Broken gem box puzzle

[Krunder]

Blue: "One of the other statements is false."
White: "One of the other boxes contains gems."
Black: "if you replace the word 'one' in the other two statements with 'both' they will both be false"

If Black is true, "both of the other statements is false" would be false because black is true "both of the other boxes contains gems" would be false because rules.
So if black is true, then black is true.
Then if white is true, then either blue or black has the gems, and then blue is false.
If white is false, it has the gems, and blue is true.

If Black is false, "both of the other statements is false" would be true, because as mentioned previously, "both of the other boxes has gems" is obviously false.
Then blue is true, because black is false.
White is either false and has the gems, or is true, and one of the others has the gems.

Am I missing something?

[Badgerlord]

In moments like these... I look at the cheaper answer... only one box has an indicator of where the gems ARE. And it says "One of the other two" Which makes it impossible to tell which other the other 2. So we look at the other statements. One of the other statements is false. If it means that Black is false, then we look Blue again. One of the other statements is false... it could be a lie. Meaning BOTH the other statements are false. And this to ME tells me that Blue Is true. White is a lie. Black doesn't matter. The gems are therefore in the White box.

That is my best guess. I don't try to overthink these puzzles and often just pick a box at random if I feel it makes my head spin, but, my first thing to look for is look for any indicator of where the gems are, and go from there.

[Dan]

Hmm, yeah, can't see what else would point to the gem location beyond "If white is true, you can't pick, so it must be false and the gems are there"

[Handful_of_Dust]

Black can never be true, because to be true, the statement "Both of the other statements are false" (which includes the statement on the black box) would have to be true. Knowing that black must be false, we know that at least one of the other two statements will be true when you replace the word "One" with the word "Both." The white box cannot possibly be true when you replace the word one with both because only one box can contain the gems, therefore, the blue box must be the one that is true when you replace one with both. This means that the statement "Both of the other statements are false" must be true, and so the white box must contain the gems, because we now know that "One of the other boxes contains gems" is a false statement.

Ignore the above, I misread one of the statements.

[Vardis]

Originally posted by Handful_of_Dust:

Black can never be true, because to be true, the statement "Both of the other statements are false" (which includes the statement on the black box) would have to be true. Knowing that black must be false, we know that at least one of the other two statements will be true when you replace the word "One" with the word "Both." The white box cannot possibly be true when you replace the word one with both because only one box can contain the gems, therefore, the blue box must be the one that is true when you replace one with both. This means that the statement "Both of the other statements are false" must be true, and so the white box must contain the gems, because we now know that "One of the other boxes contains gems" is a false statement.

Per the OP, Black's statement being true would mean that Blue's modified statement would have to be false, not true.
F/T/T looks like a valid scenario to me, but doesn't tell us where the gems are.
T/F/F is also valid, and puts the gems in White.
I wish we had a screenshot of the messages, to be sure they're accurate.

[Handful_of_Dust]

Originally posted by Vardis:

Per the OP, Black's statement being true would mean that Blue's modified statement would have to be false, not true.

Whoops, crud, you're right. I misread the black statement.

Originally posted by Vardis:

I wish we had a screenshot of the messages, to be sure they're accurate.

Yeah, agreed. Something seems off on this one.

[Krunder]

I gave in and opened white because, like badger said, at least that's a valid option that gives them a location.

And it did have the gems, so I guess hurrah, but still... it hurts a little :D

[Nokturnal]

"Blue: "One of the other statements is false."
White: "One of the other boxes contains gems."
Black: "if you replace the word 'one' in the other two statements with 'both' they will both be false""

Black can be true, obviously if white said Both of the other boxes, it'd go against game rules. Though, If Blue said Both of the other Statements are false, that could be true or false. So we have to analyze this more thoroughly.

Obviously we need something to be true or false to indicate where the gems are. If White is true, that doesn't point us anywhere. Just to a 50/50 chance. If white is false though, that means White has the gems.

Blue and Black being true or false doesn't really matter in the long run. As long as at least 1 of them is True and White is false, we know where the gems are.

In the end though, knowing the gems are in White. Black and Blue end up both being true though and White is the only false.

[Dan]

I do think there might be some overthinking going on here - White is the only information about gem location - and we know gems can only ever be in one box.
If it's true, then you lack any additional information to choose - so it's false by default, and the gems are in the white box.
I don't think you need the other two statements at all. The puzzle will never make you guess - so a solution that required you to guess is automatically not the solution, so it must be the other solution.
If one of the other statements did mention gem location, then it would complicate matters.

[Crut]

You're not considering what the "both" statement is telling you.
Consider
If black is false
Like you deduced, the modified blue statement 'both of the other statements are false.' is a true statement. (if it wasn't, then black would be false) Which means the original white box is false, not the "both" one. Ergo, the gems are in white.
if black is true,
look at the original Blue statement. If blue was false, then white would have to be true (because one box must be true). But that would mean blue was true. ergo, white must be false in this case too, gems are here.

[meep]

I know it seems like you can assume Black to be True or False and end up with two valid solutions, but you actually cannot - it has to be False.

The confusion comes when trying to analyse Black's hypothetical on Blue.

The important thing people are overlooking here is that you do not have enough information to determine the status of Black at this this stage. You instinctively want to try two branches, one where Black is True, one where it is False, but you instead have to realise that Black is INCONCLUSIVE as of now. You can consider it null or blank.

So, when you analyse the hypothetical Blue statement of "Both of the other statements is false", this too evaluates to INCONCLUSIVE, as we cannot deduce the status of Black.

This therefore means that the Black statement is False, as we cannot prove that changing both Blue and White's statements results in False.

But wait, this only tells us about one box. How do we find out where the gems are?

This where it gets fun:

You now know that Black is False, i.e. changing the statements on the other box does not result in them both being False. It can be True-True, False-True, True-False but NOT False-False.

So again, consider the hypothetical scenario it suggests. We know that the hypothetical White must be False, as two boxes cannot contain gems.

Therefore, we can deduce that hypothetical Blue must be True!

This is what tells us that the White box's original statement is False, i.e. that it contains the gems.

Important to note: Blue's original statement does not mean that ONLY one of the other statements is False.

[Dan]

Originally posted by meep:

...
...
...
This is what tells us that the White box's original statement is False, i.e. that it contains the gems.
...

This is what I'm getting at though - there's a much simpler thing that tells us White box's statement-as-written is false - the fact that if it were true, there'd be no other information available to tell you which box the gems are in. Given that single fact alone, it must be false, and therefore you know where the gems are.
You don't even need to test the true/false status of the other two boxes, you don't need to substitute "both" into other statements, etc. - neither of the other boxes reference the gem location, therefore they can be discounted entirely.

Assuming I haven't missed something, that is.

[meep]

Originally posted by Dan:

Originally posted by meep:

...
...
...
This is what tells us that the White box's original statement is False, i.e. that it contains the gems.
...

This is what I'm getting at though - there's a much simpler thing that tells us White box's statement-as-written is false - the fact that if it were true, there'd be no other information available to tell you which box the gems are in. Given that single fact alone, it must be false, and therefore you know where the gems are.
You don't even need to test the true/false status of the other two boxes, you don't need to substitute "both" into other statements, etc. - neither of the other boxes reference the gem location, therefore they can be discounted entirely.

Assuming I haven't missed something, that is.

No you are right, in this case, and in some of the other Parlor puzzles I think, you can "skip" to the solution by realising only one box being in a particular state tells you where the gems are. The reason I gave my longwinded explanation is to demonstrate why Black cannot be True, for anyone confused on how you are supposed to eliminate the FTT (and TFT) scenario.

The reason for highlighting it here is that there are other Parlor puzzles, without a way to "skip", where you cannot assume a statement to be True/False, and instead have to treat it as inconclusive.

[BobThePenguin]

If we start by looking at Black, you can rewrite it as:

The following two statements are false: "White and Black are both false" and "White and Black both contain gems".

The second sub-statement is always false since only one box can contain gems. So we can reduce Black's statement to:
The following statement is false: "White and Black are both false".

If we assume Black is false, that would mean that the sub-statement "White and Black are both false" is true, so White and Black are both false and Blue is true. The gems are in White.

If we assume Black is true, then Black's sub-statement "White and Black are both false" is false, meaning "White and Black are not both false", which gives us no further information since we already assumed Black is true. That means that either White or Blue or both are false. If we assume Blue is false, then White must be true and we're left with a 50/50 choice for gem location, so that cannot be correct. If we assume Blue is true, then White must be false and the gems are in White.

Or you can skip all that logic deduction and notice that only White gives information as to the whereabouts of the gems, so it must be false and the gems are in White.

[meep]

Originally posted by BobThePenguin:

If we assume Black is true, then Black's sub-statement "White and Black are both false" is false, meaning "White and Black are not both false", which gives us no further information since we already assumed Black is true. That means that either White or Blue or both are false. If we assume Blue is false, then White must be true and we're left with a 50/50 choice for gem location, so that cannot be correct. If we assume Blue is true, then White must be false and the gems are in White.

You're right that this is enough to lead you to choose the TFT solution over the FTT one, but it's unsatisfying as the latter is still technically valid as per the rules (although in reality, neither are).

Instead, you need to prove that Blue is True and that Black is False, and achieve a TFF solution.

But how do you prove Black to be False?

Well, when we first start looking for contradictions in the puzzle, we can't immediately disprove Blue, so we move onto White.
We can't disprove that either, so we move onto Black.

Black's state is also immediately unprovable.
So, when we try and analyse the hypothetical it suggests, of course changing White to "Both" returns False as we know, but the outcome of changing Blue cannot be deduced.
We can't branch off in two paths, one where Black is True and another where it is False, because neither actually prove whether Black is T/F.
No, since we don't know the outcome of Black, then we don't know the outcome of changing Blue to "Both".
Therefore, changing both Blue and White to "Both" does not result in False-False, so at this stage we can deduce that Black is False.

Black being False means that changing the verbiage of Blue and White to "Both" cannot result in both being False.
We know that changing White to "Both" must return False, therefore changing Blue to "Both" must return True.
This now tells us that the boxes are TFF, i.e. White is False, and that the gems are inside of it.