CGMO 2011 Problem 7

There are $n$ boxes ${B_1},{B_2},\ldots,{B_n}$ from left to right, and there are $n$ balls in these boxes. If there is at least $1$ ball in ${B_1}$ , we can move one to ${B_2}$ . If there is at least $1$ ball in ${B_n}$ , we can move one to ${B_{n - 1}}$ . If there are at least $2$ balls in ${B_k}$ , $2 \leq k \leq n - 1$ we can move one to ${B_{k - 1}}$ , and one to ${B_{k + 1}}$ . Prove that, for any arrangement of the $n$ balls, we can achieve that each box has one ball in it.