Logic nonsense/paradox I'm not sure if this is a paradox or a nonsense or neither of both.
Anyway this is the "problem" if we can call it like that:

A: B is True
B: A is False


How can you solve it?
It is not much different from the plain liar paradox:

A: A is false


There are various more or less contrived "philosophical" attempts to resolve it, but by far the most common resolution is to deny that the statement means anything in the first place; therefore it is also meaningless to ask for its truth value.

The formalist mathematical approach to the paradox is to note that a sentence that speaks directly and explicitly about its own truth value (or, as in your original example, a set of sentences that speak mutually about each other's truth value) cannot be formalized in the systems that are commonly used as foundations for mathematics.

I fact, it is a classical result (by Tarski) that it is not possible for a sentence in formal mathematics to express "A is true" (at the same metalevel) of any (symbolically given) sentence $A$.

(See also here for additional discussion about the unwillingness of mathematics to deal with such self-reference).
I've found most satisfactory (i.e. true) the interpretation that each sentence implicitly states that is true.

Meaningful interpretation presumes true premises. The statement A:A is false converts to 'A is true and A is false' rendering it meaningless. In that sentence we are not talking about anything that exists. We are talking about nothing.

Sadly this interpretation also invalidates many branches of mathematics and e.g. the classical Banach-Tarksi -paradox, which btw was rejected by the founders due to the same idea that mathematical truths should also corresponds to reality.