4-68 In a competition with $n(n \geqslant 2)$ participants lasting $k$ days, each participant's score each day is exactly $1,2, \cdots, n$ (no two participants have the same score), and after $k$ days, each participant's total score is 26. Try to find all possible values of $(n, k)$.

In a competition with $n(n \geqslant 2)$ participants lasting $k$ days, each participant's score each day is exactly $1,2, \cdots, n$ (no two participants have the same score), and after $k$ days, each participant's total score is 26. Try to find all possible values of $(n, k)$.
[Sol] Calculate the total score of all players in two different ways, we get
$$k(1+2+\cdots+n)=26 n$$

So $k(n+1)=52$.
And $52=2^{2} \times 13$, which has 6 factors: $1,2,4,13,26,52$. Since $n+1 \geqslant 3, k \geqslant$ 2 (if there is only one day, it is impossible for all players to have the same total score), there are only three solutions:
$$(n, k)=(25,2),(12,4),(3,13)$$

All three solutions are possible: