1. It is easy to know that the total sum of $k$ days is
$$k \times \frac{1}{2} n(n+1)=26 n .$$

Therefore, $k(n+1)=52$.
Thus, $(n, k)=(51,1),(25,2),(12,4)$, $(3,13)$. Only $(51,1)$ is impossible to achieve.
For $(25,2)$, we have
$$\begin{array}{l} (26,26, \cdots, 26)=(1,2, \cdots, 24,25) \\ +(25,24, \cdots, 2,1) . \end{array}$$

For $(12,4)$, we have
$$\begin{array}{l} (26,26, \cdots, 26)=2(1,2, \cdots, 11,12) \\ +2(12,11, \cdots, 2,1) . \end{array}$$

For $(3,13)$, we have
$$\begin{array}{l} (26,26,26)=(1,2,3)+2(2,3,1) \\ +2(3,1,2)+3(1,3,2)+2(3,2,1) \\ +3(2,1,3) \end{array}$$