Brain Teasers
Where's the Gold?
Two boxes are labeled "A" and "B". The sign on box A says, "The sign on box B is true and the gold is in box A". The sign on box B says, "The sign on box A is false and the gold is in box A". Assuming there is gold in one of the boxes, which box contains the gold?
1) Box A
2) Box B
3) You can't determine from the information given.
1) Box A
2) Box B
3) You can't determine from the information given.
Answer
3) You can't determine from the information given.The sign on box A says, "The sign on box B is true and the gold is in box A". The sign on box B says, "The sign on box A is false and the gold is in box A". The following argument can be made: If the statement on box A is true, then the statement on box B is true, since that is what the statement on box A says. However, the statement on box B states that the statement on box A is false, which contradicts the original assumption. Therefore, the statement on box A must be false. This implies that either the statement on box B is false or that the gold is in box B. If the statement on box B is false, then either the statement on box A is true (which it cannot be) or the gold is in box B. Either way, the gold is in box B.
However, there is a hidden assumption in this argument, namely, that each statement must be either true or false. This assumption leads to paradoxes; for example, consider the statement: "This statement is false." If it is true, it is false; if it is false, it is true. The only way out of the paradox is to deny that the statement is either true or false and label it meaningless instead. Both of the statements on the boxes are therefore meaningless and nothing can be concluded from them.
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Comments
There is no paradox here. There are compound statements - if either is false then the whole statement is false. Both statements are false - the only true part is that the statement on A is false. The gold is in box B.
I don't understand.Sign B might be wrong. Sign A is saying Sign B is right even though Sign B is saying Sign A is wrong. If A is false and B is true then however both say that A has the gold. The answer could be that A has the gold.I don't know if this theory is right but I am positive about tis teaser.
I strongly agree. Even though the first statement in Box B is false it wouldnt matter because both boxes indicate that it's in Box A.
i didnt understand
Gizzer's right. (I think.) If the gold is in box B, then both statements are false (even though part of one of them is true).
For the logically-minded: there are three variables here:
P: The statement on box A is true.
Q: The statement on box B is true.
R: The gold is in box A.
(Since the gold is in either box A or box B, if R is false, then the gold is in box B.)
The following are known to be true:
P if and only if (Q and R).
Q if and only if (~P and R).
Note: If P is true, then ~P is false, and vice-versa.
How did I get these? Notice the second parts are the statements of the boxes. Obviously, if the statement on the box is true, then the statement on the box is true, and if the statement is false, then it is false.
There are eight possibilities:
P Q R
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
The first biconditional is only true if truth values are assigned to P, Q, and R in one of the following ways:
P Q R
T T T
F T F
F F T
F F F
But of those ways, only the last will cause the second biconditional to be true:
P Q R
F F F
Meaning the statements on both boxes are false, and the gold is in box B. There is no contradiction.
Gosh, that was a mouthful.
P: The statement on box A is true.
Q: The statement on box B is true.
R: The gold is in box A.
(Since the gold is in either box A or box B, if R is false, then the gold is in box B.)
The following are known to be true:
P if and only if (Q and R).
Q if and only if (~P and R).
Note: If P is true, then ~P is false, and vice-versa.
How did I get these? Notice the second parts are the statements of the boxes. Obviously, if the statement on the box is true, then the statement on the box is true, and if the statement is false, then it is false.
There are eight possibilities:
P Q R
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
The first biconditional is only true if truth values are assigned to P, Q, and R in one of the following ways:
P Q R
T T T
F T F
F F T
F F F
But of those ways, only the last will cause the second biconditional to be true:
P Q R
F F F
Meaning the statements on both boxes are false, and the gold is in box B. There is no contradiction.
Gosh, that was a mouthful.
Sorry if that was difficult to understand; for some reason, all of the times I hit the Enter/REtuen key did not show up. Maybe my comment was too long...
Poker, I think you found the contradiction yourself. Your conclusion
"Meaning the statements on both boxes are false, and the gold is in box B. " is true. Yet, it is possible to take 2 boxes, label them as in the teaser, put the gold in box A and you won't find the gold in box B no matter how much logic you put in it
"Meaning the statements on both boxes are false, and the gold is in box B. " is true. Yet, it is possible to take 2 boxes, label them as in the teaser, put the gold in box A and you won't find the gold in box B no matter how much logic you put in it
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