I don't understand this Parlor puzzle and it might rely on a fallacy.

[SirRastusBear]

EDIT: This is resolved. I thought the note in the parlor said that one box must be true and that one box must be false, not that all statements in one box must be true and all in one box must be false, so I thought a box with one false statement and one true would ultimately count as a false box, meaning the other two could be true.

I was watching someone's stream and am confused by this parlor puzzle. Here's the arrangement:

Blue box: "The black box is true and it contains the gems"
White box: "The black box is false" and "The above statement is false"
Black box: "The blue box is empty if it is false"

Let's start by saying that the statement "The black box is false" in the white box has to be false. If it were true, that would mean the black box is false, the white box would also be false (since the other statement in it would become false), and the blue box would also be false since it states that the black one is true.

All boxes can't be false, so the statement "The black box is false" has to be false and the black box HAS TO BE TRUE, right?

Following that, if we assume the blue box is false, that means the black box is empty (since we already know the black box is true, the part of the statement that would be false would be the black box containing the gems). Then, because of the true statement in the black box, the blue box would also be empty, and the gems are in the white box.

However, if we assume the blue box is true, then that would mean that the black box contains the gems. I think the point of the puzzle is that you should believe that, because the black box statement is true, then it should not be possible for the blue box to be true and ALSO be empty, but this reasoning relies on a fallacy called "affirming the consequent". In other words, if the statement "The blue box is empty if it is false" is true, then this means that the blue box can either be true and have the gems, false and empty, or true and empty, but not false and have the gems.

In the stream, the player opened the black box following a similar reasoning and the gems weren't there. I assume the gems were in the white box, but the whole puzzle seems ambiguous and that it is equally possible for the gems to be in the white or the black box. Am I missing something, or is the puzzle wrong?

If I'm right about this, the black box should have said "The blue box is empty ONLY if it is false."

[Vardis]

Start with the white box, since it's self-referential. The white box can't display only true statements, since it claims one of its own statements is false. The white box also can't display only false statements, for the same reason.

Thus one of the black box or blue box has to display only true statements and the other only false. The blue box says that the black box is true, so since they can't both be true, we know blue is false, black is true, and white is neither (all the time).

Since blue is false, the statement "The black box is true and it contains the gems" is false. Since the black box is true, it then must not contain the gems to make that statement false.

Since black is true, we know blue doesn't have the gems, since blue is false and thus must be empty per black's statement. Thus the gems should be in white.

[Aris]

The white box is neither true or false. It contains 2 statements and they can't be both true or false at the same time.

The rules state "There's at least one box that only has true statements" and "There's at least ont box that only has false statements". The white box can't fullfill neither of these, since it has 2 statements and one is true and the other is false.

That means that you can pretty much ignore the white box.

I don't have much more time to look into the rest, so not sure if what I just said explains everyting, I'll look into that later.

[SirRastusBear]

Originally posted by Vardis:

Start with the white box, since it's self-referential. The white box can't display only true statements, since it claims one of its own statements is false. The white box also can't display only false statements, for the same reason.

Ohh I understand now. I thought the note in the parlor said that one box must be true and that one box must be false, not that all statements in one box must be true and all in one box must be false, so I thought a box with one false statement and one true would ultimately count as a false box, meaning it was possible in this case for both blue and black boxes to be true. Thanks for the help!

[GuyYouMetOnline]

The black box can't be false because if it was, the statement that's it's true would be false, leaving no box with only true statements. So the black box is true, and the blue box is false. But since the black box is true, the only option is that the false part on the blue box is that the black box has the gems. So we know, then, that the gems aren't in the black box, and since we know they're not in a false blue box, they can only be in the white box.

One trick to be aware of is that if there are multiple statements on a box, they don't all have to be true or false; there can be a mix as long as there's still one box with only true statements and one with only false statements.

EDIT: Didn't notice someone else had already explained that.