Question 37
Suppose $n$ small balls have been placed into $n$ numbered boxes $B_1, B_2, \dots, B_n$. Each time we can select a box $B_k$ and do the following operations:
(1) If $k=1$ and there is at least one ball in $B_1$, move one ball from $B_1$ into $B_2$.
(2) If $k=n$ and there is at least one ball in $B_n$, move one ball from $B_n$ into $B_{n-1}$.
(3) If $2 \le k \le n-1$ and there are at least two balls in $B_k$, move one ball from $B_k$ into $B_{k+1}$ and one ball into $B_{k-1}$, respectively.
Prove the following: no matter how the balls are distributed among the boxes originally, it is always realizable to let each box contain exactly one ball by finite operations.
(1) If $k=1$ and there is at least one ball in $B_1$, move one ball from $B_1$ into $B_2$.
(2) If $k=n$ and there is at least one ball in $B_n$, move one ball from $B_n$ into $B_{n-1}$.
(3) If $2 \le k \le n-1$ and there are at least two balls in $B_k$, move one ball from $B_k$ into $B_{k+1}$ and one ball into $B_{k-1}$, respectively.
Prove the following: no matter how the balls are distributed among the boxes originally, it is always realizable to let each box contain exactly one ball by finite operations.