3 There are $n$ players participating in a competition lasting $k$ days, where the scores of the $n$ players on any given day are exactly a permutation of $1,2,3, \cdots, n$. If at the end of the $k$th day, each player's total score is 26. Find all possible values of $(n, k)$.


3. From the condition "each player's total score is 26", we think of calculating the sum of scores $S$ of $n$ players after $k$ days. On one hand, the score each day is $1+2+3+\cdots+n$, so $S=k(1+2+\cdots+n)$. On the other hand, each player scores 26 points, thus $S=26n$. Therefore, $k(n+1)=52$. So $(n, k)=$ $(51,1) 、(25,2) 、(12,4) 、(3,13)$. When $(n, k)=(51,1)$, the total scores of each player are distinct, which is a contradiction, so it is discarded. From the following construction, we know that the other 3 cases are possible. When $(n, k)=(25,2)$, the rank set of the $i$-th player is $A_{i}=\{i, 26-i\}(i=1,2, \cdots, 25)$. When $(n, k)=(12,4)$, the rank set of the $i$-th player is $A_{i}=\{i, 13-i, i, 13-i\}(i=1,2, \cdots, 12)$. When $(n, k)=(3,13)$, the rank sets of each player are $A_{1}=$ $\{2,3,1\} \cup\{1,3,1,3, \cdots, 1,3\} 、 A_{2}=\{3,1,2\} \cup\{2,2,2,2, \cdots, 2, 2\} 、 A_{3}=\{1,2,3\} \cup\{3,1,3,1, \cdots, 3,1\}$