Question 5
Let $n\geq3$ be an integer. We say that an arrangement of the numbers
$1$, $2$, …, $n^2$ in an $n \times n$ table is row-valid
if the numbers in each row can be permuted to form an arithmetic progression,
and column-valid if the numbers in each column
can be permuted to form an arithmetic progression.
For what values of $n$ is it possible to transform any row-valid arrangement
into a column-valid arrangement by permuting the numbers in each row?
Proposed by Ankan Bhattacharya. One nonofficial editorial point; manual essay grading.
$1$, $2$, …, $n^2$ in an $n \times n$ table is row-valid
if the numbers in each row can be permuted to form an arithmetic progression,
and column-valid if the numbers in each column
can be permuted to form an arithmetic progression.
For what values of $n$ is it possible to transform any row-valid arrangement
into a column-valid arrangement by permuting the numbers in each row?
Proposed by Ankan Bhattacharya. One nonofficial editorial point; manual essay grading.