On a pandiagonal Latin square, what is a broken diagonal?

Question

After reading "No pandiagonal latin squares with order divisible by 3?" I didn't understand what a "broken diagonal" is.

Thanks

Answer 1

Broken diagonals are collections of $n$ cells - one on each row and column. They are parallel to one of the main diagonals but they wrap around when the diagonal reaches a border of the square. So either a set of cells in positions $(i,i+k)$ for $i=1,2,\ldots,n$, and a fixed $k$, or in positions $(i,i-k)$. The arithmetic on the column indices is done modulo $n$. Changing the value of $k$ gives another broken diagonal.

For example in the following latin square the red cells form one broken diagonal and the green cells another one. The brown number is in their intersection. $$ \pmatrix{ 0&\color{Red}2&4&6&\color{Green}1&3&5\cr 1&3&\color{Red}5&\color{Green}0&2&4&6\cr 2&4&\color{Green}6&\color{Red}1&3&5&0\cr 3&\color{Green}5&0&2&\color{Red}4&6&1\cr \color{Green}4&6&1&3&5&\color{Red}0&2\cr 5&0&2&4&6&1&\color{Brown}3\cr \color{Red}6&1&3&5&0&\color{Green}2&4\cr } $$

Answer 2

My illustration for broken diagonals

Please see this thread https://boinc.progger.info/odlk/forum_thread.php?id=178