red card.

Then, the judge asked the same question of B, who then flashed a black card.

Then, the judge asked B a second question: “Are A and C of the same type?” (meaning both knights or both knaves). B flashed a red card.

Finally, the judge asked C a curious question: “Will you flash a red card in answer to this question?” C then flashed a red card.

Is the defendant innocent or guilty?

Here's solution from the book.

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    $\begingroup$ The trouble with these kinds of self-referential questions is that they reveal ambiguities in the rules of the game. For example, dropping the cards for a minute, if I ask "Will you answer 'Yes' to this question?", then no matter what you say, you're being truthful. So what does a knight answer? And what does a knave do? If someone answers, say, "Yes", does that mean they're a knight (and they chose their answer at random, since both would work), since a knave shouldn't have answered at all? $\endgroup$ – Jack M Apr 17 at 15:27
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    $\begingroup$ @JackM Certainly puzzles like this are susceptible to ambiguities. But I don't think there's any issue with your example. Based on what we've been told in the setup, there's no reason to think that everyone will be able to answer every question, or will have only one way to answer any question. If I ask "Will you answer 'Yes' to this question?", a knave cannot give an answer. A knight might answer "Yes" or "No", or might not answer at all. But a knave cannot answer. So if we're told that a question is answered, this sometimes allows us to deduce something. $\endgroup$ – Alex Kruckman Apr 17 at 17:17
  • $\begingroup$ I guess my point is that I don't know exactly how knights and knaves behave (do they always answer when they can? when they have multiple truthful answers, what do they do? etc.) to make correct deductions. $\endgroup$ – Alex Kruckman Apr 17 at 17:19

I agree that the information given in the puzzle is inconsistent. Here's my reasoning:

Assume first that red means yes. Then C answered the last question truthfully, so C is a knight. Now if B is a knight, A and C are the same type, so A is a knight. And if B is a knave, then A and C are different types, so A is a knave.

Now assume that red means no. Then C answered the last question untruthfully, so C is a knave. Now if B is a knight, A and C are different types, so A is a knight. And if B is a knave, then A and C are the same type, so A is a knave.

In any of the four cases above, A and B have the same type, which contradicts the fact that they disagreed on the first two questions.


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