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 $(n, k)$.


[Solution] Using two different methods to calculate the total score of all participants, we get
$$
k(1+2+\cdots+n)=26 n,
$$

so $k(n+1)=52$.
Since $52=2^{2} \times 13$, there are 6 factors: $1,2,4,13,26,52$. Given that $n+1 \geqslant 3, k \geqslant 2$ (if there is only one day, it is impossible for all participants to have the same total score), there are only three sets of solutions:
$$
(n, k)=(25,2),(12,4),(3,13) \text {. }
$$

These three sets of solutions are all possible:
\begin{tabular}{|c|c|c|}
\hline & & $n=25$ \\
\hline \begin{tabular}{l} 
k
\end{tabular} & $1, \quad 2,3, \cdots$ & 25 \\
\hline 2 & $25,24,23, \cdots$ & 1 \\
\hline
\end{tabular}
\begin{tabular}{l|lllll} 
& & & $n=12$ \\
\hline$k$ & 1, & 2, & 3, & $\cdots$, & 12 \\
12, & 11, & 10, & $\cdots$, & 1 \\
4 & $1, \quad 2$, & 3, & $\cdots$, & 12 \\
12, & 11, & 10, & $\cdots$, & 1
\end{tabular}
\begin{tabular}{l|lllllllllllll}
\multicolumn{11}{c}{$k=13$} \\
\hline$n$ & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 \\
$\|$ & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 & 1 & 1 & 1 & 1 & 2 \\
3 & 3 & 3 & 3 & 3 & 1 & 1 & 1 & 1 & 3 & 2 & 2 & 2 & 1
\end{tabular}