The first mathematician has $x$ and the second $y$. The first one thinks the second one has either $6^{12/x}$ or $6^{12}/x$. The second one thinks the first one has $6^{12/y}$ or $6^{12}/y$. I derived an answer based on that but I'm not sure if it's correct. The answer is:
The first one has $6^{2(3+\sqrt(6))}$ and the second one has $6^{12/(2(3+\sqrt(6)))}$
to be edited...