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X and Y (Posted on 2011-01-04) Difficulty: 3 of 5
There is an island where every inhabitant is either of Type X or Type Y. One of the types always tells the truth, and the other type always lies, but you are not sure which is which. You meet two inhabitants of this island, A and B.
A: B is of Type X.
B: We are both of Type Y.
Which type tells the truth, which type lies, and what are A and B?

See The Solution Submitted by Math Man    
Rating: 4.0000 (2 votes)

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Solution re: spoiler | Comment 2 of 3 |
(In reply to spoiler by Ady TZIDON)

This answer is correct.  It can be reasoned about as follows:

1. A's and B's statements contradict each other, therefore they cannot both be true.  Therefore, either exactly one is true or they are both lies.

2. If B's statement were true, then A's statement would have to be false.  This creates a contradiction since if they are both Y their statements must have the same truth value.  Therefore B's statement is false and they cannot both by Y.

3. Given that B's statement is false, consider A's statement. If it were also false, then B would have to be of type Y.  However, if B is of type Y and B lies, then since A is a liar he would also be type Y.  This is a contradiction, since we know they are not both of type Y from (2) above.  Therefore, A's statement must be true.  B is of Type X and lies, meaning that A is of type Y and tells the truth.


  Posted by Oren on 2011-01-04 20:48:39
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