Portia casket logic problem. Two boxes, one is gold and the other silver. The sign on the gold box reads, "The portrait is not here." The sign on the silver one reads, "Exactly one of the two statements is true." Guess where the portrait is. What I want to see is how you reason the problem.
 A: The portrait could be in either box because anyone could write all kinds of nonsense on a box and it wouldn't affect what's inside!
This is obvious when you think about it for a moment.  But what is not at all obvious is why the seemingly logical argument that the portrait is in the gold box fails.  Indeed, if the inscription on the silver box is true, then the portrait is in the gold box, and if the inscription on the silver box is false, then the portrait is in the gold box.  The only explanation is that the seemingly harmless assumption "either the inscription on the silver box is true, or the inscription on the silver box is false" is not justified.
To understand this explanation, let's first consider the more fundamental "liar paradox." Consider the sentence $\Phi$ which says "this sentence is false."  You can even write it on a box if you like.  If $\Phi$ is true, then it is false, which is a contradiction.  If it is false, then it is true, and again we have a contradiction.  Therefore the assumption "$\Phi$ is true or $\Phi$ is false" leads to a contradiction; in other words, this assumption is wrong somehow.
Although classical mathematical logic admits a rule (the law of excluded middle) saying "$\psi$ is true or $\psi$ is false" for every sentence $\psi$, we must not interpret this rule out of context.  Here "every sentence $\psi$" only applies to sentences in the realm of mathematical logic.
Some English sentences such as "How tall are you?" or "One cup of coffee, please" would not qualify; we cannot deem them either true or false.  One might hope that the sentence "this sentence is false" might qualify, because it seems to make an assertion of fact.  But as we have seen, we can't allow such self-referential statements without being led into contradiction.
Returning to the question, let's observe that the inscription on the silver box is equivalent to the statement $\Phi$ saying "the portrait is not in the gold box if and only if this statement is false."  The argument from this point on can be rephrased as follows.

Assume toward a contradiction that the portrait is not in the gold box. Then $\Phi$ is equivalent to the statement "this statement is false," and when we attempt to evaluate the truth value of this statement, we are led into contradiction.  Therefore the portrait is in the gold box.

So we see that the liar paradox has been cleverly concealed so that it figures into a proof by contradiction, rather than leading to an outright contradiction.  But the mistake is the same as always: we carelessly assumed that a self-referential statement could be assigned a truth value.
To make it clear why this is a mistake, notice that an analogous argument (if it were valid) could be used to prove anything whatsoever.
For example, consider another sentence $\Phi'$ saying "the portrait is not on the moon if and only if this statement is false."  (Again, you can write this on a box if you like.)  Then we can argue as follows.

Assume toward a contradiction that the portrait is not on the moon. Then $\Phi'$ is equivalent to the statement "this statement is false," and when we attempt to evaluate the truth value of this statement, we are led into contradiction.  Therefore the portrait is on the moon.


EDIT:  To see the invalidity of the other answers more directly, let's consider a slight variation of the original problem:

Suppose we have two boxes, one gold and the other silver. The statement on the gold box says "the portrait is not on the moon." The statement on the silver one says "exactly one of these two statements is true." Where is the portrait?

If the statement on the sliver box is true, then the statement on the gold box is false, so the portrait is on the moon.  If the statement on the silver box is false, then again the statement on the gold box is false, so the portrait is also on the moon in this case.  In either case the portrait is on the moon, so the portrait must be on the moon.  If this argument were valid, you could prove the portrait to be anywhere you like, simply by replacing the word "here" (or the words "on the moon") on the gold box with any other location!  So clearly the argument cannot be valid.
A: Suppose the Silver one is true, then the Gold one must be false, so it's in the gold box.
Suppose the Silver one is false, then obviously we can't have both statements be true, so again the gold one must be false. So again it is in the gold box.
A: One possibility is that the portrait is in the gold box. If the silver one is true, then the gold one is false. If the silver one is false, then the gold one is false. If the gold one is true, that is not possible.
Edit: The other possibility is to ignore what the signs say and the portrait could be in any box.
