Number-Theoretic Coin Puzzle

Question

There are three piles of coins. You are allowed to move coins from one pile to another, but only if the number of coins in the destination pile is doubled. For example, if the first pile has 6 coins and the second pile has 4 coins, then you may move 4 coins from the first pile to the second pile — no more or less. Prove that by repeating moves of this sort, it is possible to empty one of the piles.

Despite my best efforts, I have been unable to find a general solution strategy. Any pointers would be much appreciated.

Background: A special case of the puzzle appeared in The Moscow Puzzles: 359 Mathematical Recreations.

Starting with piles $(11,7,6)$, can you move coins (as specified above) so that you have piles $(8,8,8)$?

Yes, $(11,7,6)\to(4,14,6)\to(4,8,12)\to(8,8,8)$ works.

Answer 1

The solution is detailed in this paper by Peter Winkler: http://math.dartmouth.edu/~pw/papers/algor.ps

Answer 2

If I understand correctly, then this game is the same as a question featured a while ago by USAMTS (year 23, round 1, question 5). They called the game "Tristack Solitaire" and the goal was to set two piles equal to each other. Once that goal is accomplished, then it simply takes one additional move to set one pile to zero.

Here is the solutions document for the relevant puzzle set, which provides two full proofs and the algorithm. (This is problem #5, so it is towards the end).