# Thread: Game theory of marbles

1. ## Game theory of marbles

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.)

2. ## Re: Game theory of marbles

the game is not unwinnable for every choice of a,b,c since one scenario is the following:

a = 1, b = 2, c = 3.

move 1: move 2 marbles from pile c to pile b. now pile 1 has 1, pile 2 has 4, and pile 3 has 1, game over.

this suggests that you try to show the game is winnable.

3. ## Re: Game theory of marbles

Thanks for responding, Deveno. However, I realized that my problem statement was incorrect. Thus, your insight is unfortunately nonapplicable (sorry for the error)

4. ## Re: Game theory of marbles

ah, so you must prove whether or not some intitial combination is unwinnable.