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We can now solve The Hardest Logic Puzzle Ever.
Your first move is to find a god who you can be certain is not Random, and
hence is either True or False.
To do so, turn to A and ask Question 1: Does da mean yes iff, you are True iff B is Random? If A is True or False and you get the answer da, then as we have seen, B is Random, and therefore C is either True or False; but if A is True or False and you get the answer ja, then B is not Random, therefore B is either True or False.
But what if A is Random?
If A is Random, then neither B nor C is Random!
So if A is Random and you get the answer da, C is not Random (neither is B,
but that’s irrelevant), and therefore C is either True or False; and if A is Random and you get the answer ja, B is not random (neither is C, irrelevantly), and therefore B is either True or False.
Thus, no matter whether A is True, False, or Random, if you get the answer da
to Question 1, C is either True or False, and if you get the answer ja, B is either True or False!
Now turn to whichever of B and C you have just discovered is either True or
False — let us suppose that it is B (if it is C, just interchange the names B and C in what follows) — and ask Question 2: Does da mean yes iff Rome is in Italy? True will answer da, and False will answer ja. Thus, with two questions, you have either identified B as True or identified B as False.
For our third and last question, turn again to B, whom you have now either
identified as True or identified as False, and ask Question 3: Does da mean yes iff A is Random?
Suppose B is True. Then if you get the answer da, then A is Random, and
therefore A is Random, B is True, C is False, and you are done; but if you get the answer ja, then A is not Random, so A is False, B is true, C is Random, and you are again done.
Suppose B is False. Then if you get the answer da, then since B speaks falsely, A is not Random, and therefore A is True, B is False, C is Random, and you are done; but if we get ja, then A is Random, and thus B is False, and C is True, and you are again done.
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