How can you deal with a puzzle where some or all of the statements are falsehoods?
Actually, it's not too bad. A known lie is just as useful as a known truth. You just have to think about what the opposite of the false statement is. For example, if you know that the statement "Either chest A or B contains gold" is false, then you know that neither chest A nor chest B can contain gold.
Let's go through a full example.
In this problem, the three clues in the bulleted list above the chests are all guaranteed to be true. On the other hand, the statement on each of the three chests might either be true or false. Our goal is to deduce if the statement on each chest is true or false and, ultimately, to figure out whether or not there's gold in each chest.
You can turn the buttons on and off by tapping them.
The statements on the chests give mixed messages. First, let's look for the statements that seem to contradict each other:
The statements on chests A and B are both about the contents in chest A. Since the chest either contains gold or not, one of these two statements must be true and the other must be false. They cannot both be true and they cannot both be false.
The statements on chests A and C cannot both be false. To make statement A is false, there would need to be gold in chest A. To make statement C false, there would need to be gold in chest C. However, because we know that there is gold only in one chest, these two statements cannot both be false.
We can use either of these facts as a starting point to determine where the gold is. Here are the steps we used to finish the problem. (You can also finish it yourself using the buttons in the problem above.)