Ethel , Felicia and Gabrielle live on an island inhabitated by three types: the Knights, the Liars and the Weirdos.
Each is either a Knight who always tells the truth, a Liar who always lies , or a Weirdo who may do either - that is, a Weirdo chooses whether to speak truth or lie for each statement.
Ethel says : "If we all belong to the same type, then that type is the Liar."
Felicia says: "If just one of us belongs to a different type from each of the others, then that one is a Liar."
Gabrielle says : "If each of us belongs to a different type from each of the others, then I am a Liar."
Whose type can you deduce with absolute certainty?
Suppose Ethel is a liar. Then, her statement is false. For an if statement to be false, the first part has to be true and the second part has to be false. Therefore, they are all of the same type, but not all liars. However, Ethel is a liar, so that is a contradiction. Therefore, Ethel is not a liar.
Suppose Gabrielle is a liar. Then, they are all of different types and she is not a liar. That is a contradiction, so Gabrielle is not a liar.
Suppose Felicia is a knight. Then, if one is a different type from the others, then that one is a liar. However, Ethel and Gabrielle are not liars, so none of them are liars. That implies that there is not one of them that is different from the others. Therefore, they are either all of the same type or all of different types. They cannot all be of different types because there are no liars. Therefore, they are all of the same type, so they are all knights. However, Ethel would be lying, so they cannot all be knights. Therefore, Felicia is not a knight.
Suppose Felicia is a liar. Then, one of them is a different type from the others, and that one is not a liar. Therefore, Felicia is not the only liar. However, we already proved that Ethel and Gabrielle are not liars. That is a contradiction, so Felicia is not a liar. Therefore, she is a weirdo. Felicia is the one we know. Felicia is a weirdo.
Posted by Math Man
on 2012-04-23 20:13:23