$13 \cdot 12$ On the two pans of a balance, there are $k$ weights each, numbered from 1 to $k$, and the left pan is heavier. If any two weights with the same number are exchanged, the right pan always becomes heavier or the pans balance. Find all possible values of $k$.


[Solution] From the given information, each weight on the left pan is heavier than the corresponding weight on the right pan, denoted by the weight difference as $m_{i}, i=1,2, \cdots, k$. Since by swapping a pair of weights with the same number, the right pan becomes heavier or balanced, we have
$$
m_{i} \geqslant \sum_{j \neq i} m_{j}, i=1,2, \cdots, k .
$$

Summing the above equation for $i$ from 1 to $k$ yields
$$
\sum_{i=1}^{k} m_{i} \geqslant \sum_{i=1}^{k} \sum_{j \neq i} m_{j}=(k-1) \sum_{i=1}^{k} m_{i} \text {. }
$$

This leads to $k \leqslant 2$. When $k=1$, it is obviously true; when $k=2$, the two weights in each pan can be equal. Therefore, the values of $k$ that satisfy the requirements of the problem are only two: 1 and 2.