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- If the third box is true, then both first and second box contain gems, which would break the rules of the puzzle, and thus is impossible.
- The other two boxes are completely unimportant in truth values, as the boxes "claim" to be true is what matters for the third. As far as their truth values go, they're beside the point, and are entirely ambiguous. "This statement is false" is a paradox as it claims falsehood, but a statement claiming truth is not a paradox -- if "This statement is true" is true, then it's self-evident, and if it is false, then given it says it's true, it is inaccurate, and thus false, which is clear and logically sound.
Situation 1: Box 1 contains gems. In this case, Box 2, that claims to be true, does not contain gems, so the statement "Boxes that claim to be true contain gems" is false -- only 1 box that claims to be true contains gems, so "Boxes" do not.
Situation 2: Box 2 contains gems. In this case, Box 1, that claims to be true, does not contain gems, so the statement "Boxes that claim to be true contain gems" is false -- only 1 box that claims to be true contains gems, so "Boxes" do not.
Situation 3: Box 3 contains gems. In this case, no box that claims to be true contains gems, so the statement "Boxes that claim to be true contain gems" is false.