OBM 2001 Problem 6

A one-player game is played as follows: There is a bowl at each integer on the $Ox$ -axis. All the bowls are initially empty, except for that at the origin, which contains $n \geq 2$ stones. A move is either

(A) to remove two stones from a bowl and place one in each of the two adjacent bowls, or

(B) to remove a stone from each of two adjacent bowls and to add one stone to the bowl immediately to their left.

Show that only a finite number of moves can be made and that the final position (when no more moves are possible) is independent of the moves made (for a given $n$ ).