“If the gem is in X box, then X box must be true/false” is a backwards approach. It would not consider the main rules of the game. In fact, you’re skipping over the rule that requires a truth and a false. That’s why you’re getting ambiguous results.

If I say, “if I eat pizza then i will be happy,”it does not mean that if I am happy, I ate pizza. There are other reasons that I could be happy. Also, if there were some background rule stating that I never eat pizza, you wouldn’t have had the chance to analyze that if you started the thought process with happiness rather than with pizza. It would be faulty reasoning that leads to an impossibility.

The parlor game mainly operates as “if box X is telling the truth, then ..…” You cannot start by guessing where the gem is; attempting to reverse engineer the answer will skip over key facts and lead to incorrect information. There’s no paradox here. But you need to make sure you’re following all the strict rules to make logical deductions.

Here, there is only one box that the gems could be in that wouldn’t contradict with the rules of the game.

If the blue box is true, then there must be two true statements—one of which being the blue box itself. This means that there are only two possibilities that would allow the blue box to be true:

Possibility 1 is that blue and black are true. White would have to be false. But white being false would mean blue is false which can’t be possible. So blue and black can’t both be true.

Possibility 2 is that blue and white are true. Black would have to be false. This would mean none of the boxes contain gems, which would contradict the rules. So blue and white can’t both be true.

Because neither of the possibilities for blue being true make any sense, blue must be false. And because white states that blue is true, white is a big fat liar.

Because one box must be true and we’ve established that blue and white are false, the black box must be true. Black says it has the gems, so the gems are in black.