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 $6$ and
>! the second one has $6$