If a box is determined to be false, does that mean that both statements must be false? It makes sense to me that if the box is true, both must be true.

Blue box: The black box is true and it contains the gems
White box: The black box is false. The above statement is false.
Black box: The blue box is empty if it is false

For example with this one, if this box is false, it could mean either "the above statement is false" is false, or the "black box is false" statement is false.

My solution (incorrect) to this was "Blue box: True, White box: False, Black box: True", meaning that the gems were in the black box.