Consider the 24 Game with a deck of cards that forms a set $ \Omega = \{1, 2, ..., 13\}$, and we try to compute 24 using addition, subtraction, division and multiplication in Rational numbers $\mathbb{Q}$ for any set $E$ that is 4 samples with replacement from $\Omega$. Now, is there a general condition, maybe from the group theory, that specifies the conditions that $E$ must satisfy so that it is solvable?
It is trivial to actually test if any specific $E$ is solvable, by an exhaustive brute force search for all possible combinations of arithmetic operations and their order. However, is it also possible to determine solvability without such a brute force attempt? My hunch is, this could be answered using number theory, or Galois theory.
This might relate to another post, but I couldn't understand the answer in that post. Maybe it was unrelated.
