An associative magic square is a magic square with the additional property that numbers symmetric to the center sum up to $n^2+1$. For example, the square $\pmatrix{6&9&12&7\\3&16&13&2\\15&4&1&14\\10&5&8&11}$ is such a square.
My questions :
Is there an easy proof that associative magic squares of size n do not exist, if $n \equiv 2\ (mod\ 4)$?
Is there an associative magic square of any size n, as long as $n \neq 2\ (mod\ 4)$ ?
I found a summary for the number of associative magic squares, but it only went upto $n = 10$.
How can an associative magic square be constructed ?
A proof of the fact that there are no associative magic squares of orders of the form $n = 4k+2$, attributed to C. Planck (1919), is given on the page http://budshaw.ca/Associative.html. It goes as follows:
Let $S$ be such a square, and let $m = n/2 = 2k+1$ and $\mu = m(n^2+1)$ be the half-order and the magic constant, respectively. Note that both $m$ and $\mu$ are odd. Partition $S$ into four subsquares of side $m$, and call the sums of the elements in the upper left, the upper right, and the lower left subsquares $A$, $B$, and $C$, respectively. Because each row and column of $S$ sums to $\mu$, we have $A+B = A+C = m \mu$. Because $S$ is associative, $B+C = m^2 (n^2 + 1) = m \mu$ as well. Hence $A=B=C=m\mu/2$ are not integers, a contradiction. So no such magic square exists.
The same page also gives links to pages describing construction of associative magic squares of orders of the form $n = 2k+1$ and $n = 4k$.