Question 65
For a positive integer $n$, we say that an array with three rows and $n$ columns is *magical* if there exists a positive integer $k$, $1 \le k \le n$, such that
* the first row is filled with numbers $1, 2, \ldots, n$ in that order
* the second row is filled with numbers $k, k+1, \ldots, n, 1, 2, \ldots, k-1$ in that order
* the third row is filled with numbers between $1$ and $n$ in such order that the sums of the three numbers in each column are all the same.
Determine all positive integers $n$ for which a magical array exists, and for each such $n$ determine the number of magical arrays. (Ireland)
* the first row is filled with numbers $1, 2, \ldots, n$ in that order
* the second row is filled with numbers $k, k+1, \ldots, n, 1, 2, \ldots, k-1$ in that order
* the third row is filled with numbers between $1$ and $n$ in such order that the sums of the three numbers in each column are all the same.
Determine all positive integers $n$ for which a magical array exists, and for each such $n$ determine the number of magical arrays. (Ireland)