A pandiagonal Latin square of order $n$ is an assignment of the numbers $\{0,\ldots,n-1\}$ to the cells of an $n \times n$ grid such that no row, column, or diagonal of any length contains the same number twice.
For instance, here is an image of a pandiagonal Latin square of order $5$ (using colors in place of numbers).
The excellent puzzle book "1000 Playthinks" makes the claim, without proof, that a pandiagonal Latin square of order $n$ exists if and only if $n$ is not a multiple of $2$ or $3$.
The closest I have been able to find to a proof of this claim is in these notes, which give a nice proof that pandiagonal Latin squares do not exist for even $n$. However, they use a stricter definition which requires that there are no repeats along broken (wrap-around) diagonals.
Does anyone have a proof of the full claim?
