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These puzzles are similar to some that appear in Smullyan’s book Forever Undecided. The inhabitants of the planet GP are either green or purple. Green people always tell the truth and purple people never tell the truth. Consider two people named X and Y . 1. X says: I am purple and Y is purple. What color is X? What color is Y ? 2. X says: I am purple or Y is green. What color is X? What color is Y ? 3. X says: If I am green, then Y is green. What color is X? What color is Y ? 4. X says: I am green if and only if Y is green. What color is Y ?

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    What is this for? Are you just presenting us with a fun challenge, or do you need help, or what? What's the context here? – 2017-01-23
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    These kinds of questions are incredibly misleading. Consider "X is a cat that always tells the truth. X says 'I am a dog'. What kind of animal is X?" In your question, it is best to think of it as "Can it be proven that X is purple? Can it be proven that X is green"? It is possible that 0, 1, or both of the prior claims can be proven from the premises given, so you should consider them individually. – 2017-01-23
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    Welcome to Math.SE! If you review [ask], you'll find a guide to posting the Question in a way better suited to learning mathematics for future Readers. In particular I'd recommend posting just one problem rather than a batch all at once. – 2017-01-23

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  • Statement: I am purple and Y is purple.

X cannot be green, because then his statement would be a lie which cannot be.

So X is purple. So what he says is a lie, hence "Y is purple" is false. So Y is green.

  • Statement: I am purple or Y is green.

If X were green, the statement has to be true, and as the first part is false in that case, "Y is green" must be true.

If X were purple, his statement must be false, but "I am purple" is already true. So contardiction, we are in the previouse case: X green, Y green.

  • Statement: X says: If I am green, then Y is green ("I am green" $\rightarrow$ "Y is green")

If X is green, the statement is true. And the antecedent is true too, hence "Y is green" is then true as well.

If X were purple, the "if" statement is false. THis only happens when the first statement "I am green" is true and "Y is green" is false. But this cannot be the case: he cannot be green and purple at once. So this is ruled out again. So X is green and Y is green.

  • Statement: I am green iff Y is green.

If X were green, this would be true and so Y has to be green as well, as X is.

If X were purple the equivalence is false and Y would be green (as X purple and Y purple makes the equivalence true). So Y is green regardless of X's colour.

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    #2 should be X green, Y green. – 2017-01-24
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    @AlexKruckman indeed, I corrected it already. – 2017-01-24