The first mathematician: Frodo The second mathematician: Saruman Frodo has $x$ and Saruman has $y$. Frodo thinks that Saruman has either $6^{12/x}$, $6^{12}/x$ or $6^{12-x}$. Saruman thinks that Frodo has $6^{12/y}$, $6^{12}/y$ 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 $6^{6}$ and the second one has $6^{6}$