A diagonal $D$ is chosen on a $n$ x $n$ grid. The diagonal goes from the upper left corner to the lower right corner of the grid dividing it into to congruent triangular regions as shown in the picture below.
The squares of $D$ and the squares of one of the triangular regions are coloured with $n$ different colors (the colors for $D$ and of the regions are the same) and then the other triangular region is coloured symmetrically to the part that has already been coloured as if D were a mirror. Here is a case for $n=3$ (but it is not valid in this case since all of the colors aren't used).
If $3 \leq n \leq 25 $, for which values of $n$ is it possible to color a $n$ x $n$ this way if no colors can be repeated in $D$, in any row and on any column.
I could just find that $n$ had to be an odd number. Since no colors can be repeated on a column, each of the $n$ colors appear once in each column, therefore each color appears $n$ times. Now suppose you place color $A$ in the upper left corner (on the diagonal), and let's say that color appears "$x$" times in one of the triangular regions, as both regions are symmetric it also appears $x$ times in the other regions, now that color appears $2x+1$ times, and since $n$ = (# of times a specific color appears), $n=2x+1$, therefore $n$ is odd.
I do not know how to put the tag to this question, so I'll put game theory, let me know if there is a tag for this question.