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.
Best Answer
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.