In a game, there exist three piles of marbles, each pile with $\displaystyle a$, $\displaystyle b$, and $\displaystyle c$ marbles respectively, where $\displaystyle a,b,c$ are natural numbers and $\displaystyle a \neq b \neq c$. At each turn, you can double the number of marbles in one pile by transporting marbles from one other larger pile (relative to the pile that is going to be doubled, before the doubling). The game is won when any two of the piles have an equal number of marbles.

Either show that the game can be won from any starting $\displaystyle a,b,c$, or prove that this is not the case. (In particular, prove that the game cannot be won from every starting $\displaystyle a,b,c$.)

A pretty problem, but enlightenment does not come...

EDIT: Sorry, I recalled the problem from memory, and your post catalyzed my editing of it (your interpretation of what I posted was correct, but I did not remember the problem correctly.)