A problem from the National Bank of New Zealand Competition 2000, via Crux Mathematicorum, November 2006:
Humanity is visited by three alien races, the Kweens, the Ozdaks, and the Merkuns. Kweens always speak the truth, and Ozdaks always lie. In any group of aliens, a Merkun never speaks first; when it does speak, it tells the truth if the previous statement was a lie and lies if the previous statement was truthful. The three alien races can tell one another apart, but to humans they all look the same. A delegation of three aliens visits Earth. At least one of them is a Kween. When they arrive they make the following statements, in order:
First alien: The second alien is a Merkun.
Second alien: The third alien is not a Merkun.
Third alien: The first alien is a Merkun.
Which aliens can we be sure are Kweens?
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Merkuns never speak first, so the first alien is either a Kween or an Ozdak. Suppose it’s a Kween. That means the first statement is true and the second alien is a Merkun, which means that the second statement is false and thus the third alien is a Merkun. But this would require the third statement to be true (since we’re supposing that the third alien is a Merkun and the second statement is false), and that can’t be the case, as we know that a Merkun won’t speak first in a group. So the first alien is an Ozdak, and thus the second is either a Kween or an Ozdak. If it’s a Kween then the third is an Ozdak (the third can’t be a Kween), and if it’s an Ozdak then the third is a Merkun. The latter can’t be the case, because a Merkun would speak the truth after a lie is uttered, and we know that the third statement can’t be true. So the first and third aliens are Ozdaks and the second is a Kween.
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