Gummy bear brands

Logic Level 1

You bought 3 packages of gummy bears from 2 different brands, Gummez and GetAddictiveCandyHere.

You aren’t sure whether you got 2 from Gummez and one from GetAddictiveCandyHere, or 2 from GetAddictiveCandyHere and one from Gummez.

However, they have some labeling.

Package 1 says that it was made by Gummez.

Package 2 says it was made by Gummez if Package 1 was made by GetAddictiveCandyHere.

Package 3 says that both other packages are lying.

You know that Gummez labeling is always true, and GetAddictiveCandyHere labeling is always false.

Given this information, how many packages were made by Gummez?

2 1

2 solutions

Richard Desper
Dec 13, 2019

All of the following triples are possible:

(Gummez, Gummez, GetAddictiveCandyHere) First statement is true. Second statement is also true, since "if" clause is not triggered. Third statement is false.
Note: second statement is logically equivalent to "Package 1 is Gummez or Package 2 is Gummez."

(GetAddictiveCandyHere, Gummez, GetAddictiveCandyHere)

First statement is false. Second statement is true. Third statement is false.

(GetAddictiveCandyHere, GetAddictiveCandyHere, Gummez)

First statement is false. Second statement is false. Third statement is true.

Douglas Foster
Dec 12, 2019

I don't think this is solveable. A quick proof would be to show that there are two possible solutions.

Solution 1: Package 1 and 2 are Addictive and are lying, while Package 3 is Gummez and is truthful. There are no contradictions as package one says it is Gummez but is lying, which makes it Addictive and that checks out, while package two says it is made by Gummez because package one is Addictive, but it is lying and is Addictive, so that checks out.

Solution 2: Package 1 and 2 are Gummez and are telling the truth, while Package 3 is Addictive and lying. There are again no contradictions as package one claims to be Gummez and it is since it is telling the truth. Package 2 is ambiguous since no information is given in the event that Package 1 is Gummez. (You only know that in the event package 1 is Addictive, then package 2 claims to be Gummez, not the other way around, if this statement instead said "Package 2 is the opposite brand of package 1" then this would not work) Therefore Package 2 could be telling the truth, there is no way to claim it is not, so it could be Gummez. Lastly Package 3 is Addictive and lying. Since package 1 and 2 are telling the truth, then package 3 is obviously lying in order to claim both others are lying.

I believe statement 2 should be changed to state that Package 2 is always the opposite brand as Package 1 and then I think the logic would follow that there is always one Gummez and two Addictive's

There's no solution two to be considered, because if package 1 is really Gummez as it claimed, then the statement on package 2 must be false if it is also a Gummez, making it a contradiction as Gummez never lies.

Saya Suka - 1 year, 6 months ago

This is not correct. The statement on Package 2 is considered to be true if its "if" clause is not triggered.

Logically, A \implies B = not( A and (not B))

Richard Desper - 1 year, 6 months ago

I'm not sure I follow? If package 1 is Gummez then nothing is claimed by package 2. Package 2 only states that if P1 = Addictive, then P2=Gummez.

However it DOESNT say anything about what happens if P1=Gummez. Therefore if P1 is Gummez the P2 statement cannot be proven or disproven.

If statement 2 said P2=Gummez IF AND ONLY IF P1=Addictive, then I would agree with you.

Douglas Foster - 1 year, 6 months ago

Yes, but as the question was posted, "You know that Gummez labeling is always true, and GetAddictiveCandyHere labeling is always false.".

Saya Suka - 1 year, 6 months ago

Suppose package 2 is manufactured by Gummez, then the statement must be true. So what happens is P1 Addy --> P2 Gummy. Since P2 is Gummy and telling the truth, we know that the Third Statement is false, leading us to the answer (A, G, A), as what you explained as your solution one.
Now suppose package 2 is manufactured by GetAddictiveCandyHere, then the statement must be false. So what happens now is P1 Addy --> P2 Addy. Since both are Addies, we now know that the Third Statement is true, the only Gummy in the batch, leading us to the answer (A, A, G). This is the other situation with a same final answer to the original question.

Saya Suka - 1 year, 6 months ago

I don't understand how you are determining the brand of P1 based on the brand of P2. Nothing in P2 say that if it is made by Gummez, then P1 is Addy. It says that if P1 is Addy then P2 is Gummez. That logic only goes one direction, but you are using it in the reverse direction.

Compare it to saying "if you get above a 90%, then you pass". That doesn't mean that if you pass you got above a 90, just that you got something above a 60% (F)

I do think your logic is the intended logic by the writer, but in its current form it doesn't follow. There needs to be an "if and only if" in statement 2

Douglas Foster - 1 year, 6 months ago

@Douglas Foster – Okay, just read an answer from another question, I didn't know of vacuous truths before. Thanks.

Saya Suka - 1 year, 6 months ago

Gummy bear brands

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