Question: For which orders $n$ does there exist a Latin square of order $n$ which remains a Latin square after replacing each row by its square (when the rows are interpreted as permutations)?
If $L=(l_{ij})$ then row $i$ corresponds to the permutation $j \mapsto l_{ij}$.
For example, the rows of this (randomly generated) Latin square $$\begin{bmatrix} 1 & 3 & 4 & 2 \\ 4 & 2 & 1 & 3 \\ 3 & 1 & 2 & 4 \\ 2 & 4 & 3 & 1 \\ \end{bmatrix} \leftrightarrow \begin{bmatrix} (234) \\ (143) \\ (132) \\ (124) \\ \end{bmatrix} $$ and if we square the permuations on the right, we get $$ \begin{bmatrix} (234)^2 \\ (143)^2 \\ (132)^2 \\ (124)^2 \\ \end{bmatrix} = \begin{bmatrix} (243) \\ (134) \\ (123) \\ (142) \\ \end{bmatrix} \leftrightarrow \begin{bmatrix} 1 & 4 & 2 & 3 \\ 3 & 2 & 4 & 1 \\ 2 & 3 & 1 & 4 \\ 4 & 1 & 3 & 2 \\ \end{bmatrix}.$$
A randomly generated order-$5$ example is: $$\begin{bmatrix} 1 & 4 & 2 & 5 & 3 \\ 4 & 5 & 3 & 2 & 1 \\ 3 & 1 & 5 & 4 & 2 \\ 2 & 3 & 4 & 1 & 5 \\ 5 & 2 & 1 & 3 & 4 \\ \end{bmatrix} \leftrightarrow \begin{bmatrix} (2453) \\ (1425) \\ (1352) \\ (1234) \\ (1543) \\ \end{bmatrix} $$ After squaring the rows, we get $$ \begin{bmatrix} (2453)^2 \\ (1425)^2 \\ (1352)^2 \\ (1234)^2 \\ (1543)^2 \\ \end{bmatrix} = \begin{bmatrix} (25)(34) \\ (12)(45) \\ (15)(23) \\ (13)(24) \\ (14)(35) \\ \end{bmatrix} \leftrightarrow \begin{bmatrix} 1 & 5 & 4 & 3 & 2 \\ 2 & 1 & 3 & 5 & 4 \\ 5 & 3 & 2 & 4 & 1 \\ 3 & 4 & 1 & 2 & 5 \\ 4 & 2 & 5 & 1 & 3 \\ \end{bmatrix} $$
After generating $60000$ random Latin squares of order $6$, my computer didn't find an example.
This question is something I happen to be thinking about. I haven't put too much thought into solving this; it's just a curious distraction vaguely related to a research problem.